Tensors in Geometry and Continuum Media

[22] Banerjee N, Sen S. Does Brans-Dicke theory always yield general relativity in the infinite omega limit? Physical Review D. 1997;**56**:1334-1337

*Advances on Tensor Analysis and Their Applications*

[23] Bhadra A, Nandi KK. Omega dependence of the scalar field in Brans-Dicke theory. Physical Review D. 2001;

[24] Faraoni V, Côté J. Two new approaches to the anomalous limit of Brans-Dicke theory to Einstein gravity. Physical Review D. 2019;**99**(6):064013

**64**:087501

**46**

**Chapter 4**

Form

**Abstract**

basis of governing ones easily.

also about the divergent form of transport Eqs.

**2. Continuity equation**

particles *V*

**49**

!

divergence form

**1. Introduction**

Fluid Motion Equations in Tensor

In the current chapter, some applications of tensor analysis to fluid dynamics are presented. Governing equations of fluid motion and energy are obtained and analyzed. We shall discuss about continuity equation, equation of motion, and mechanical energy transport equation and four forms of energy equation. Finally, we shall talk about the divergence from transfer equations of different parameters of motion. The tensor form of equations has advantages over the component form: these are, first, compact writing of equations and, second, independency from reference frames, etc. Moreover, it allows to obtain new forms of equations on the

**Keywords:** stress tensor, Navier–stokes equation, energy, continuity, vorticity,

The mathematical model of moving fluid includes a set of equations, which are

usually written as transport equations of main physical parameters—density, velocity, energy, etc. These equations are conservation laws in fluid flows. Traditionally the component form of the equations is usually used, but at the same time, the componentless form (Gibbs approach) could be applied to obtain and transform these equations. In this chapter several main conservation laws are discussed and represented in tensor form, which has many advantages against usually used component form, like simplicity and compactness, independence on reference frames, less errors in transformations, etc. Below we obtain and analyze continuity and momentum equations and vorticity and energy transport equations, and we discuss

Continuity equation is the mass conservation law for a fluid flow and is presented as a scalar equation, which connects density *ρ* and velocity of fluid

> *ρ* þ *ρ*∇ ! � *V* !

¼ 0, (1)

, and for any liquid it could be written as

*∂ρ ∂t* þ *V* ! � ∇ !

*Dmitry Nikushchenko and Valery Pavlovsky*

#### **Chapter 4**

## Fluid Motion Equations in Tensor Form

*Dmitry Nikushchenko and Valery Pavlovsky*

#### **Abstract**

In the current chapter, some applications of tensor analysis to fluid dynamics are presented. Governing equations of fluid motion and energy are obtained and analyzed. We shall discuss about continuity equation, equation of motion, and mechanical energy transport equation and four forms of energy equation. Finally, we shall talk about the divergence from transfer equations of different parameters of motion. The tensor form of equations has advantages over the component form: these are, first, compact writing of equations and, second, independency from reference frames, etc. Moreover, it allows to obtain new forms of equations on the basis of governing ones easily.

**Keywords:** stress tensor, Navier–stokes equation, energy, continuity, vorticity, divergence form

#### **1. Introduction**

The mathematical model of moving fluid includes a set of equations, which are usually written as transport equations of main physical parameters—density, velocity, energy, etc. These equations are conservation laws in fluid flows. Traditionally the component form of the equations is usually used, but at the same time, the componentless form (Gibbs approach) could be applied to obtain and transform these equations. In this chapter several main conservation laws are discussed and represented in tensor form, which has many advantages against usually used component form, like simplicity and compactness, independence on reference frames, less errors in transformations, etc. Below we obtain and analyze continuity and momentum equations and vorticity and energy transport equations, and we discuss also about the divergent form of transport Eqs.

#### **2. Continuity equation**

Continuity equation is the mass conservation law for a fluid flow and is presented as a scalar equation, which connects density *ρ* and velocity of fluid particles *V* ! , and for any liquid it could be written as

$$\frac{\partial \rho}{\partial t} + \left(\overrightarrow{\mathbf{V}} \cdot \overrightarrow{\mathbf{V}}\right)\rho + \rho \overrightarrow{\mathbf{V}} \cdot \overrightarrow{\mathbf{V}} = \mathbf{0},\tag{1}$$

where ∇ ! ¼ *e* ! *i ∂ <sup>∂</sup>xi* ¼ *e* ! 1 *∂ <sup>∂</sup>x*<sup>1</sup> þ *e* ! 2 *∂ <sup>∂</sup>x*<sup>2</sup> þ *e* ! 2 *∂ <sup>∂</sup>x*<sup>3</sup> is the Hamilton operator and the dot is a symbol of a scalar product.

Hereinafter, the Einstein summation convention is used by default. It also can be written in two other equivalent forms [1, 2]:

$$\frac{d\rho}{dt} + \rho \vec{\nabla} \cdot \vec{V} = 0 \text{ and } \frac{\partial \rho}{\partial t} + \vec{\nabla} \cdot \rho \vec{V} = 0. \tag{2}$$

In the case of incompressible fluid, we could obtain its simplified form:

$$
\overrightarrow{\nabla} \cdot \overrightarrow{V} = \mathbf{0}.\tag{3}
$$

The tensor ∇

! *ρV* !

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

> ∇ ! *ρV* ! ¼ *e* ! *k ∂ ∂xk ρVj e* ! *<sup>j</sup>* <sup>¼</sup> *<sup>∂</sup> ∂xk ρVj e* ! *k e* ! *j* ¼

can be represented as

*Vj* þ *ρ*

þ *tr* ∇ ! *ρ* ⊗ *V* ! þ *ρ*∇ ! *V*

⊗ ∇! *ρ* � �; *<sup>V</sup>*

i.e., convective derivative is equal to trace of corresponding tensor.

� � *<sup>e</sup>*

*∂Vj ∂xk*

Convective derivatives of density and pressure (and any another scalar

! *k e* ! *<sup>j</sup>* ¼ ∇ ! *ρ* ⊗ *V* ! þ *ρ*∇ ! *V* ! *:*

h i !

! � ∇ � � !

In addition, for the divergence of the product of scalar and vector functions, we

� � !

� *σ* þ *ρ f* !

!

**3. Equations of motion of fluid with constant and variable properties**

tensor *σ*. In accordance with Newton's law, tensor *σ* for an incompressible fluid is

where *p* is the pressure; *μ* is the fluid shear (dynamic) viscosity; and

!

� *S* ¼ �∇ ! *p* þ ∇ !

Then equation of motion of incompressible fluid (Navier–Stokes equation) at

*p* þ *μ*Δ*V* ! þ ∇ !

!*<sup>T</sup>* � � is the rate of strain tensor. Due to relation <sup>∇</sup>

*μ* � 2*S* þ 2*μ*∇

*<sup>p</sup>* <sup>¼</sup> *tr V*!

: *E:*

⊗ ∇! *p* � �,

, (12)

� *σ* is the divergence of stress

! � *V* !

!

*μ* � 2*S* þ *μ*Δ*V*

!

*μ* � 2*S* þ *ρ f*

¼ 0, when

*:* (14)

*:* (15)

*σ* ¼ �*pE* þ 2*μS*, (13)

(10)

¼ 0*:* (11)

<sup>¼</sup> *<sup>∂</sup><sup>ρ</sup> ∂xk*

Finally, continuity equation can be written in form

*∂ρ ∂t*

*<sup>ρ</sup>* <sup>¼</sup> *tr V*!

∇ ! � *ρV* ! ¼ ∇ ! *ρ* ⊗ *V* ! þ *ρ*∇ ! *V*

The equation of a motion in terms of stress [4, 5] is

*ρ dV* ! *dt* <sup>¼</sup> <sup>∇</sup> !

is the body force per unit mass and ∇

*μ* 6¼ *const* divergence of stress tensor *σ* is written as

*ρ dV* ! *dt* ¼ �<sup>∇</sup> !

quantitatives) also can be written in tensor form:

*V* ! � ∇ � � !

can obtain the following relation:

where *f* !

> ∇ !

*μ* 6¼ *const* is

**51**

� *σ* ¼ �∇ ! *p* þ ∇ !

*<sup>S</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> ∇ ! *V* ! þ ∇ ! *V*

Let us apply gradient operator to continuity equation (Eq. (1)):

$$
\overrightarrow{\nabla}\frac{\partial\rho}{\partial t} + \overrightarrow{\nabla}\left[\left(\overrightarrow{V}\cdot\overrightarrow{\nabla}\right)\rho\right] + \overrightarrow{\nabla}\left[\rho\overrightarrow{\nabla}\cdot\overrightarrow{V}\right] = \mathbf{0}.
$$

As a result, we obtain vector equation:

$$\frac{\partial}{\partial t}\vec{\nabla}\rho + \vec{\nabla}\rho \cdot \vec{\nabla}\vec{V}^T + \vec{\nabla}\cdot\vec{\nabla}\vec{\rho} + \left(\vec{\nabla}\cdot\vec{V}\right)\vec{\nabla}\rho + \rho\vec{\nabla}\left(\vec{\nabla}\cdot\vec{V}\right) = \mathbf{0},\tag{4}$$

which could be written in a more compact form:

$$\frac{d}{dt}\vec{\nabla}\rho + \vec{\nabla}\vec{V}\cdot\vec{\nabla}\rho + \left(\vec{\nabla}\cdot\vec{V}\right)\vec{\nabla}\rho + \rho\vec{\nabla}\left(\vec{\nabla}\cdot\vec{V}\right) = \mathbf{0} \tag{5}$$

or by a little bit different way:

$$\frac{d}{dt}\vec{\nabla}\rho + \vec{\nabla}\vec{V}\cdot\vec{\nabla}\rho + \vec{\nabla}\left(\rho\vec{\nabla}\cdot\vec{V}\right) = \mathbf{0}.\tag{6}$$

These equations contain gradient of vector *V* ! divergence, which [3] equal to

$$
\vec{\nabla} \left( \vec{\nabla} \cdot \vec{V} \right) = \Delta \vec{V} + \vec{\nabla} \times \vec{\nabla} \times \vec{V}.
$$

For incompressible fluid the left part of this relation is equal to zero; therefore for rotation of velocity vector, we can write:

$$
\overrightarrow{\nabla} \times \overrightarrow{\nabla} \times \overrightarrow{\nabla} = -\Delta \overrightarrow{V}.\tag{7}
$$

In the case of compressible fluid in accordance with Eq. (6), we have additional terms in the right part of the equation:

$$\vec{\nabla} \times \vec{\nabla} \times \vec{V} = -\left(\Delta \vec{V} + \frac{1}{\rho} \frac{d}{dt} \vec{\nabla} \rho + \frac{1}{\rho} \vec{\nabla} \vec{V} \cdot \vec{\nabla} \rho + \frac{\left(\vec{\nabla} \cdot \vec{V}\right)}{\rho} \vec{\nabla} \rho\right). \tag{8}$$

Continuity equation can be also written in tensor form:

$$\frac{1}{3}tr\left(\frac{\partial\rho}{\partial t}\underline{E}\right) + tr\left(\vec{\nabla}\rho\vec{V}\right) = 0.\tag{9}$$

where ∇ ! ¼ *e* ! *i ∂ <sup>∂</sup>xi* ¼ *e* ! 1 *∂ <sup>∂</sup>x*<sup>1</sup> þ *e* ! 2 *∂ <sup>∂</sup>x*<sup>2</sup> þ *e* ! 2 *∂*

symbol of a scalar product.

Hereinafter, the Einstein summation convention is used by default.

<sup>¼</sup> 0 and *<sup>∂</sup><sup>ρ</sup>*

*ρ* h i þ ∇ ! *ρ*∇ ! � *V* h i !

> ∇ ! *ρ* þ *ρ*∇ ! ∇ ! � *V* � � !

> > !

¼ �Δ*V* !

¼ Δ*V* ! þ ∇ ! � ∇ ! � *V* ! *:*

For incompressible fluid the left part of this relation is equal to zero; therefore

In the case of compressible fluid in accordance with Eq. (6), we have additional

þ *tr* ∇ ! *ρV* � � !

∇ ! *ρ* þ *ρ*∇ ! ∇ ! � *V* � � !

In the case of incompressible fluid, we could obtain its simplified form:

∇ ! � *V* !

Let us apply gradient operator to continuity equation (Eq. (1)):

*∂t* þ ∇ ! � *ρV* !

It also can be written in two other equivalent forms [1, 2]:

*dρ dt* <sup>þ</sup> *<sup>ρ</sup>*<sup>∇</sup> ! � *V* !

*Advances on Tensor Analysis and Their Applications*

∇ ! *∂ρ ∂t* þ ∇ ! *V* ! � ∇ � � !

As a result, we obtain vector equation:

*d dt* <sup>∇</sup> ! *ρ* þ ∇ ! *V* ! � ∇ ! *ρ* þ ∇ ! � *V* � � !

or by a little bit different way:

which could be written in a more compact form:

*d dt* <sup>∇</sup> ! *ρ* þ ∇ ! *V* ! � ∇ ! *ρ* þ ∇ ! *ρ*∇ ! � *V* � � !

These equations contain gradient of vector *V*

for rotation of velocity vector, we can write:

terms in the right part of the equation:

∇ ! � ∇ ! � *V* !

**50**

∇ ! ∇ ! � *V* � � !

¼ � Δ*V* ! þ 1 *ρ d dt* <sup>∇</sup> ! *ρ* þ 1 *ρ* ∇ ! *V* ! � ∇ ! *ρ* þ

> 1 3 *tr ∂ρ ∂t E* � �

Continuity equation can be also written in tensor form:

0 @ ∇ ! � ∇ ! � *V* !

*∂ ∂t* ∇ ! *ρ* þ ∇ ! *ρ* � ∇ ! *V* !*<sup>T</sup>* þ *V* ! � ∇ ! ∇ ! *ρ* þ ∇ ! � *V* � � !

*<sup>∂</sup>x*<sup>3</sup> is the Hamilton operator and the dot is a

¼ 0*:* (3)

¼ 0*:*

¼ 0*:* (2)

¼ 0, (4)

¼ 0 (5)

¼ 0*:* (6)

divergence, which [3] equal to

*:* (7)

∇ ! *ρ*

¼ 0*:* (9)

1

A*:* (8)

∇ ! � *V* � � !

*ρ*

The tensor ∇ ! *ρV* ! can be represented as

$$\begin{split} \overrightarrow{\nabla} \rho \overrightarrow{V} &= \overrightarrow{\varepsilon}\_{k} \frac{\partial}{\partial \mathbf{x}\_{k}} \rho V\_{j} \overrightarrow{\mathbf{e}}\_{j} = \frac{\partial}{\partial \mathbf{x}\_{k}} \rho V\_{j} \overrightarrow{\mathbf{e}}\_{k} \overrightarrow{\mathbf{e}}\_{j} = \\\\ &= \left[ \frac{\partial \rho}{\partial \mathbf{x}\_{k}} V\_{j} + \rho \frac{\partial V\_{j}}{\partial \mathbf{x}\_{k}} \right] \overrightarrow{\mathbf{e}}\_{k} \overrightarrow{\mathbf{e}}\_{j} = \overrightarrow{\nabla} \rho \otimes \overrightarrow{V} + \rho \overrightarrow{\nabla} \overrightarrow{V}. \end{split} \tag{10}$$

Finally, continuity equation can be written in form

$$\frac{\partial \rho}{\partial t} + tr \left[ \vec{\nabla} \rho \otimes \vec{V} + \rho \vec{\nabla} \vec{V} \right] = \mathbf{0}.\tag{11}$$

Convective derivatives of density and pressure (and any another scalar quantitatives) also can be written in tensor form:

$$(\vec{V} \cdot \vec{\nabla})\rho = \text{tr}(\vec{V} \otimes \vec{\nabla}\rho); \quad \left(\vec{V} \cdot \vec{\nabla}\right)p = \text{tr}\left(\vec{V} \otimes \vec{\nabla}p\right),$$

i.e., convective derivative is equal to trace of corresponding tensor.

In addition, for the divergence of the product of scalar and vector functions, we can obtain the following relation:

$$
\overrightarrow{\nabla} \cdot \rho \overrightarrow{V} = \left( \overrightarrow{\nabla} \rho \otimes \overrightarrow{V} + \rho \overrightarrow{\nabla} \overrightarrow{V} \right) : \underline{E}.
$$

#### **3. Equations of motion of fluid with constant and variable properties**

The equation of a motion in terms of stress [4, 5] is

$$
\rho \frac{d\vec{V}}{dt} = \vec{\nabla} \cdot \underline{\sigma} + \rho \vec{f} \,, \tag{12}
$$

where *f* ! is the body force per unit mass and ∇ ! � *σ* is the divergence of stress tensor *σ*. In accordance with Newton's law, tensor *σ* for an incompressible fluid is

$$
\underline{\sigma} = -p\underline{E} + 2\mu\underline{S},
\tag{13}
$$

where *p* is the pressure; *μ* is the fluid shear (dynamic) viscosity; and *<sup>S</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> ∇ ! *V* ! þ ∇ ! *V* !*<sup>T</sup>* � � is the rate of strain tensor. Due to relation <sup>∇</sup> ! � *V* ! ¼ 0, when *μ* 6¼ *const* divergence of stress tensor *σ* is written as

$$
\overrightarrow{\nabla} \cdot \underline{\sigma} = -\overrightarrow{\nabla}p + \overrightarrow{\nabla}\mu \cdot 2\underline{\mathfrak{S}} + 2\mu \overrightarrow{\nabla} \cdot \underline{\mathfrak{S}} = -\overrightarrow{\nabla}p + \overrightarrow{\nabla}\mu \cdot 2\underline{\mathfrak{S}} + \mu \Delta \overrightarrow{\mathcal{V}}.\tag{14}
$$

Then equation of motion of incompressible fluid (Navier–Stokes equation) at *μ* 6¼ *const* is

$$
\rho \frac{d\vec{V}}{dt} = -\vec{\nabla}p + \mu \Delta \vec{V} + \vec{\nabla}\mu \cdot 2\vec{\mathbf{S}} + \rho \vec{f} \,. \tag{15}
$$

Additional term ∇ ! *μ* � 2*S* relates to changing of shear viscosity. In Cartesian coordinates Eq. (15) has the form:

$$
\rho \frac{dV\_i}{dt} = -\frac{\partial p}{\partial \mathbf{x}\_i} + \mu \Delta V\_i + \frac{\partial \mu}{\partial \mathbf{x}\_j} \left(\frac{\partial V\_j}{\partial \mathbf{x}\_i} + \frac{\partial V\_i}{\partial \mathbf{x}\_j}\right) + \rho f\_i.
$$

In case of compressible fluid with variable viscosity, the equation will contain a term with divergence ∇ ! � *V* ! , which is not equal to zero now. Rheological relation is this case has the form [2]:

$$
\underline{\sigma} = -p\underline{E} - \frac{2}{3}\mu \left( \vec{\nabla} \cdot \vec{\nabla} \right) \underline{E} + 2\mu \underline{S}.\tag{16}
$$

∇ ! � *ρ ∂V* ! *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>∇</sup> ! *ρ* � *∂V* ! *∂t* þ *ρ ∂ ∂t* ∇ ! � *V* ! ;

∇ ! � *ρ V* ! � ∇ � � !

∇ ! � �∇ ! *<sup>p</sup>*<sup>0</sup> � � ¼ �Δ*p*<sup>0</sup>

∇ ! � *μ*Δ*V* ! ¼ ∇ ! *μ* � Δ*V* !

∇ ! � *μ*∇ ! ∇ ! � *V* � � !

∇ ! � ∇ ! *μ* � 2*S* � � <sup>¼</sup> <sup>∇</sup>

In this case ∇

*ρ d dt* <sup>∇</sup> ! � *V* � � !

! � *ρ f* ! ¼ ∇ ! *ρ* � *f* ! þ *ρ*∇ ! � *f* !

*μ* 6¼ *const*, we obtain scalar equation:

þ ∇ ! *ρ* � *dV* ! *dt* <sup>þ</sup> *<sup>ρ</sup>*<sup>∇</sup> ! *V* ! : ∇ ! *V* !

*V* ! ¼ ∇ ! *ρ* � *V* ! � ∇ � � !

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

;

¼ ∇ ! *μ* � ∇ ! ∇ ! � *V* � � !

! ∇ ! *μ* : ∇ ! *V* ! þ ∇ ! *μ* � ∇ ! ∇ ! � *V* � � !

Function *U* is linear; therefore Δ*U* ¼ 0 and

In the case of incompressible fluid, we have

*ρ*∇ ! *V* ! : ∇ ! *V* !

if also *μ* ¼ *const*, then

compressibility.

vector *V* !

**53**

þ *μ*Δ ∇ ! � *V* � � ! ;

*V* h i !

þ *ρ*∇ ! *V* ! : ∇ ! *V* !

þ *μ*Δ ∇ ! � *V* � � ! ;

If the fluid motion occurs in gravity force field, then there is potential *U* ¼ *gz*,

¼ �∇ ! *ρ* � ∇ !

As a result of applied divergence operation to Navier–Stokes equation at

¼ �Δ*p* þ ∇

*ρ*∇ ! *V* ! : ∇ ! *V* !

*ρ dV* ! *dt* ¼ �<sup>∇</sup> !

8 >><

>>:

∇ ! � *V* ! ¼ 0 !

Now we consider the general case of fluid motion, taking into account its

The set of equations of motion of an incompressible fluid contains two— Navier–Stokes and continuity (one vector equation and one scalar equation) [2, 3]:

> *p* þ *μ*Δ*V* ! þ *ρ f* !

This set of two equations is closed: it contains two unknown quantities—velocity

and pressure per two equations. The set describes laminar flows; in

turbulent flows it becomes unclosed because Reynolds stress tensor appeared.

¼ �∇ ! *ρ* � ∇ ! *U:*

¼ �Δ*p*<sup>0</sup> þ ∇

þ2*μ*Δ ∇ ! � *V* � � !

� 2*μ*Δ*V* ! þ ∇ ! ∇ !

where *z* is the vertical coordinate and the body force per unit mass is *f*

∇ ! � *ρ f* ! þ *ρ* ∇ ! � *V* � � !

þ ∇ ! ∇ ! *μ* : ∇ ! *V* !*<sup>T</sup>* þ ∇ ! *μ* � Δ*V* ! *:*

*U* � *ρΔU:*

!

2*μ* � Δ*V* þ ∇

þ ∇ ! ∇ !

¼ �Δ*p:* (23)

! ∇ ! � *V*

� � � � !

: 2*μS* � ∇ ! *ρ* � ∇ ! *U:* (21)

: 2*μS*; (22)

*:* (24)

*V* ! � ∇ � � ! ;

!

¼ �∇ ! *U*.

þ

Let us introduce the denotation:

$$p + \frac{2}{3}\mu \left(\vec{\nabla} \cdot \vec{V}\right) = p';\tag{17}$$

then we can write Eq. (16) as

$$
\underline{\sigma} = -p'\underline{E} + 2\mu \underline{\text{S}}.\tag{18}
$$

Divergence of this tensor at *μ* 6¼ *const* is

$$
\overrightarrow{\nabla} \cdot \underline{\sigma} = -\overrightarrow{\nabla}p' + \overrightarrow{\nabla}\mu \cdot 2\underline{\mathfrak{S}} + \mu\Delta\overrightarrow{\dot{V}} + \mu\vec{\nabla}\left(\overrightarrow{\nabla} \cdot \overrightarrow{\dot{V}}\right).
$$

As a result, equation of motion of compressible fluid with variable viscosity has the form:

$$
\rho \frac{d\vec{V}}{dt} = -\vec{\nabla}p' + \mu \Delta \vec{V} + \mu \vec{\nabla} \left(\vec{\nabla} \cdot \vec{V}\right) + \vec{\nabla}\mu \cdot \mathfrak{L}\mathfrak{S} + \rho \vec{f} \,, \tag{19}
$$

in Cartesian coordinates

$$
\rho \left[ \frac{\partial V\_i}{\partial t} + V\_j \frac{\partial V\_i}{\partial \mathbf{x}\_j} \right] = -\frac{\partial p'}{\partial \mathbf{x}\_i} + \mu \Delta V\_i + \mu \frac{\partial}{\partial \mathbf{x}\_i} \left( \frac{\partial V\_j}{\partial \mathbf{x}\_j} \right) + \frac{\partial \mu}{\partial \mathbf{x}\_j} \left( \frac{\partial V\_j}{\partial \mathbf{x}\_i} + \frac{\partial V\_i}{\partial \mathbf{x}\_j} \right) + \rho f\_i.
$$

If we represent fluid particle acceleration as the sum of local and convective terms, then (Eq. (19)) will take the form:

$$
\rho \frac{\partial \vec{V}}{\partial t} + \rho \left(\vec{\mathbf{V}} \cdot \vec{\nabla}\right) \vec{\mathbf{V}} = -\vec{\nabla}p' + \mu \Delta \vec{\mathbf{V}} + \mu \vec{\nabla} \left(\vec{\nabla} \cdot \vec{\nabla}\right) + \vec{\nabla}\mu \cdot 2\underline{\mathbf{S}} + \rho \vec{\overline{f}}\,,\tag{20}
$$

considering viscosity variability is especially important for turbulent flow modeling using the Boussinesq hypothesis with turbulent viscosity *μt*.

Let us apply divergence operation to the Navier–Stokes equation for compressible fluid with variable viscosity. With this purpose we shall apply operation ∇ ! � to each vector term of Eq. (20):

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

Additional term ∇

term with divergence ∇

this case has the form [2]:

!

*Advances on Tensor Analysis and Their Applications*

coordinates Eq. (15) has the form:

*ρ dVi dt* ¼ � *<sup>∂</sup><sup>p</sup> ∂xi*

Let us introduce the denotation:

then we can write Eq. (16) as

Divergence of this tensor at *μ* 6¼ *const* is

� *σ* ¼ �∇ ! *p*<sup>0</sup> þ ∇ !

¼ � *<sup>∂</sup>p*<sup>0</sup> *∂xi*

*p*<sup>0</sup> þ *μ*Δ*V* ! þ *μ*∇ ! ∇ ! � *V* !

þ *μ*Δ*Vi* þ *μ*

*p*<sup>0</sup> þ *μ*Δ*V* ! þ *μ*∇ ! ∇ ! � *V* !

modeling using the Boussinesq hypothesis with turbulent viscosity *μt*. Let us apply divergence operation to the Navier–Stokes equation for compressible fluid with variable viscosity. With this purpose we shall apply

to each vector term of Eq. (20):

∇ !

*ρ dV* ! *dt* ¼ �<sup>∇</sup> !

*∂Vi ∂xj*

terms, then (Eq. (19)) will take the form:

*V* ! ¼ �∇ !

in Cartesian coordinates

þ *ρ V* ! � ∇ !

! � 

the form:

*ρ ∂Vi ∂t* þ *Vj*

*ρ ∂V* ! *∂t*

operation ∇

**52**

! � *V* !

*μ* � 2*S* relates to changing of shear viscosity. In Cartesian

*∂Vj ∂xi* þ *∂Vi ∂xj* 

¼ *p*<sup>0</sup>

! þ *μ*∇ ! ∇ ! � *V* ! *:*

> þ ∇ !

> > þ *∂μ ∂xj*

> > > þ ∇ !

*μ* � 2*S* þ *ρ f*

*∂Vj ∂xi* þ *∂Vi ∂xj* 

*μ* � 2*S* þ *ρ f*

!

!

, (19)

þ *ρfi :*

, (20)

, which is not equal to zero now. Rheological relation is

þ *ρf i :*

*E* þ 2*μS:* (16)

; (17)

*E* þ 2*μS:* (18)

*∂μ ∂xj*

In case of compressible fluid with variable viscosity, the equation will contain a

þ *μ*Δ*Vi* þ

3 *μ* ∇ ! � *V* !

*σ* ¼ �*p*<sup>0</sup>

*μ* � 2*S* þ *μ*Δ*V*

As a result, equation of motion of compressible fluid with variable viscosity has

*∂ ∂xi*

If we represent fluid particle acceleration as the sum of local and convective

considering viscosity variability is especially important for turbulent flow

*∂Vj ∂xj* 

*<sup>σ</sup>* ¼ �*pE* � <sup>2</sup>

*p* þ 2 3 *μ* ∇ ! � *V* !

$$\begin{aligned} &\vec{\nabla}\cdot\rho\frac{\partial\vec{\nabla}}{\partial t}=\vec{\nabla}\rho\cdot\frac{\partial\vec{\nabla}}{\partial t}+\rho\frac{\partial}{\partial t}\vec{\nabla}\cdot\vec{\nabla};\\ &\vec{\nabla}\cdot\rho\left(\vec{\nabla}\cdot\vec{\nabla}\right)\vec{\nabla}=\vec{\nabla}\rho\cdot\left[\left(\vec{\nabla}\cdot\vec{\nabla}\right)\vec{\nabla}\right]+\rho\vec{\nabla}\vec{\nabla}:\vec{\nabla}\vec{V}+\rho\left(\vec{\nabla}\cdot\vec{\nabla}\right)\left(\vec{\nabla}\cdot\vec{\nabla}\right);\\ &\vec{\nabla}\cdot\left(-\vec{\nabla}p'\right)=-\Delta p';\\ &\vec{\nabla}\cdot\mu\Delta\vec{\nabla}=\vec{\nabla}\mu\cdot\Delta\vec{\nabla}+\mu\Delta\left(\vec{\nabla}\cdot\vec{\nabla}\right);\\ &\vec{\nabla}\cdot\mu\vec{\nabla}\left(\vec{\nabla}\cdot\vec{\nabla}\right)=\vec{\nabla}\mu\cdot\vec{\nabla}\left(\vec{\nabla}\cdot\vec{\nabla}\right)+\mu\Delta\left(\vec{\nabla}\cdot\vec{\nabla}\right);\\ &\vec{\nabla}\cdot\left(\vec{\nabla}\mu\cdot\Delta\mathbf{S}\right)=\vec{\nabla}\vec{\nabla}\mu\cdot\vec{\nabla}\vec{\nabla}+\vec{\nabla}\mu\cdot\vec{\nabla}\left(\vec{\nabla}\cdot\vec{\nabla}\right)+\vec{\nabla}\mu\cdot\vec{\Delta}\vec{\nabla}.\end{aligned}$$

If the fluid motion occurs in gravity force field, then there is potential *U* ¼ *gz*, where *z* is the vertical coordinate and the body force per unit mass is *f* ! ¼ �∇ ! *U*. In this case ∇ ! � *ρ f* ! ¼ ∇ ! *ρ* � *f* ! þ *ρ*∇ ! � *f* ! ¼ �∇ ! *ρ* � ∇ ! *U* � *ρΔU:*

Function *U* is linear; therefore Δ*U* ¼ 0 and

$$
\overrightarrow{\nabla} \cdot \rho \overrightarrow{f} = -\overrightarrow{\nabla} \rho \cdot \overrightarrow{\nabla} U.
$$

As a result of applied divergence operation to Navier–Stokes equation at *μ* 6¼ *const*, we obtain scalar equation:

$$\begin{split} \rho \frac{d}{dt} \left( \vec{\nabla} \cdot \vec{\nabla} \right) + \vec{\nabla} \rho \cdot \frac{d \vec{\nabla}}{dt} + \rho \vec{\nabla} \vec{\nabla} : \vec{\nabla} \vec{V} &= -\Delta p' + \vec{\nabla} 2\mu \cdot \left( \Delta V + \vec{\nabla} \left( \vec{\nabla} \cdot \vec{V} \right) \right) + \\ &+ 2\mu \Delta \left( \vec{\nabla} \cdot \vec{\nabla} \right) + \vec{\nabla} \vec{\nabla} : 2\mu \underline{S} - \vec{\nabla} \rho \cdot \vec{\nabla} U. \end{split} \tag{21}$$

In the case of incompressible fluid, we have

$$
\rho \overrightarrow{\nabla} \overrightarrow{V} : \overrightarrow{\nabla} \overrightarrow{V} = -\Delta p + \overrightarrow{\nabla} \cdot 2\mu \Delta \overrightarrow{V} + \overrightarrow{\nabla} \overrightarrow{\nabla} : 2\mu \underline{S}; \tag{22}
$$

if also *μ* ¼ *const*, then

$$
\rho \overrightarrow{\nabla} \overrightarrow{V} : \overrightarrow{\nabla} \overrightarrow{V} = -\Delta p.\tag{23}
$$

Now we consider the general case of fluid motion, taking into account its compressibility.

The set of equations of motion of an incompressible fluid contains two— Navier–Stokes and continuity (one vector equation and one scalar equation) [2, 3]:

$$\begin{cases} \rho \frac{d\vec{V}}{dt} = -\vec{\nabla}p + \mu \Delta \vec{V} + \rho \vec{f} \\ \vec{\nabla} \cdot \vec{V} = \mathbf{0} \end{cases} \tag{24}$$

This set of two equations is closed: it contains two unknown quantities—velocity vector *V* ! and pressure per two equations. The set describes laminar flows; in turbulent flows it becomes unclosed because Reynolds stress tensor appeared.

In case of compressible flows at *ρ* 6¼ *const*, divergence of velocity is ∇ ! � *V* ! 6¼ 0, and the Navier–Stokes equation (Eq. (19)) of a fluid motion at *μ* ¼ *const* has the form:

$$
\rho \frac{d\vec{V}}{dt} = -\vec{\nabla}p' + \mu \Delta \vec{V} + \mu \vec{\nabla} \left(\vec{\nabla} \cdot \vec{\nabla}\right) + \rho \vec{f} \,, \tag{25}
$$

*<sup>ω</sup>*<sup>1</sup> <sup>¼</sup> *<sup>ω</sup><sup>x</sup>* <sup>¼</sup> <sup>1</sup>

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

*<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> *<sup>ω</sup><sup>y</sup>* <sup>¼</sup> <sup>1</sup>

*<sup>ω</sup>*<sup>3</sup> <sup>¼</sup> *<sup>ω</sup><sup>z</sup>* <sup>¼</sup> <sup>1</sup>

<sup>2</sup> ∇ ! *V* ! � ∇ ! *V*

Vorticity vector *ω*

say that tensor <sup>Ω</sup> <sup>¼</sup> <sup>1</sup>

expression is valid:

Ω : <sup>3</sup>

**55**

*<sup>ε</sup>* � � <sup>¼</sup> <sup>1</sup> 2 ∇ ! *V* ! � ∇ ! *V* !*<sup>T</sup>* � � : <sup>3</sup>

> ¼ 1 2

> > þ 1 2 *εji*<sup>3</sup>

þ 1 2

¼ �2*ω*<sup>1</sup> *e* !

The same for Eq. (31):

3 *<sup>ε</sup>* � � � *<sup>ω</sup>*

to zero; when *i* ¼ 1, *j* ¼ 2 they are

1 2 *∂Vs ∂xt*

The matrix of components of this tensor is

Levi-Civita tensor <sup>3</sup>*<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup>ijk <sup>e</sup>*

following relations are satisfied:

And vice versa, vector *ω*

associated with spin tensor.

Let us prove expression Eq. (30):

*∂Vj ∂xi*

� *<sup>∂</sup>Vi ∂xj* � �*εjik <sup>e</sup>*

> *∂Vj ∂xi*

*∂V*<sup>3</sup> *∂x*<sup>1</sup>

2

2

2

! *i e* ! *j e* !

Ω : <sup>3</sup>

*<sup>ε</sup>* � � <sup>¼</sup> <sup>3</sup>

3 *<sup>ε</sup>* � � � *<sup>ω</sup>*

*<sup>ε</sup>* � � <sup>¼</sup> <sup>1</sup> 2

In Eq. (30) spin tensor Ω is translated to the vector *ω*

! *<sup>k</sup>* <sup>¼</sup> <sup>1</sup> 2 *εji*<sup>1</sup>

> ! <sup>3</sup> <sup>¼</sup> <sup>1</sup> 2

� *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>3</sup> þ *∂V*<sup>3</sup> *∂x*<sup>1</sup>

<sup>2</sup> � 2*ω*<sup>3</sup> *e* !

*<sup>ε</sup>ktsε*12*<sup>k</sup>* <sup>¼</sup> <sup>1</sup>

2 *∂Vs ∂xt*

values for all *i*, *j* ¼ 1, 2, 3 could be obtained by the same way.

� � *<sup>e</sup>*

� *<sup>∂</sup>Vi ∂xj* � � *<sup>e</sup>*

> � *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>3</sup>

<sup>1</sup> � 2*ω*<sup>2</sup> *e* !

! ¼ *εijk e* ! *i e* ! *j e* ! *<sup>k</sup>* � *ω<sup>s</sup> e* !

*∂Vz <sup>∂</sup><sup>y</sup>* � *<sup>∂</sup>Vy ∂z* � �

> *∂Vx <sup>∂</sup><sup>z</sup>* � *<sup>∂</sup>Vz ∂x*

*∂Vy <sup>∂</sup><sup>x</sup>* � *<sup>∂</sup>Vx ∂y* � �

� �

!*<sup>T</sup>* � � is associated to vector *<sup>ω</sup>*

*<sup>ε</sup>* � � : <sup>Ω</sup> ¼ �2*<sup>ω</sup>*

! ¼ *ω* ! � <sup>3</sup>

> *∂Vj ∂xi*

> > *∂Vj ∂xi*

*∂V*<sup>2</sup> *∂x*<sup>3</sup>

> ! <sup>2</sup> þ 1 2

*<sup>s</sup>* ¼ *ωkεijk e*

Let us descry components of this second-rank tensor: when *i* ¼ *j* they are equal

*<sup>ε</sup>*3*ts* <sup>¼</sup> <sup>1</sup> 2

<sup>3</sup> ¼ �2*ω*

� *<sup>∂</sup>Vi ∂xj* � � *<sup>e</sup>*

� *<sup>∂</sup>Vi ∂xj* � � *<sup>e</sup>*

> � *<sup>∂</sup>V*<sup>3</sup> *∂x*<sup>2</sup>

! ¼ �∇ ! � *V* ! *:*

> ! *i e* ! *<sup>j</sup>* <sup>¼</sup> <sup>1</sup> 2 *∂Vs ∂xt*

> > *∂V*<sup>2</sup> *∂x*<sup>1</sup>

� *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>2</sup> � �;

! *i e* ! *<sup>j</sup>* : *εpqk e* ! *p e* ! *q e* ! *<sup>k</sup>* ¼

! <sup>1</sup> þ 1 2 *εji*<sup>2</sup>

� *<sup>∂</sup>V*<sup>3</sup> *∂x*<sup>2</sup> þ *∂V*<sup>2</sup> *∂x*<sup>3</sup>

� � *<sup>e</sup>*

*∂V*<sup>1</sup> *∂x*<sup>2</sup>

� *<sup>∂</sup>V*<sup>2</sup> *∂x*<sup>1</sup>

¼ 1 2

¼ 1 2

¼ 1 2

! and spin tensor Ω are connected with each other through

! ¼ �∇ ! � *V* !

! is associated with tensor Ω since the following

*∂V*<sup>3</sup> *∂x*<sup>2</sup>

*∂V*<sup>1</sup> *∂x*<sup>3</sup>

*∂V*<sup>2</sup> *∂x*<sup>1</sup>

� *<sup>∂</sup>V*<sup>2</sup> *∂x*<sup>3</sup>

9

>>>>>>>>=

*:* (29)

, because the

! is

*:* (30)

>>>>>>>>;

� *<sup>∂</sup>V*<sup>3</sup> *∂x*<sup>1</sup>

� *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>2</sup>

� �

� �

� �

*<sup>k</sup>*. These quantities are mutually associated. They

! <sup>¼</sup> <sup>1</sup> 2 ∇ ! � *V* !

*<sup>ε</sup>* � � <sup>¼</sup> <sup>Ω</sup>*:* (31)

!; in Eq. (31) vector *ω*

*∂Vj ∂xi*

> ! <sup>1</sup>þ

� *<sup>∂</sup>V*<sup>2</sup> *∂x*<sup>1</sup> þ *∂V*<sup>1</sup> *∂x*<sup>2</sup>

� � *<sup>e</sup>*

*εktsεijk e* ! *i e* ! *j:*

� *<sup>∂</sup>Vi ∂xj* � � *<sup>e</sup>*

! <sup>2</sup>þ

> ! <sup>3</sup> ¼

where *p*<sup>0</sup> is defined by Eq. (17). Continuity equation is written in the form of Eq. (1). If *ρ* 6¼ *const*, the set of equations (Eq. (25) and Eq. (1)) becomes unclosed, because density will also be unknown. To close the set of equation, energy equation is used, which contains one more unknown scalar quantity—temperature *T*. To determine temperature *T*, state equation is used; usually in fluid dynamics, it is the Mendeleev-Clapeyron equation. Energy equation could be written as the equation of specific internal energy transport:

$$
\rho \frac{du}{dt} = -\overrightarrow{\nabla} \cdot \overrightarrow{q} + \underline{\sigma} : \overrightarrow{\nabla} \overrightarrow{V} + \rho q\_s,\tag{26}
$$

where *u* is the specific internal energy (for ideal gas it could be expressed with the help of the isochore heat capacity, *du* ¼ *cvdT*); *q* ! is the heat flux vector (in laminar flow by Fourier's law, *q* ! ¼ �*λ*<sup>∇</sup> ! *T*); *λ* is the thermal conductivity of the material; and *qs* is the heat flux from internal or external sources.

Mendeleev-Clapeyron equation has the form

$$p = \rho RT,\tag{27}$$

where *R* is the universal gas constant. In case of real gas or fluid, the state equation becomes more complicated.

For liquids it usually supposes *ρ = const*. This condition is applicable for gas motions and also in case the velocities of gas particles are less than 1/3 of sound velocity.

Eqs. (1), (25), (26), and (27) are valid for laminar regime of motion. In case of turbulent regime in these equations, correlations will appear, caused by velocity, density, and temperature pulsations. For closure of the set of equations of turbulent motion, additional relations are required.

#### **4. Vorticity vector and its associated tensor**

Vorticity *ω* ! is a vector quantity, which characterizes velocity field:

$$
\overrightarrow{\boldsymbol{\phi}} = \frac{1}{2} \boldsymbol{rot} \overrightarrow{\boldsymbol{V}} = \frac{1}{2} \overrightarrow{\boldsymbol{\nabla}} \times \overrightarrow{\boldsymbol{V}},\tag{28}
$$

in component form

$$
\overrightarrow{\boldsymbol{\alpha}} = \frac{1}{2} \frac{\partial V\_j}{\partial \mathbf{x}\_i} \overrightarrow{\boldsymbol{e}}\_{kij} \overrightarrow{\boldsymbol{e}}\_{k},
$$

where *εijk* is the Levi-Civita tensor in component form. Components of *ω* ! vector are

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

In case of compressible flows at *ρ* 6¼ *const*, divergence of velocity is ∇

*p*<sup>0</sup> þ *μ*Δ*V* ! þ *μ*∇ ! ∇ ! � *V* !

*ρ dV* ! *dt* ¼ �<sup>∇</sup> !

*Advances on Tensor Analysis and Their Applications*

of specific internal energy transport:

(in laminar flow by Fourier's law, *q*

equation becomes more complicated.

motion, additional relations are required.

**4. Vorticity vector and its associated tensor**

*ω* ! <sup>¼</sup> <sup>1</sup> 2 *rotV*!

> *ω* ! <sup>¼</sup> <sup>1</sup> 2 *∂Vj ∂xi εkij e* ! *k*,

velocity.

Vorticity *ω*

vector are

**54**

in component form

*ρ du dt* ¼ �<sup>∇</sup> ! � *q* ! <sup>þ</sup> *<sup>σ</sup>* : <sup>∇</sup> ! *V* ! þ *ρqs*

with the help of the isochore heat capacity, *du* ¼ *cvdT*); *q*

Mendeleev-Clapeyron equation has the form

material; and *qs* is the heat flux from internal or external sources.

form:

and the Navier–Stokes equation (Eq. (19)) of a fluid motion at *μ* ¼ *const* has the

where *p*<sup>0</sup> is defined by Eq. (17). Continuity equation is written in the form of Eq. (1). If *ρ* 6¼ *const*, the set of equations (Eq. (25) and Eq. (1)) becomes unclosed, because density will also be unknown. To close the set of equation, energy equation is used, which contains one more unknown scalar quantity—temperature *T*. To determine temperature *T*, state equation is used; usually in fluid dynamics, it is the Mendeleev-Clapeyron equation. Energy equation could be written as the equation

where *u* is the specific internal energy (for ideal gas it could be expressed

! ¼ �*λ*<sup>∇</sup> !

where *R* is the universal gas constant. In case of real gas or fluid, the state

For liquids it usually supposes *ρ = const*. This condition is applicable for gas motions and also in case the velocities of gas particles are less than 1/3 of sound

Eqs. (1), (25), (26), and (27) are valid for laminar regime of motion. In case of turbulent regime in these equations, correlations will appear, caused by velocity, density, and temperature pulsations. For closure of the set of equations of turbulent

! is a vector quantity, which characterizes velocity field:

where *εijk* is the Levi-Civita tensor in component form. Components of *ω*

¼ 1 2 ∇ ! � *V* ! ! � *V* ! 6¼ 0,

, (25)

, (26)

! is the heat flux vector

, (28)

!

*T*); *λ* is the thermal conductivity of the

*p* ¼ *ρRT*, (27)

þ *ρ f* !

$$o\nu\_1 = o\nu\_x = \frac{1}{2} \left(\frac{\partial V\_x}{\partial \mathbf{y}} - \frac{\partial V\_y}{\partial \mathbf{z}}\right) = \frac{1}{2} \left(\frac{\partial V\_3}{\partial \mathbf{x}\_2} - \frac{\partial V\_2}{\partial \mathbf{x}\_3}\right)$$

$$o\nu\_2 = o\nu\_y = \frac{1}{2} \left(\frac{\partial V\_x}{\partial \mathbf{z}} - \frac{\partial V\_x}{\partial \mathbf{x}}\right) = \frac{1}{2} \left(\frac{\partial V\_1}{\partial \mathbf{x}\_3} - \frac{\partial V\_3}{\partial \mathbf{x}\_1}\right)$$

$$o\nu\_3 = o\nu\_x = \frac{1}{2} \left(\frac{\partial V\_y}{\partial \mathbf{x}} - \frac{\partial V\_x}{\partial \mathbf{y}}\right) = \frac{1}{2} \left(\frac{\partial V\_2}{\partial \mathbf{x}\_1} - \frac{\partial V\_1}{\partial \mathbf{x}\_2}\right)$$

Vorticity vector *ω* ! and spin tensor Ω are connected with each other through Levi-Civita tensor <sup>3</sup>*<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup>ijk <sup>e</sup>* ! *i e* ! *j e* ! *<sup>k</sup>*. These quantities are mutually associated. They say that tensor <sup>Ω</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> ∇ ! *V* ! � ∇ ! *V* !*<sup>T</sup>* � � is associated to vector *<sup>ω</sup>* ! <sup>¼</sup> <sup>1</sup> 2 ∇ ! � *V* ! , because the following relations are satisfied:

$$
\underline{\mathfrak{Q}} : \left( \underline{\mathfrak{k}} \right) = \left( \underline{\mathfrak{k}} \right) : \underline{\mathfrak{Q}} = -\underline{\mathfrak{A}} \vec{\mathfrak{o}} = -\vec{\nabla} \times \vec{\mathcal{V}}.\tag{30}
$$

And vice versa, vector *ω* ! is associated with tensor Ω since the following expression is valid:

$$\left(\begin{smallmatrix}\mathfrak{z}\\\mathfrak{z}\end{smallmatrix}\right)\cdot\overrightarrow{a}=\overrightarrow{a}\cdot\left(\begin{smallmatrix}\mathfrak{z}\\\mathfrak{z}\end{smallmatrix}\right)=\underline{\mathfrak{Q}}.\tag{31}$$

In Eq. (30) spin tensor Ω is translated to the vector *ω* !; in Eq. (31) vector *ω* ! is associated with spin tensor.

Let us prove expression Eq. (30):

Ω : <sup>3</sup> *<sup>ε</sup>* � � <sup>¼</sup> <sup>1</sup> 2 ∇ ! *V* ! � ∇ ! *V* !*<sup>T</sup>* � � : <sup>3</sup> *<sup>ε</sup>* � � <sup>¼</sup> <sup>1</sup> 2 *∂Vj ∂xi* � *<sup>∂</sup>Vi ∂xj* � � *<sup>e</sup>* ! *i e* ! *<sup>j</sup>* : *εpqk e* ! *p e* ! *q e* ! *<sup>k</sup>* ¼ ¼ 1 2 *∂Vj ∂xi* � *<sup>∂</sup>Vi ∂xj* � �*εjik <sup>e</sup>* ! *<sup>k</sup>* <sup>¼</sup> <sup>1</sup> 2 *εji*<sup>1</sup> *∂Vj ∂xi* � *<sup>∂</sup>Vi ∂xj* � � *<sup>e</sup>* ! <sup>1</sup> þ 1 2 *εji*<sup>2</sup> *∂Vj ∂xi* � *<sup>∂</sup>Vi ∂xj* � � *<sup>e</sup>* ! <sup>2</sup>þ þ 1 2 *εji*<sup>3</sup> *∂Vj ∂xi* � *<sup>∂</sup>Vi ∂xj* � � *<sup>e</sup>* ! <sup>3</sup> <sup>¼</sup> <sup>1</sup> 2 *∂V*<sup>2</sup> *∂x*<sup>3</sup> � *<sup>∂</sup>V*<sup>3</sup> *∂x*<sup>2</sup> � *<sup>∂</sup>V*<sup>3</sup> *∂x*<sup>2</sup> þ *∂V*<sup>2</sup> *∂x*<sup>3</sup> � � *<sup>e</sup>* ! <sup>1</sup>þ þ 1 2 *∂V*<sup>3</sup> *∂x*<sup>1</sup> � *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>3</sup> � *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>3</sup> þ *∂V*<sup>3</sup> *∂x*<sup>1</sup> � � *<sup>e</sup>* ! <sup>2</sup> þ 1 2 *∂V*<sup>1</sup> *∂x*<sup>2</sup> � *<sup>∂</sup>V*<sup>2</sup> *∂x*<sup>1</sup> � *<sup>∂</sup>V*<sup>2</sup> *∂x*<sup>1</sup> þ *∂V*<sup>1</sup> *∂x*<sup>2</sup> � � *<sup>e</sup>* ! <sup>3</sup> ¼ ¼ �2*ω*<sup>1</sup> *e* ! <sup>1</sup> � 2*ω*<sup>2</sup> *e* ! <sup>2</sup> � 2*ω*<sup>3</sup> *e* ! <sup>3</sup> ¼ �2*ω* ! ¼ �∇ ! � *V* ! *:*

The same for Eq. (31):

$$\left(^{3}\underline{\varepsilon}\right)\cdot\stackrel{\cdot}{\alpha} = \varepsilon\_{\vec{\imath}\vec{k}}\overrightarrow{\varepsilon\_{i}}\overrightarrow{\varepsilon\_{j}}\overrightarrow{\varepsilon\_{k}}\cdot\alpha\_{\vec{\imath}}\overrightarrow{\varepsilon\_{s}} = \alpha\_{k}\varepsilon\_{\vec{\imath}\vec{k}}\overrightarrow{\varepsilon\_{i}}\overrightarrow{\varepsilon\_{j}} = \frac{1}{2}\frac{\partial V\_{s}}{\partial \mathfrak{x}\_{\mathsf{l}}}\varepsilon\_{\vec{\imath}\vec{k}}\overrightarrow{\varepsilon\_{i}}\overrightarrow{\varepsilon\_{j}}.$$

Let us descry components of this second-rank tensor: when *i* ¼ *j* they are equal to zero; when *i* ¼ 1, *j* ¼ 2 they are

$$\frac{1}{2}\frac{\partial V\_s}{\partial \mathbf{x}\_t}\varepsilon\_{kts}\varepsilon\_{12k} = \frac{1}{2}\frac{\partial V\_s}{\partial \mathbf{x}\_t}\varepsilon\_{3t} = \frac{1}{2}\left(\frac{\partial V\_2}{\partial \mathbf{x}\_1} - \frac{\partial V\_1}{\partial \mathbf{x}\_2}\right)\varepsilon\_{12k}$$

values for all *i*, *j* ¼ 1, 2, 3 could be obtained by the same way. The matrix of components of this tensor is

$$\begin{pmatrix} 0 & \frac{1}{2} \left( \frac{\partial V\_2}{\partial \mathbf{x}\_1} - \frac{\partial V\_1}{\partial \mathbf{x}\_2} \right) & \frac{1}{2} \left( \frac{\partial V\_3}{\partial \mathbf{x}\_1} - \frac{\partial V\_1}{\partial \mathbf{x}\_3} \right) \\\\ \frac{1}{2} \left( \frac{\partial V\_1}{\partial \mathbf{x}\_2} - \frac{\partial V\_2}{\partial \mathbf{x}\_1} \right) & 0 & \frac{1}{2} \left( \frac{\partial V\_2}{\partial \mathbf{x}\_1} - \frac{\partial V\_1}{\partial \mathbf{x}\_2} \right) \\\\ \frac{1}{2} \left( \frac{\partial V\_1}{\partial \mathbf{x}\_3} - \frac{\partial V\_3}{\partial \mathbf{x}\_1} \right) & \frac{1}{2} \left( \frac{\partial V\_2}{\partial \mathbf{x}\_3} - \frac{\partial V\_3}{\partial \mathbf{x}\_2} \right) & 0 \end{pmatrix}.$$

It can be seen that it is a matrix of components of the antisymmetric tensor Ω, which means that relation (31) is valid.

It is easy to see also that

$$
\overrightarrow{\boldsymbol{\phi}} \cdot \underline{\boldsymbol{\Omega}} = \mathbf{0}, \overrightarrow{\boldsymbol{\phi}} \times \underline{\boldsymbol{\Omega}} = \mathbf{0}, \overrightarrow{\boldsymbol{\nabla}} \cdot \overrightarrow{\boldsymbol{\phi}} = \mathbf{0}.\tag{32}
$$

We could rewrite the second term of the left part in this form:

h i !

� ∇

*ω* ! � 2 *ω*

*ω* ! � *ω* ! � ∇ � � !

! � ∇ � � !

*V* !

It is necessary to note that in the case of compressible fluid and at *μ* 6¼ *const*, this

This transport equation can be written in another form, considering the equality

. Then

If we apply divergence operation to Eq. (36), then for incompressible fluid we

! *ω* ! � ∇ � � !

*ω* ! � *ω* ! � ∇ � � !

" #

*ω* ! h i � <sup>∇</sup>

and, finally, we have 0 ¼ 0, i.e., we shall not obtain a new expression.

As a result, we can obtain transport equation of vortices in an incompressible

*V* ! þ ∇ ! � *V* � � !

> ! � *V* !

! � ∇ � � !

*V* ! *:*

*V* !

¼ *ν*Δ*ω*

¼ *ν*Δ*ω*

*V* ! ¼ ∇ ! � *ν*Δ*ω* !;

*V* h i !

> ∇ ! � *ω* ! � �; <sup>∇</sup>

∇ ! � *V* � � ! ,

¼ 0*:*

! � *ω* ! � ∇ � � !

*V* h i !

, (39)

¼ *ν*Δ*ω*

∇ ! � *V* � � ! � *V* ! ∇ ! � 2*ω* ! � �,

¼ 0); therefore, this term

!, (36)

!*:* (37)

!*:* (38)

� ∇ ! � *V* � � !

*∂ω* ! *∂t* þ *V* ! � ∇ � � !

equation becomes much more complicated.

*V* ! ¼ ∇ ! � *ω* ! � *V* � � !

∇ ! � *<sup>∂</sup><sup>ω</sup>* ! *∂t* þ *V* ! � ∇ � � !

! ¼ 0, we have

∇ ! � *V* ! � ∇ � � !

*ω* ! h i <sup>¼</sup> <sup>∇</sup>

! *V* ! : ∇ ! *ω* ! þ *V* ! � ∇ � � !

¼ ∇ ! *V* ! : ∇ ! *ω* ! þ *ω* ! � ∇ � � !

For the second power of vorticity, we can write

*<sup>ω</sup>*<sup>2</sup> <sup>¼</sup> <sup>∇</sup> ! *V* !*<sup>T</sup>* : ∇ ! *V* ! � ∇ ! *V* ! : ∇ ! *V* !

! ¼ 0, and fluid is incompressible (∇

2 *V* ! � ∇ � � !

*dω* ! *dt* � *<sup>ω</sup>*

*∂ω* ! *∂t* þ ∇ ! � *ω* ! � *V* � � !

viscous fluid, which is named as the generalized Helmholtz equation:

*V* ! � ∇ � � !

finally can be written as

but ∇ ! � *ω*

*V* ! � ∇ � � !

obtain

**57**

*ω* ! � *ω* ! � ∇ � � !

because ∇ ! � *ω*

On the other hand,

∇ ! � *V* ! � ∇ � � !

∇ ! � *V* � � !

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

or in more compact form:

#### **5. Vorticity transport equation**

Rotation of convective acceleration of a fluid particle could be written as

$$
\overrightarrow{\nabla} \times \left[ \left( \overrightarrow{\nabla} \cdot \overrightarrow{\nabla} \right) \overrightarrow{V} \right] = 2 \left( \overrightarrow{\boldsymbol{\omega}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\boldsymbol{V}} - 2 \left( \overrightarrow{\boldsymbol{V}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\boldsymbol{\alpha}}.\tag{33}
$$

Evidence of this equation can be performed by writing convective acceleration according to the formula:

$$\left(\overrightarrow{\mathbf{V}} \cdot \overrightarrow{\mathbf{V}}\right)\overrightarrow{\mathbf{V}} = \overrightarrow{\mathbf{V}}\frac{\overrightarrow{\mathbf{V}}^2}{2} - \overrightarrow{\mathbf{V}} \times \left(\overrightarrow{\mathbf{V}} \times \overrightarrow{\mathbf{V}}\right). \tag{34}$$

Gradient of vorticity is the pseudo tensor of rank 2:

$$\vec{\nabla}\vec{a}\vec{\rho} = \vec{\nabla}\left(\frac{\mathbf{1}}{2}\vec{\nabla} \times \vec{V}\right) = \vec{e}\_s \frac{\partial}{\partial \mathbf{x}\_i} \left(\frac{\mathbf{1}}{2}\frac{\partial V\_j}{\partial \mathbf{x}\_i} \varepsilon\_{k\vec{\eta}} \vec{e}\_k\right) = \frac{\mathbf{1}}{2} \frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_i} \varepsilon\_{k\vec{\eta}} \vec{e}\_s \otimes \vec{e}\_k. \tag{35}$$

Trace of this tensor is *tr*∇ ! *ω* ! ¼ 0. It is also possible to distinguish the symmetric and antisymmetric parts of this tensor:

$$\frac{1}{4} \left( \frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_i} \varepsilon\_{k\vec{n}j} + \frac{\partial^2 V\_j}{\partial \mathbf{x}\_k \partial \mathbf{x}\_i} \varepsilon\_{s\vec{j}} \right) \overrightarrow{\bar{e}}\_s \otimes \overrightarrow{e}\_k , \frac{1}{4} \left( \frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_i} \varepsilon\_{k\vec{n}j} - \frac{\partial^2 V\_j}{\partial \mathbf{x}\_k \partial \mathbf{x}\_i} \varepsilon\_{s\vec{j}} \right) \overrightarrow{e}\_s \otimes \overrightarrow{e}\_k .$$

Let us assume that the fluid is incompressible, *μ = const*, and its motion occurs in the field of potential mass forces. In this case with the help of Eq. (34), we can obtain the Navier–Stokes equation in the form:

$$\frac{\partial \overrightarrow{V}}{\partial t} + \overrightarrow{\nabla} \left( \frac{V^2}{2} + P + U \right) + \left( \overrightarrow{\nabla} \times \overrightarrow{V} \right) \times \overrightarrow{V} = \nu \Delta \overrightarrow{V};$$

now we apply curl operation (*rot* ¼ ∇ ! �) to the left and right parts of this equation:

$$\frac{\partial}{\partial t} \left( \vec{\nabla} \times \vec{V} \right) + \vec{\nabla} \times \left[ \left( \vec{\nabla} \times \vec{V} \right) \times \vec{V} \right] = \nu \Delta \left( \vec{\nabla} \times \vec{V} \right) \dots$$

0

*Advances on Tensor Analysis and Their Applications*

� �

� *<sup>∂</sup>V*<sup>2</sup> *∂x*<sup>1</sup>

� *<sup>∂</sup>V*<sup>3</sup> *∂x*<sup>1</sup> � � 1

*ω*

! � Ω ¼ 0, *ω*

*V* h i !

> *V* ! ¼ ∇ ! *V* !<sup>2</sup>

> > 1 2 *∂Vj ∂xi εkij e* ! *k*

*∂V*<sup>1</sup> *∂x*<sup>2</sup>

*∂V*<sup>1</sup> *∂x*<sup>3</sup>

which means that relation (31) is valid.

**5. Vorticity transport equation**

according to the formula:

Trace of this tensor is *tr*∇

and antisymmetric parts of this tensor:

*εkij* þ

*∂V* ! *∂t* þ ∇ ! *V*<sup>2</sup>

*∂ ∂t* ∇ ! � *V* � � !

� �

*∂*2 *Vj ∂xk∂xi*

obtain the Navier–Stokes equation in the form:

now we apply curl operation (*rot* ¼ ∇

∇ ! *ω* ! ¼ ∇ ! 1 2 ∇ ! � *V* ! � �

1 4

equation:

**56**

*∂*2 *Vj ∂xs∂xi* ∇ ! � *V* ! � ∇ � � !

> *V* ! � ∇ � � !

Gradient of vorticity is the pseudo tensor of rank 2:

¼ *e* ! *s ∂ ∂xs*

> ! *ω*

> > *εsij*

*e* ! *<sup>s</sup>* ⊗ *e* ! *k*, 1 4

<sup>2</sup> <sup>þ</sup> *<sup>P</sup>* <sup>þ</sup> *<sup>U</sup>* � �

> þ ∇ !

� ∇ ! � *V* � � !

1 2

0

BBBBBBBBB@

It is easy to see also that

1 2 1 2

2

*∂V*<sup>2</sup> *∂x*<sup>1</sup>

*∂V*<sup>2</sup> *∂x*<sup>3</sup>

� *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>2</sup> � � 1

> � *<sup>∂</sup>V*<sup>3</sup> *∂x*<sup>2</sup>

� �

It can be seen that it is a matrix of components of the antisymmetric tensor Ω,

! � Ω ¼ 0, ∇

Rotation of convective acceleration of a fluid particle could be written as

¼ 2 *ω* ! � ∇ � � !

Evidence of this equation can be performed by writing convective acceleration

<sup>2</sup> � *<sup>V</sup>* ! � ∇ ! � *V* � � !

� �

! � *ω*

*V* ! � 2 *V* ! � ∇ � � !

¼ 1 2 *∂*2 *Vj ∂xs∂xi*

*∂*2 *Vj ∂xs∂xi*

Let us assume that the fluid is incompressible, *μ = const*, and its motion occurs in

þ ∇ ! � *V* � � !

h i !

� *V*

the field of potential mass forces. In this case with the help of Eq. (34), we can

!

! ¼ 0. It is also possible to distinguish the symmetric

*<sup>ε</sup>kij* � *<sup>∂</sup>*<sup>2</sup>

� *V* !

¼ *ν*Δ ∇ ! � *V* � � !

� �

<sup>0</sup> <sup>1</sup>

2

2

*∂V*<sup>3</sup> *∂x*<sup>1</sup>

*∂V*<sup>2</sup> *∂x*<sup>1</sup>

� *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>3</sup>

1

CCCCCCCCCA *:*

! ¼ 0*:* (32)

*ω*

*εkij e* ! *<sup>s</sup>* ⊗ *e* !

*Vj ∂xk∂xi*

¼ *ν*Δ*V* ! ;

�) to the left and right parts of this

*εsij*

*:*

*e* ! *<sup>s</sup>* ⊗ *e* ! *k:*

!*:* (33)

*:* (34)

*<sup>k</sup>:* (35)

� *<sup>∂</sup>V*<sup>1</sup> *∂x*<sup>2</sup>

� �

� �

0

We could rewrite the second term of the left part in this form:

$$
\left(\overrightarrow{\mathbf{V}} \cdot \overrightarrow{\mathbf{V}}\right)\left(\overrightarrow{\nabla} \times \overrightarrow{\mathbf{V}}\right) - \left[\left(\overrightarrow{\nabla} \times \overrightarrow{\mathbf{V}}\right) \cdot \overrightarrow{\nabla}\right] \overrightarrow{\mathbf{V}} + \left(\overrightarrow{\nabla} \times \overrightarrow{\mathbf{V}}\right)\left(\overrightarrow{\nabla} \cdot \overrightarrow{\mathbf{V}}\right) - \overrightarrow{\mathbf{V}}\left(\overrightarrow{\nabla} \cdot 2\overrightarrow{\boldsymbol{\omega}}\right),
$$

but ∇ ! � *ω* ! ¼ 0, and fluid is incompressible (∇ ! � *V* ! ¼ 0); therefore, this term finally can be written as

$$2\left(\overrightarrow{V}\cdot\overrightarrow{\nabla}\right)\overrightarrow{\boldsymbol{\phi}} - 2\left(\overrightarrow{\boldsymbol{\phi}}\cdot\overrightarrow{\nabla}\right)\overrightarrow{V} .$$

As a result, we can obtain transport equation of vortices in an incompressible viscous fluid, which is named as the generalized Helmholtz equation:

$$\frac{\partial \overrightarrow{\boldsymbol{\alpha}}}{\partial t} + \left(\overrightarrow{\boldsymbol{V}} \cdot \overrightarrow{\boldsymbol{\nabla}}\right) \overrightarrow{\boldsymbol{\alpha}} - \left(\overrightarrow{\boldsymbol{\alpha}} \cdot \overrightarrow{\boldsymbol{\nabla}}\right) \overrightarrow{\boldsymbol{V}} = \nu \Delta \overrightarrow{\boldsymbol{\alpha}},\tag{36}$$

or in more compact form:

$$\frac{d\overrightarrow{a\phi}}{dt} - \left(\overrightarrow{\phi} \cdot \overrightarrow{\nabla}\right)\overrightarrow{V} = \nu \Delta \overrightarrow{\phi}.\tag{37}$$

It is necessary to note that in the case of compressible fluid and at *μ* 6¼ *const*, this equation becomes much more complicated.

This transport equation can be written in another form, considering the equality *V* ! � ∇ � � ! *ω* ! � *ω* ! � ∇ � � ! *V* ! ¼ ∇ ! � *ω* ! � *V* � � ! . Then

$$\frac{\partial \overrightarrow{a\rho}}{\partial t} + \overrightarrow{\nabla} \times \left(\overrightarrow{a\rho} \times \overrightarrow{V}\right) = \nu \Delta \overrightarrow{a\rho}.\tag{38}$$

If we apply divergence operation to Eq. (36), then for incompressible fluid we obtain

$$\overrightarrow{\nabla} \cdot \left[ \frac{\partial \overrightarrow{\boldsymbol{\phi}}}{\partial t} + \left( \overrightarrow{\boldsymbol{V}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\boldsymbol{\phi}} - \left( \overrightarrow{\boldsymbol{\phi}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\boldsymbol{V}} \right] = \overrightarrow{\nabla} \cdot \nu \Delta \overrightarrow{\boldsymbol{\phi}};$$

because ∇ ! � *ω* ! ¼ 0, we have

$$\overrightarrow{\nabla} \cdot \left[ \left( \overrightarrow{\nabla} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\dot{\boldsymbol{\alpha}}} \right] - \overrightarrow{\nabla} \left[ \left( \overrightarrow{\boldsymbol{\alpha}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\dot{\boldsymbol{V}}} \right] = \mathbf{0}.$$

On the other hand,

$$
\overrightarrow{\nabla} \cdot \left[ \left( \overrightarrow{\nabla} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\boldsymbol{\omega}} \right] = \overrightarrow{\nabla} \overrightarrow{\mathbf{V}} : \overrightarrow{\nabla} \overrightarrow{\boldsymbol{\omega}} + \left( \overrightarrow{\mathbf{V}} \cdot \overrightarrow{\nabla} \right) \left( \overrightarrow{\nabla} \cdot \overrightarrow{\boldsymbol{\omega}} \right) ; \, \overrightarrow{\nabla} \cdot \left[ \left( \overrightarrow{\boldsymbol{\omega}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\mathbf{V}} \right] \, \, \overrightarrow{\mathbf{V}} $$
 
$$= \overrightarrow{\nabla} \overrightarrow{\mathbf{V}} : \overrightarrow{\nabla} \overrightarrow{\boldsymbol{\omega}} + \left( \overrightarrow{\boldsymbol{\omega}} \cdot \overrightarrow{\nabla} \right) \left( \overrightarrow{\nabla} \cdot \overrightarrow{\mathbf{V}} \right) ,$$

and, finally, we have 0 ¼ 0, i.e., we shall not obtain a new expression. For the second power of vorticity, we can write

$$
\rho \mathbf{a}^2 = \overrightarrow{\nabla} \overrightarrow{\mathbf{V}}^T : \overrightarrow{\nabla} \overrightarrow{\mathbf{V}} - \overrightarrow{\nabla} \overrightarrow{\mathbf{V}} : \overrightarrow{\nabla} \overrightarrow{\mathbf{V}}, \tag{39}
$$

and now, if we scalar multiply transport equation of vortices by *ω* !:

$$
\overrightarrow{\boldsymbol{\phi}} \cdot \left[ \frac{\overrightarrow{\partial \boldsymbol{\phi}}}{\partial t} + \left( \overrightarrow{\boldsymbol{V}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\boldsymbol{\phi}} - \left( \overrightarrow{\boldsymbol{\phi}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{\boldsymbol{V}} \right] = \overrightarrow{\boldsymbol{\phi}} \cdot \nu \Delta \overrightarrow{\boldsymbol{\phi}} \overrightarrow{\boldsymbol{\varphi}}
$$

then we obtain scalar transport equation of *ω*2:

$$\frac{d\boldsymbol{\alpha}^{2}}{dt} - \left(\overrightarrow{\boldsymbol{\alpha}} \otimes \overrightarrow{\boldsymbol{\alpha}}\right) : \overrightarrow{\boldsymbol{\nabla}}\overrightarrow{\boldsymbol{V}} = \nu \Delta \boldsymbol{\alpha}^{2} - \nu \overrightarrow{\boldsymbol{\nabla}} \overrightarrow{\boldsymbol{\alpha}}^{T} : \overrightarrow{\boldsymbol{\nabla}} \overrightarrow{\boldsymbol{\alpha}}.\tag{40}$$

∇ ! *f* � ∇ ! *φ* ¼ ∇ !

*∂φ ∂x*<sup>3</sup>

*∂f ∂x*<sup>1</sup> � *∂f ∂x*<sup>3</sup>

� *∂f ∂x*<sup>2</sup>

� � *<sup>e</sup>*

*∂ ∂xi*

> ! <sup>1</sup> þ

> > ! <sup>3</sup> ¼

> > > *∂f ∂x*<sup>3</sup>

The vector product of gradients of scalar functions also can be written as

� � *<sup>e</sup>*

*∂φ ∂x*<sup>2</sup>

*∂φ ∂x*<sup>2</sup>

! 1 þ

> ! 3,

*∂φ ∂x*<sup>1</sup>

*∂xi f ∂φ ∂xj* ¼ *εkij*

*f ∂φ ∂xj* � � *<sup>e</sup>* ! <sup>2</sup> þ *ε*3*ij*

> *∂ ∂x*<sup>3</sup> *f ∂φ ∂x*<sup>1</sup> � *∂ ∂x*<sup>1</sup> *f ∂φ ∂x*<sup>3</sup>

*∂φ ∂x*<sup>1</sup> � *∂f ∂x*<sup>1</sup>

*<sup>ε</sup>* � � ¼ �<sup>∇</sup> !

> ¼ ∇ !

*f ∂φ ∂xk* � �*εki*<sup>2</sup> *<sup>e</sup>*

*∂ ∂xi*

> ! <sup>1</sup> þ

Mechanical energy balance equation can be obtained as a scalar product of each

! : ! 3*:*

*f ∂φ ∂xk* � �*εkij <sup>e</sup>*

> *∂f ∂x*<sup>1</sup> � *∂φ ∂x*<sup>3</sup>

> > ! *f* � ∇ !

� *<sup>f</sup> <sup>∂</sup><sup>φ</sup> ∂xk εkij e* ! *i e* ! *j*

� � *<sup>e</sup>*

Really, the left part of this equation is a vector:

*<sup>φ</sup>* <sup>¼</sup> *<sup>∂</sup><sup>f</sup> ∂x*<sup>2</sup>

þ

� *<sup>f</sup> <sup>∂</sup><sup>φ</sup> ∂xj e* ! *<sup>j</sup>* ¼ *e* ! *<sup>i</sup>* � *e* ! *j* � � *∂*

*f ∂φ ∂xj* � � *<sup>e</sup>* ! <sup>1</sup> þ *ε*2*ij*

� � *<sup>e</sup>*

� *∂f ∂x*<sup>3</sup>

� � *<sup>e</sup>*

� � *<sup>e</sup>*

*∂φ ∂x*<sup>2</sup>

*f* � ∇ ! *f* ⊗ ∇! *φ*

� *<sup>f</sup> <sup>∂</sup><sup>φ</sup> ∂xk εkij e* ! *i e* ! *<sup>j</sup>* <sup>¼</sup> *<sup>∂</sup> ∂xi*

*f ∂φ ∂xk* � �*εki*<sup>1</sup> *<sup>e</sup>*

� � : <sup>3</sup>

� �

! <sup>1</sup> þ

� *∂f ∂x*<sup>2</sup> � *∂φ ∂x*<sup>3</sup>

> � *∂f ∂x*<sup>1</sup> � *∂φ ∂x*<sup>2</sup>

� � *<sup>e</sup>*

� � *<sup>e</sup>*

! <sup>1</sup> þ

∇ ! *f* � ∇ !

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

and the right part is.

! *i ∂ ∂xi*

> *∂ ∂xi*

> > *∂ ∂x*<sup>1</sup> *f ∂φ ∂x*<sup>2</sup> � *∂ ∂x*<sup>2</sup> *f ∂φ ∂x*<sup>1</sup>

*∂φ ∂x*<sup>3</sup>

Here we have in component form:

! � *<sup>f</sup> <sup>∂</sup><sup>φ</sup> ∂xs e* ! *<sup>s</sup>* � *εkij e* ! *k e* ! *i e* ! *j*

¼ *e* ! *t ∂ ∂xt*

<sup>¼</sup> *<sup>∂</sup> ∂xi*

<sup>¼</sup> *<sup>∂</sup><sup>f</sup> ∂x*<sup>3</sup> � *∂φ ∂x*<sup>2</sup>

þ

**6. Mechanical energy equation**

member of Eq. (12) on velocity vector *V*

*∂f ∂x*<sup>2</sup> � *∂φ ∂x*<sup>1</sup>

It is easy to see that this is equal to expression for ∇

¼ *ε*1*ij*

<sup>¼</sup> *<sup>∂</sup> ∂x*<sup>2</sup> *f ∂φ ∂x*<sup>3</sup> � *∂ ∂x*<sup>3</sup> *f ∂φ ∂x*<sup>2</sup>

þ

<sup>¼</sup> *<sup>∂</sup><sup>f</sup> ∂x*<sup>2</sup>

∇ ! *f* � ∇ ! *<sup>φ</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> <sup>∇</sup> ! *φ* ⊗ ∇!

∇ !

**59**

� *f*∇ ! *φ* � �� 3 *ε* h i <sup>¼</sup> <sup>∇</sup>

Therefore, Eq. (44) is valid.

∇ ! � *f*∇ ! *φ* � � <sup>¼</sup> *<sup>e</sup>* � *f*∇ ! *φ*

> *∂f ∂x*<sup>3</sup>

*∂φ ∂x*<sup>1</sup>

*∂ ∂xi*

*∂ ∂xi*

� � *<sup>e</sup>*

*∂φ ∂x*<sup>3</sup>

! <sup>2</sup> þ

� *f*∇ ! *φ* � � � <sup>3</sup>

� �

! *<sup>j</sup>* ¼

! <sup>2</sup> þ

� *∂f ∂x*<sup>1</sup>

*f ∂φ ∂xj* � � *<sup>e</sup>* ! *<sup>k</sup>* ¼

*f ∂φ ∂xj* � � *<sup>e</sup>* ! <sup>3</sup> ¼

> ! 2

> > *∂f ∂x*<sup>1</sup>

*∂φ ∂x*<sup>2</sup>

*<sup>ε</sup>* � � h i*:* (45)

¼

*f ∂φ ∂xk* � �*εki*<sup>3</sup> *<sup>e</sup>*

! <sup>3</sup> ¼

! 2

*φ* with the minus sign.

*∂ ∂xi*

� *∂f ∂x*<sup>3</sup> � *∂φ ∂x*<sup>1</sup>

� � *<sup>e</sup>*

� *∂f ∂x*<sup>2</sup>

� � *<sup>e</sup>*

*∂φ ∂x*<sup>1</sup>

! 3*:*

� � *<sup>e</sup>*

� �*:* (44)

*∂φ ∂x*<sup>3</sup>

! 2

For incompressible fluid ∇ ! � *S* ¼ ∇ ! � Ω because ∇ ! � *<sup>S</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> Δ*V* ! þ 1 2 ∇ ! ∇ ! � *V* � � ! and ∇ ! � <sup>Ω</sup> <sup>¼</sup> <sup>1</sup> <sup>2</sup> Δ*V* ! � 1 2 ∇ ! ∇ ! � *V* � � ! *:*

As we already know *V* ! � ∇ ! � *V* � � ! ¼ �2*V* ! � Ω; therefore, Eq. (34) can be written as

$$\left(\vec{\boldsymbol{V}} \cdot \vec{\boldsymbol{\nabla}}\right)\vec{\boldsymbol{V}} = \vec{\boldsymbol{\nabla}}\frac{\boldsymbol{V}^{2}}{2} + 2\vec{\boldsymbol{V}} \cdot \underline{\boldsymbol{\Omega}}.\tag{41}$$

Let us write one more equation:

$$
\begin{split}
\vec{\nabla} \cdot \left[ \vec{V} \times \left( \vec{\nabla} \times \vec{\nabla} \right) \right] &= -\vec{\nabla} \cdot \left[ \vec{V} \cdot 2\mathfrak{Q} \right] = \vec{\nabla} \cdot \left[ \vec{V} \cdot \vec{\nabla} \vec{V}^T - \vec{V} \cdot \vec{\nabla} \vec{V} \right] = \\ \\ &= \vec{\nabla} \cdot \left[ V\_i \frac{\partial V\_i}{\partial \mathbf{x}\_j} - V\_i \frac{\partial V\_j}{\partial \mathbf{x}\_i} \right] \vec{e}\_j = \frac{\partial}{\partial \mathbf{x}\_j} \left[ V\_i \frac{\partial V\_i}{\partial \mathbf{x}\_j} - V\_i \frac{\partial V\_j}{\partial \mathbf{x}\_i} \right] = \frac{\partial V\_i}{\partial \mathbf{x}\_j} \frac{\partial V\_i}{\partial \mathbf{x}\_j} + V\_i \frac{\partial^2 V\_i}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} - \\ \\ &- \frac{\partial V\_i}{\partial \mathbf{x}\_j} \frac{\partial V\_j}{\partial \mathbf{x}\_i} - V\_i \frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_j} = \vec{\nabla} \vec{V}^T \cdot \vec{\nabla} \vec{V} - \vec{\nabla} \vec{V} : \vec{\nabla} \vec{V} + \vec{V} \cdot \Delta \vec{V} - \vec{V} \cdot \vec{\nabla} \left( \vec{\nabla} \cdot \vec{\nabla} \right) .\end{split}
$$

Therefore,

$$\vec{\nabla} \cdot \left[ \vec{V} \times \left( \vec{\nabla} \times \vec{V} \right) \right] = \vec{\nabla} \vec{V}^T : \vec{\nabla} \vec{V} - \vec{\nabla} \vec{V} : \vec{\nabla} \vec{V} + \vec{V} \cdot \left[ \Delta \vec{V} - \vec{\nabla} \left( \vec{\nabla} \cdot \vec{V} \right) \right] \tag{42}$$

This equation also can be written in the next form:

$$\overrightarrow{\nabla} \cdot \left[ \overrightarrow{V} \times \left( \overrightarrow{\nabla} \times \overrightarrow{V} \right) \right] = 2\underline{\Omega} : \overrightarrow{\nabla}\overrightarrow{V} + 2\overrightarrow{V} \cdot \left( \overrightarrow{\nabla} \cdot \underline{\Omega} \right) \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\widetilde{\nabla} \,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\,\widetilde{\nabla} \,\,\,\,\,\,\,\,\,\widetilde{\$$

or

$$
\overrightarrow{\nabla} \cdot \left[ \overrightarrow{V} \times \left( \overrightarrow{\nabla} \times \overrightarrow{V} \right) \right] = 2\underline{\Omega} : \underline{\Omega} + 2\underline{\Omega} : \underline{\mathbb{S}} + 2\vec{V} \cdot \left( \overrightarrow{\nabla} \cdot \underline{\Omega} \right).
$$

One more interesting relation is

$$
\overrightarrow{V} \cdot \overrightarrow{\nabla} \overrightarrow{V} = \overrightarrow{V} \cdot \underline{\Omega} + \overrightarrow{V} \cdot \underline{\Omega} \ast
$$

but as we already mentioned, *V* ! � ∇ ! � *V* � � ! ¼ 2*V* ! � Ω; therefore,

$$
\overrightarrow{\dot{V}} \cdot \underline{\mathfrak{L}} = \overrightarrow{\nabla} \frac{\mathbf{V}^2}{2} - \frac{\mathbf{1}}{2} \overrightarrow{\dot{V}} \times \left(\overrightarrow{\nabla} \times \overrightarrow{\dot{V}}\right). \tag{43}
$$

The vector product of gradients of scalar functions gives us a vector; in terms of rotation of a vector function, we can write

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

$$
\vec{\nabla}f \times \vec{\nabla}\varphi = \vec{\nabla} \times \left( f \vec{\nabla}\varphi \right). \tag{44}
$$

Really, the left part of this equation is a vector:

$$
\begin{split}
\overrightarrow{\nabla}f \times \overrightarrow{\nabla}\varphi &= \left(\frac{\partial f}{\partial \mathbf{x}\_{2}}\frac{\partial \rho}{\partial \mathbf{x}\_{3}} - \frac{\partial f}{\partial \mathbf{x}\_{3}}\frac{\partial \rho}{\partial \mathbf{x}\_{2}}\right)\overrightarrow{e}\_{1} + \left(\frac{\partial f}{\partial \mathbf{x}\_{3}}\frac{\partial \rho}{\partial \mathbf{x}\_{1}} - \frac{\partial f}{\partial \mathbf{x}\_{1}}\frac{\partial \rho}{\partial \mathbf{x}\_{3}}\right)\overrightarrow{e}\_{2} \\ &+ \left(\frac{\partial f}{\partial \mathbf{x}\_{1}}\frac{\partial \rho}{\partial \mathbf{x}\_{2}} - \frac{\partial f}{\partial \mathbf{x}\_{2}}\frac{\partial \rho}{\partial \mathbf{x}\_{1}}\right)\overrightarrow{e}\_{3},
\end{split}
$$

and the right part is.

and now, if we scalar multiply transport equation of vortices by *ω*

*ω* ! � *ω* ! � ∇ � � !

" #

: ∇ ! *V* ! *V* !

<sup>¼</sup> *<sup>ν</sup>*Δ*ω*<sup>2</sup> � *<sup>ν</sup>*<sup>∇</sup>

<sup>2</sup> <sup>þ</sup> <sup>2</sup>*<sup>V</sup>* !

! � �

! � *<sup>S</sup>* <sup>¼</sup> <sup>1</sup> <sup>2</sup> Δ*V* ! þ 1 2 ∇ ! ∇ ! � *V* � � !

� Ω because ∇

¼ �2*V* !

*V* ! ¼ ∇ ! *V*<sup>2</sup>

¼ ∇ ! � *V* ! � ∇ ! *V* !*<sup>T</sup>* � *V* ! � ∇ ! *V*

> *e* ! *<sup>j</sup>* <sup>¼</sup> *<sup>∂</sup> ∂xj Vi ∂Vi ∂xj* � *Vi ∂Vj ∂xi*

¼ ∇ ! *V* !*<sup>T</sup>* : ∇ ! *V* ! � ∇ ! *V* ! : ∇ ! *V* ! þ *V* ! � Δ*V* ! � *V* ! � ∇ ! ∇ ! � *V* � � ! *:*

¼ 2Ω : ∇ ! *V* ! þ 2*V* ! � ∇ ! � Ω � �

¼ 2Ω : Ω þ 2Ω : *S* þ 2*V*

� Ω þ *V* ! � *S*,

The vector product of gradients of scalar functions gives us a vector; in terms of

¼ 2*V* ! ¼ *ω* ! � *ν*Δ*ω* !;

! *ω* !*<sup>T</sup>* : <sup>∇</sup> ! *ω*

*ω* ! � *<sup>∂</sup><sup>ω</sup>* ! *∂t* þ *V* ! � ∇ � � !

*Advances on Tensor Analysis and Their Applications*

*dω*<sup>2</sup> *dt* � *<sup>ω</sup>*

> ! � ∇ ! � *V* � � !

For incompressible fluid ∇

Let us write one more equation:

¼ �∇ ! � *V* ! � 2Ω h i

¼ ∇ ! � *Vi ∂Vi ∂xj* � *Vi ∂Vj ∂xi*

∇ ! � *V* ! � ∇ ! � *V* h i � � !

∇ ! � *V* ! � ∇ ! � *V* h i � � !

One more interesting relation is

but as we already mentioned, *V*

rotation of a vector function, we can write

� *<sup>∂</sup>Vi ∂xj ∂Vj ∂xi* � *Vi ∂*2 *Vj ∂xi∂xj*

> ¼ ∇ ! *V* !*<sup>T</sup>* : ∇ ! *V* ! � ∇ ! *V* ! : ∇ ! *V* ! þ *V* ! � Δ*V* ! � ∇ ! ∇ ! � *V*

This equation also can be written in the next form:

*V* ! � ∇ ! *V* ! ¼ *V* !

*V* ! ! � ∇ ! � *V* � � !

! *V*<sup>2</sup> <sup>2</sup> � <sup>1</sup> 2 *V* ! � ∇ ! � *V* � � !

� *S* ¼ ∇

∇ !

> ∇ ! � *V* ! � ∇ ! � *V* h i � � !

� <sup>Ω</sup> <sup>¼</sup> <sup>1</sup>

<sup>2</sup> Δ*V* ! � 1 2 ∇ ! ∇ ! � *V* � � ! *:*

Therefore,

∇ ! � *V* ! � ∇ ! � *V* h i � � !

or

**58**

As we already know *V*

then we obtain scalar transport equation of *ω*2:

! ⊗ *ω* ! � �

> � *S* ¼ ∇ !

!

*V* ! � ∇ � � !

� �

!:

!*:* (40)

� Ω; therefore, Eq. (34) can be written as

¼

! � ∇ ! � Ω � � *:*

� Ω; therefore,

*:* (43)

� �

� Ω*:* (41)

<sup>¼</sup> *<sup>∂</sup>Vi ∂xj ∂Vi ∂xj* þ *Vi ∂*2 *Vi ∂xj∂xj* �

h i � � !

and

(42)

∇ ! � *f*∇ ! *φ* � � <sup>¼</sup> *<sup>e</sup>* ! *i ∂ ∂xi* � *<sup>f</sup> <sup>∂</sup><sup>φ</sup> ∂xj e* ! *<sup>j</sup>* ¼ *e* ! *<sup>i</sup>* � *e* ! *j* � � *∂ ∂xi f ∂φ ∂xj* ¼ *εkij ∂ ∂xi f ∂φ ∂xj* � � *<sup>e</sup>* ! *<sup>k</sup>* ¼ ¼ *ε*1*ij ∂ ∂xi f ∂φ ∂xj* � � *<sup>e</sup>* ! <sup>1</sup> þ *ε*2*ij ∂ ∂xi f ∂φ ∂xj* � � *<sup>e</sup>* ! <sup>2</sup> þ *ε*3*ij ∂ ∂xi f ∂φ ∂xj* � � *<sup>e</sup>* ! <sup>3</sup> ¼ <sup>¼</sup> *<sup>∂</sup> ∂x*<sup>2</sup> *f ∂φ ∂x*<sup>3</sup> � *∂ ∂x*<sup>3</sup> *f ∂φ ∂x*<sup>2</sup> � � *<sup>e</sup>* ! <sup>1</sup> þ *∂ ∂x*<sup>3</sup> *f ∂φ ∂x*<sup>1</sup> � *∂ ∂x*<sup>1</sup> *f ∂φ ∂x*<sup>3</sup> � � *<sup>e</sup>* ! 2 þ *∂ ∂x*<sup>1</sup> *f ∂φ ∂x*<sup>2</sup> � *∂ ∂x*<sup>2</sup> *f ∂φ ∂x*<sup>1</sup> � � *<sup>e</sup>* ! <sup>3</sup> ¼ <sup>¼</sup> *<sup>∂</sup><sup>f</sup> ∂x*<sup>2</sup> *∂φ ∂x*<sup>3</sup> � *∂f ∂x*<sup>3</sup> *∂φ ∂x*<sup>2</sup> � � *<sup>e</sup>* ! <sup>1</sup> þ *∂f ∂x*<sup>3</sup> *∂φ ∂x*<sup>1</sup> � *∂f ∂x*<sup>1</sup> *∂φ ∂x*<sup>3</sup> � � *<sup>e</sup>* ! <sup>2</sup> þ *∂f ∂x*<sup>1</sup> *∂φ ∂x*<sup>2</sup> � *∂f ∂x*<sup>2</sup> *∂φ ∂x*<sup>1</sup> � � *<sup>e</sup>* ! 3*:*

Therefore, Eq. (44) is valid.

The vector product of gradients of scalar functions also can be written as

$$\overrightarrow{\nabla}\hat{f}\times\overrightarrow{\nabla}\rho=\frac{1}{2}\left(\overrightarrow{\nabla}\rho\otimes\overrightarrow{\nabla}\hat{f}-\overrightarrow{\nabla}f\otimes\overrightarrow{\nabla}\rho\right):\left(^{3}\underline{\mathfrak{L}}\right)=-\overrightarrow{\nabla}\cdot\left[\left(\overrightarrow{f}\overrightarrow{\nabla}\rho\right)\cdot\left(^{3}\underline{\mathfrak{L}}\right)\right].\tag{45}$$

Here we have in component form:

∇ ! � *f*∇ ! *φ* � �� 3 *ε* h i <sup>¼</sup> <sup>∇</sup> ! � *<sup>f</sup> <sup>∂</sup><sup>φ</sup> ∂xs e* ! *<sup>s</sup>* � *εkij e* ! *k e* ! *i e* ! *j* � � ¼ ∇ ! � *<sup>f</sup> <sup>∂</sup><sup>φ</sup> ∂xk εkij e* ! *i e* ! *j* � � ¼ ¼ *e* ! *t ∂ ∂xt* � *<sup>f</sup> <sup>∂</sup><sup>φ</sup> ∂xk εkij e* ! *i e* ! *<sup>j</sup>* <sup>¼</sup> *<sup>∂</sup> ∂xi f ∂φ ∂xk* � �*εkij <sup>e</sup>* ! *<sup>j</sup>* ¼ <sup>¼</sup> *<sup>∂</sup> ∂xi f ∂φ ∂xk* � �*εki*<sup>1</sup> *<sup>e</sup>* ! <sup>1</sup> þ *∂ ∂xi f ∂φ ∂xk* � �*εki*<sup>2</sup> *<sup>e</sup>* ! <sup>2</sup> þ *∂ ∂xi f ∂φ ∂xk* � �*εki*<sup>3</sup> *<sup>e</sup>* ! <sup>3</sup> ¼ <sup>¼</sup> *<sup>∂</sup><sup>f</sup> ∂x*<sup>3</sup> � *∂φ ∂x*<sup>2</sup> � *∂f ∂x*<sup>2</sup> � *∂φ ∂x*<sup>3</sup> � � *<sup>e</sup>* ! <sup>1</sup> þ *∂f ∂x*<sup>1</sup> � *∂φ ∂x*<sup>3</sup> � *∂f ∂x*<sup>3</sup> � *∂φ ∂x*<sup>1</sup> � � *<sup>e</sup>* ! 2 þ *∂f ∂x*<sup>2</sup> � *∂φ ∂x*<sup>1</sup> � *∂f ∂x*<sup>1</sup> � *∂φ ∂x*<sup>2</sup> � � *<sup>e</sup>* ! 3*:*

It is easy to see that this is equal to expression for ∇ ! *f* � ∇ ! *φ* with the minus sign.

#### **6. Mechanical energy equation**

Mechanical energy balance equation can be obtained as a scalar product of each member of Eq. (12) on velocity vector *V* ! :

*Advances on Tensor Analysis and Their Applications*

$$
\overrightarrow{\mathbf{V}} \cdot \rho \left[ \frac{\overrightarrow{\vartheta} \overrightarrow{V}}{\partial t} + \left( \overrightarrow{\mathbf{V}} \cdot \overrightarrow{\nabla} \right) \overrightarrow{V} \right] = \overrightarrow{\mathbf{V}} \cdot \left( \overrightarrow{\mathbf{V}} \cdot \underline{\underline{\sigma}} \right) + \overrightarrow{\mathbf{V}} \cdot \rho \overrightarrow{\overline{f}} \,. \tag{46}
$$

The second term of the right part of Eq. (49) in component form is

*δjkδis* ¼ *σij*

<sup>¼</sup> *<sup>∂</sup>Vi ∂xk*

*∂Vs ∂xk e* ! *<sup>j</sup>* � *e* ! *k <sup>e</sup>*

<sup>¼</sup> *<sup>∂</sup>Vi ∂xk σik:*

> *∂σik ∂xk*

, but *σki* ¼ *σik*; therefore both parts of the expression are equal to

1 *dt* <sup>∇</sup> !

� *<sup>∂</sup>Vi ∂xk σik:*

*∂Vi ∂xj*

*σik* þ *Vi*

Due to the symmetry of the stress tensor *<sup>σ</sup>* <sup>¼</sup> *<sup>σ</sup>T*, this expression is an identity (i.e., the left side is equal to the right one). Indeed, the second term of the righthand side of this relation, after re-designating the index *i* by *k* and vice versa, takes

We could simplify the last term of the right part of Eq. (46) if we introduce potential *U* of mass forces in field of gravity (*z* axis is positive upwards) as earlier

After substituting the above expressions into Eq. (46), we obtain the equation

In the left part of this equation, we observe the total mechanical energy of a fluid flow as the sum of kinetic and potential energy of the flow [6]. Often the right part of Eq. (51) is written in another form, where stress tensor is written as the sum

¼ ∇ ! � *V* ! � *σ* � *<sup>σ</sup>* : <sup>∇</sup>

! *<sup>i</sup>* � *e* ! *s* <sup>¼</sup>

! is the gravity acceleration vector in the field of

*<sup>U</sup>* � *d r*! ¼ �*<sup>ρ</sup>*

! *V* !

� *τ* : ∇ ! *V* !

*ipδjkδij e* ! *k* <sup>¼</sup>

> *<sup>k</sup>* ¼ � *<sup>∂</sup>pVk ∂xs*

> > *∂Vk ∂xk*

¼ *p*∇ ! � *V* ! *:*

! *V* !

! � *V* !

� *pVk e* !

*∂Vs ∂xk* *p* � *p*∇ ! � *V* ! *:*

*δijδjkδis* ¼ *p*

*dU*

*:* (51)

*:* (52)

*∂xk*

*pVk* ¼

*<sup>δ</sup>sk* ¼ � *<sup>∂</sup>*

*dt :* (50)

*σ* : ∇ ! *V* !

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

the form *Vk*

¼ �∇ !

gravity. Then

*V* ! � *ρ f* !

*f* ! *∂σki ∂xi*

*U*, *U* ¼ *gz* ¼ *r*

¼ �*ρV* ! � ∇ !

for mechanical energy of a fluid flow:

�∇ ! � *V* ! � *pE* <sup>þ</sup> <sup>∇</sup>

!

¼ �∇ !

¼ �*Vk*

¼ *pδij e* ! *i e* ! *j* : *∂Vs ∂xk e* ! *k e* ! *<sup>s</sup>* ¼ *p*

� *Vi e* ! *<sup>i</sup>* � *pδjk e* ! *j e* ! *k* ¼ �<sup>∇</sup>

� *pVk e* ! *k* ¼ � *<sup>e</sup>*

*∂p ∂xk* � *p ∂Vk ∂xk*

The first member is

The third member is

*pE* : ∇ ! *V* !

�∇ ! � *V* ! � *pE* ¼ �<sup>∇</sup>

**61**

each other. Thus, equality (49) is valid.

! � *<sup>g</sup>*

*ρ d dt*

*U* ¼ �*ρ*

!, where *g*

*V*2 <sup>2</sup> <sup>þ</sup> *<sup>U</sup>* 

*σ* ¼ �*pE* þ *τ*. Then the right part of Eq. (51) will have the form:

! � *V* ! � *τ* <sup>þ</sup> *pE* : <sup>∇</sup>

> ! *s ∂ ∂xs*

¼ � *V* ! � ∇ !

*d r*! *dt* � <sup>∇</sup> ! *U* ¼ �*<sup>ρ</sup>*

¼ *σij e* ! *i e* ! *j* : *∂Vs ∂xk e* ! *k e* ! *<sup>s</sup>* ¼ *σij*

> *∂Vs ∂xk*

¼ *σij*

Finally Eq. (49) in component form is

*Vk ∂σik ∂xi*

Transformations of the left part lead us to the following results:

$$\overrightarrow{V} \cdot \frac{\partial \overrightarrow{V}}{\partial t} = V\_k \overrightarrow{e}\_k \cdot \frac{\partial}{\partial t} V\_i \overrightarrow{e}\_i = V\_k \frac{\partial V\_i}{\partial t} \left(\overrightarrow{e}\_k \cdot \overrightarrow{e}\_i\right) = V\_k \frac{\partial V\_i}{\partial t} \delta\_{ki} = V\_k \frac{\partial V\_k}{\partial t} = \frac{1}{2} \frac{\partial}{\partial t} V\_k V\_k = \frac{1}{2} \frac{\partial V^2}{\partial t};$$

$$V \cdot \left[\left(\overrightarrow{V} \cdot \overrightarrow{\nabla}\right) \overrightarrow{V}\right] = V\_k \overrightarrow{e}\_k \cdot V\_j \frac{\partial}{\partial \mathbf{x}\_j} V\_i \overrightarrow{e}\_i = V\_k V\_j \frac{\partial V\_i}{\partial \mathbf{x}\_j} \left(\overrightarrow{e}\_k \cdot \overrightarrow{e}\_i\right) =$$

$$= V\_k V\_j \frac{\partial V\_i}{\partial \mathbf{x}\_j} \delta\_{ki} = V\_k V\_j \frac{\partial V\_k}{\partial \mathbf{x}\_j} = V\_j \frac{\partial}{\partial \mathbf{x}\_j} \frac{1}{2} V\_k V\_k = \left(\overrightarrow{V} \cdot \overrightarrow{\nabla}\right) \frac{V^2}{2}.$$

It is easy to see that in sum the left part is the material derivative of kinetic energy of a fluid particle—quantity *<sup>d</sup> dt V*2 <sup>2</sup> . Then Eq. (46) takes the form:

$$
\rho \frac{d}{dt} \left[ \frac{\mathbf{V}^2}{2} \right] = \vec{\mathbf{V}} \cdot \left( \vec{\nabla} \cdot \underline{\mathbf{\sigma}} \right) + \vec{\mathbf{V}} \cdot \rho \vec{f} \,. \tag{47}
$$

Usually stress tensor is defined as the sum *σ* ¼ �*pE* þ *τ*, where *τ* is the shear stress tensor. Then the first term of the right part of Eq. (46) takes the form:

$$
\overrightarrow{\boldsymbol{V}} \cdot \left( \overrightarrow{\boldsymbol{\nabla}} \cdot \underline{\boldsymbol{\sigma}} \right) = -\overrightarrow{\boldsymbol{V}} \cdot \left( \overrightarrow{\boldsymbol{\nabla}} \cdot \underline{\boldsymbol{p}} \underline{\boldsymbol{E}} \right) + \overrightarrow{\boldsymbol{V}} \cdot \left( \overrightarrow{\boldsymbol{\nabla}} \cdot \underline{\boldsymbol{\tau}} \right).
$$

Now we can represent equation of mechanical energy balance (Eq. (46)) considering ∇ ! � *pE* ¼ ∇ ! *p* as follows:

$$
\rho \frac{d}{dt} \left[ \frac{\mathbf{V}^2}{2} \right] = -\vec{\mathbf{V}} \cdot \vec{\nabla} p + \vec{\mathbf{V}} \cdot \left( \vec{\nabla} \cdot \underline{\mathbf{r}} \right) + \rho \vec{\mathbf{V}} \cdot \vec{f} \,. \tag{48}
$$

The first member of the right part of Eq. (46) is power of stresses *V* ! � ∇ ! � *σ* � �, which can be written in the form:

$$
\overrightarrow{\boldsymbol{V}} \cdot \left( \overrightarrow{\boldsymbol{\nabla}} \cdot \underline{\boldsymbol{\sigma}} \right) = \overrightarrow{\boldsymbol{\nabla}} \cdot \left( \overrightarrow{\boldsymbol{V}} \cdot \underline{\boldsymbol{\sigma}} \right) - \underline{\boldsymbol{\sigma}} : \overrightarrow{\boldsymbol{\nabla}} \overrightarrow{\boldsymbol{V}}.\tag{49}
$$

It is easy to be proven if we rewrite this expression in component form in Cartesian coordinates. In this case the left part of Eq. (49) is

$$\begin{split} \left(\overrightarrow{V} \cdot \left(\overrightarrow{\nabla} \cdot \underline{\sigma}\right) = V\_k \overrightarrow{\overline{e}}\_k \cdot \left(\overrightarrow{\overline{e}}\_i \frac{\partial}{\partial \mathbf{x}\_i} \cdot \sigma\_{\overline{g}} \overrightarrow{\overline{e}}\_s \overrightarrow{e}\_j\right) = V\_k \overrightarrow{\overline{e}}\_k \cdot \left[\frac{\partial \sigma\_{\overline{g}}}{\partial \mathbf{x}\_i} \left(\overrightarrow{\overline{e}}\_i \cdot \overrightarrow{\overline{e}}\_s\right) \overrightarrow{e}\_j\right] = 0 \\ = V\_k \overrightarrow{\overline{e}}\_k \cdot \frac{\partial \sigma\_{\overline{g}}}{\partial \mathbf{x}\_i} \overrightarrow{e}\_j = V\_k \frac{\partial \sigma\_{\overline{i}}}{\partial \mathbf{x}\_i} \left(\overrightarrow{\overline{e}}\_k \cdot \overrightarrow{e}\_j\right) = V\_k \frac{\partial \sigma\_{ik}}{\partial \mathbf{x}\_i} .\end{split}$$

The first term of the right part of Eq. (49) in component form is

$$
\begin{split}
\overrightarrow{\nabla} \cdot \left( \overrightarrow{V} \cdot \underline{\sigma} \right) &= \overrightarrow{\nabla} \cdot \left( V\_i \overrightarrow{e\_i} \cdot \sigma\_{\overrightarrow{\eta}} \overrightarrow{e\_i} \overrightarrow{e\_j} \right) = \overrightarrow{\nabla} \cdot \left[ V\_i \sigma\_{\overrightarrow{\eta}} \left( \overrightarrow{e\_i} \cdot \overrightarrow{e\_i} \right) \overrightarrow{e\_j} \right] = \overrightarrow{\nabla} \cdot \left( V\_i \sigma\_{\overrightarrow{\eta}} \overrightarrow{e\_j} \right) = 0 \\
&= \overrightarrow{\sigma}\_k \frac{\partial}{\partial \textbf{x}\_k} \cdot V\_i \sigma\_{\overrightarrow{\eta}} \overrightarrow{e\_j} = \delta\_{k\overrightarrow{\eta}} \frac{\partial}{\partial \textbf{x}\_k} V\_i \sigma\_{\overrightarrow{\eta}} = \frac{\partial}{\partial \textbf{x}\_k} V\_i \sigma\_{ik} = \frac{\partial V\_i}{\partial \textbf{x}\_k} \sigma\_{ik} + V\_i \frac{\partial \sigma\_{ik}}{\partial \textbf{x}\_k}.
\end{split}
$$

*V* ! � *ρ ∂V* ! *∂t*

*Advances on Tensor Analysis and Their Applications*

*V* h i !

energy of a fluid particle—quantity *<sup>d</sup>*

*V* ! � *∂V* ! *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *Vk <sup>e</sup>* ! *k* � *∂ ∂t Vi e* ! *<sup>i</sup>* ¼ *Vk*

> *V* � *V* ! � ∇ � � !

sidering ∇ ! þ *V* ! � ∇ � � !

*∂Vi ∂t e* ! *<sup>k</sup>* � *e* ! *i* � �

> *∂Vi ∂xj*

*V*2 2 � �

¼ *Vk e* ! *<sup>k</sup>* � *Vj*

¼ *VkVj*

*ρ d dt*

*p* as follows:

*V*2 2 � �

*V* ! � ∇ ! � *σ* � �

¼ *Vk e* ! *<sup>k</sup>* � *e* ! *i ∂ ∂xi* � *σsj e* ! *s e* ! *j*

¼ *Vk e* ! *k* � *∂σij ∂xi e* ! *<sup>j</sup>* ¼ *Vk*

� *Vs e* ! *<sup>s</sup>* � *σij e* ! *i e* ! *j*

¼ ∇ !

¼ *e* ! *k ∂ ∂xk*

Cartesian coordinates. In this case the left part of Eq. (49) is

� �

� *Viσij e* ! *<sup>j</sup>* ¼ *δkj*

*V* ! � ∇ ! � *σ* � �

*ρ d dt*

� *pE* ¼ ∇ !

which can be written in the form:

*V* ! � ∇ ! � *σ* � �

∇ ! � *V* ! � *σ* � �

**60**

" #

Transformations of the left part lead us to the following results:

*∂ ∂xj Vi e* !

> *dt V*2

¼ *V* ! � ∇ ! � *σ* � �

¼ �*V* ! � ∇ ! � *pE* � �

¼ �*V* ! � ∇ ! *p* þ *V* ! � ∇ ! � *τ* � �

The first member of the right part of Eq. (46) is power of stresses *V*

¼ ∇ ! � *V* ! � *σ* � �

� �

The first term of the right part of Eq. (49) in component form is

¼ ∇ !

> *∂ ∂xk*

It is easy to be proven if we rewrite this expression in component form in

*∂σij ∂xi e* ! *<sup>k</sup>* � *e* ! *j* � �

*δki* ¼ *VkVj*

It is easy to see that in sum the left part is the material derivative of kinetic

Usually stress tensor is defined as the sum *σ* ¼ �*pE* þ *τ*, where *τ* is the shear stress tensor. Then the first term of the right part of Eq. (46) takes the form:

Now we can represent equation of mechanical energy balance (Eq. (46)) con-

*V* ! ¼ *V* ! � ∇ ! � *σ* � �

¼ *Vk*

*∂Vi ∂t*

*<sup>i</sup>* ¼ *VkVj*

*∂Vk ∂xj*

*δki* ¼ *Vk*

*∂Vi ∂xj e* ! *<sup>k</sup>* � *e* ! *i* � �

¼ *Vj*

þ *V* ! � *ρ f* !

þ *V* ! � ∇ ! � *τ* � � *:*

> þ *ρV* ! � *f* !

¼ *Vk*

*e* ! *j*

*Viσik* <sup>¼</sup> *<sup>∂</sup>Vi ∂xk*

� *σ* : ∇ ! *V* !

¼ *Vk e* ! *<sup>k</sup>* � *<sup>∂</sup>σsj ∂xi e* ! *<sup>i</sup>* � *e* ! *s* � �

� *Vsσij e* ! *<sup>s</sup>* � *e* ! *i* � �

*Viσij* <sup>¼</sup> *<sup>∂</sup>*

h i

*∂xk*

<sup>2</sup> . Then Eq. (46) takes the form:

*∂ ∂xj* 1 2

þ *V* ! � *ρ f* !

> *∂Vk <sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>1</sup> 2 *∂ ∂t*

> > ¼

*VkVk* ¼ *V*

! � ∇ � � ! *V*<sup>2</sup>

*:* (47)

*:* (48)

! � ∇ ! � *σ* � � ,

*:* (49)

*e* ! *j*

� *Viσij e* ! *j* � �

*σik* þ *Vi*

¼

¼

*∂σik ∂xk :*

� �

*∂σik ∂xi :*

> ¼ ∇ !

*:* (46)

*VkVk* <sup>¼</sup> <sup>1</sup> 2 *∂V*<sup>2</sup> *∂t* ;

2 *:*

The second term of the right part of Eq. (49) in component form is

$$\sigma : \overrightarrow{\nabla} \overrightarrow{V} = \sigma\_{\overrightarrow{\eta}} \overrightarrow{e\_i} \overrightarrow{e\_j} : \frac{\partial V\_s}{\partial \mathbf{x}\_k} \overrightarrow{e\_k} \overrightarrow{e\_s} = \sigma\_{\overrightarrow{\eta}} \frac{\partial V\_s}{\partial \mathbf{x}\_k} \left(\overrightarrow{e\_j} \cdot \overrightarrow{e}\_k\right) \left(\overrightarrow{e\_i} \cdot \overrightarrow{e}\_s\right) = 0$$

$$= \sigma\_{\overrightarrow{\eta}} \frac{\partial V\_s}{\partial \mathbf{x}\_k} \delta\_{jk} \delta\_{is} = \sigma\_{\overrightarrow{\eta}} \frac{\partial V\_i}{\partial \mathbf{x}\_j} = \frac{\partial V\_i}{\partial \mathbf{x}\_k} \sigma\_{ik}.$$

Finally Eq. (49) in component form is

$$V\_k \frac{\partial \sigma\_{ik}}{\partial \mathbf{x}\_i} = \frac{\partial V\_i}{\partial \mathbf{x}\_k} \sigma\_{ik} + V\_i \frac{\partial \sigma\_{ik}}{\partial \mathbf{x}\_k} - \frac{\partial V\_i}{\partial \mathbf{x}\_k} \sigma\_{ik} \dots$$

Due to the symmetry of the stress tensor *<sup>σ</sup>* <sup>¼</sup> *<sup>σ</sup>T*, this expression is an identity (i.e., the left side is equal to the right one). Indeed, the second term of the righthand side of this relation, after re-designating the index *i* by *k* and vice versa, takes the form *Vk ∂σki ∂xi* , but *σki* ¼ *σik*; therefore both parts of the expression are equal to each other. Thus, equality (49) is valid.

We could simplify the last term of the right part of Eq. (46) if we introduce potential *U* of mass forces in field of gravity (*z* axis is positive upwards) as earlier *f* ! ¼ �∇ ! *U*, *U* ¼ *gz* ¼ *r* ! � *<sup>g</sup>* !, where *g* ! is the gravity acceleration vector in the field of gravity. Then

$$\overrightarrow{\mathbf{V}} \cdot \rho \overrightarrow{\mathbf{f}} = -\rho \overrightarrow{\mathbf{V}} \cdot \overrightarrow{\mathbf{V}} \mathbf{U} = -\rho \frac{d \overrightarrow{r}}{dt} \cdot \left(\overrightarrow{\mathbf{V}} U\right) = -\rho \frac{1}{dt} \left(\overrightarrow{\mathbf{V}} U \cdot d \overrightarrow{r}\right) = -\rho \frac{d U}{dt}.\tag{50}$$

After substituting the above expressions into Eq. (46), we obtain the equation for mechanical energy of a fluid flow:

$$
\rho \frac{d}{dt} \left(\frac{\vec{V}^2}{2} + U\right) = \vec{\nabla} \cdot \left(\vec{V} \cdot \underline{\sigma}\right) - \underline{\sigma} : \vec{\nabla}\vec{V}.\tag{51}
$$

In the left part of this equation, we observe the total mechanical energy of a fluid flow as the sum of kinetic and potential energy of the flow [6]. Often the right part of Eq. (51) is written in another form, where stress tensor is written as the sum *σ* ¼ �*pE* þ *τ*. Then the right part of Eq. (51) will have the form:

$$-\overrightarrow{\nabla} \cdot \left(\overrightarrow{V} \cdot p\underline{E}\right) + \overrightarrow{\nabla} \cdot \left(\overrightarrow{V} \cdot \underline{\underline{r}}\right) + p\underline{E} : \overrightarrow{\nabla}\overrightarrow{V} - \underline{\underline{r}} : \overrightarrow{\nabla}\overrightarrow{V} . \tag{52}$$

The first member is

$$- \overrightarrow{\nabla} \cdot \left( \overrightarrow{V} \cdot p \underline{E} \right) = - \overrightarrow{\nabla} \cdot \left( V\_i \overrightarrow{e}\_i \cdot p \delta\_{jk} \overrightarrow{e}\_j \overrightarrow{e}\_k \right) = - \overrightarrow{\nabla} \cdot \left( \overrightarrow{V}\_i p \delta\_{jk} \delta\_{lj} \overrightarrow{e}\_k \right) =$$

$$= - \overrightarrow{\nabla} \cdot \left( p V\_k \overrightarrow{e}\_k \right) = - \overrightarrow{e}\_s \frac{\partial}{\partial \mathbf{x}\_s} \cdot p V\_k \overrightarrow{e}\_k = - \frac{\partial p V\_k}{\partial \mathbf{x}\_s} \delta\_{kl} = - \frac{\partial}{\partial \mathbf{x}\_k} p V\_k = 0$$

$$= - V\_k \frac{\partial p}{\partial \mathbf{x}\_k} - p \frac{\partial V\_k}{\partial \mathbf{x}\_k} = - \left( \overrightarrow{V} \cdot \overrightarrow{\nabla} \right) p - p \overrightarrow{\nabla} \cdot \overrightarrow{\nabla}.$$

The third member is

$$p\underline{E}: \vec{\nabla V} = p\delta\_{\vec{\eta}} \vec{e}\_i \vec{e}\_j : \frac{\partial V\_s}{\partial \mathbf{x}\_k} \vec{e}\_k \vec{e}\_s = p\frac{\partial V\_s}{\partial \mathbf{x}\_k} \delta\_{\vec{\eta}} \delta\_{\vec{k}} \delta\_{\vec{s}} = p\frac{\partial V\_k}{\partial \mathbf{x}\_k} = p\vec{\nabla} \cdot \vec{V} . \vec{e}\_j$$

When we substitute these terms in Eq. (52) and then in Eq. (51), the equation for mechanical energy of a fluid flow will take the form:

$$
\rho \frac{d}{dt} \left(\frac{\mathbf{V}^2}{2} + \mathbf{U}\right) = -\left(\overrightarrow{\mathbf{V}} \cdot \overrightarrow{\mathbf{V}}\right)p + \overrightarrow{\mathbf{V}}\left(\overrightarrow{\mathbf{V}} \cdot \underline{\mathbf{r}}\right) - \boldsymbol{\tau} : \overrightarrow{\mathbf{V}} \vec{\mathbf{V}}.\tag{53}
$$

∇ ! *V*<sup>2</sup> <sup>2</sup> � *<sup>V</sup>* ! � ∇ ! � *V* � � !

where *<sup>ν</sup>* <sup>¼</sup> *<sup>μ</sup>*

considering ∇

! � *V* !

In incompressible fluid *ρ* ¼ *const*, ∇

∇ ! *V*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>p</sup> ρ* <sup>þ</sup> *gz* � � <sup>¼</sup> *<sup>V</sup>*

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

> ¼ � <sup>1</sup> *ρ* ∇ ! *p* þ *ν*∇ ! ∇ ! � *V* � � !

*<sup>ρ</sup>* is the kinematic (momentum) viscosity of fluid.

! � *V* !

! � ∇ ! � *V* � � !

The gradient of total mechanical energy of a fluid particle *<sup>E</sup>* <sup>¼</sup> *<sup>V</sup>*<sup>2</sup>

¼ 0 and using Eq. (55) and the relation:

¼ *V* ! ∇ ! � *V* � � !

¼ �∇ ! *p* þ ∇ !

Using concepts of identity tensor *<sup>I</sup>* and Levi-Civita tensor <sup>3</sup>*<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup>ijk <sup>e</sup>*

<sup>þ</sup> *pI* � *<sup>ρ</sup>UI* <sup>þ</sup> *ρν* <sup>3</sup>

write members of the right part in the divergence form, and as a result, the whole Navier–Stokes equation for steady flow of incompressible fluid can be written as

h i � � !

The last term of Eq. (59) can be considered in a more simple form due to

form of the Navier–Stokes equation for steady flow of an incompressible fluid is

The first law of thermodynamics connects internal energy, heat, and work. In

þ ð Þ *p* þ *ρgz I* � *μS*

h i <sup>¼</sup> <sup>0</sup>*:* (60)

! ¼ ∇ ! � ∇ ! *V* ! ¼ ∇ !

It is possible to obtain the divergence form of Eq. (54) for incompressible fluid

depends on vortex structure of the flow. When ∇ � *V*

∇ ! � *V* ! ⊗ *V* � � !

Then Eq. (54) will take the form:

∇ ! � *ρV* ! ⊗ *V* � � !

∇ ! � *ρV* ! ⊗ *V* !

relation for incompressible fluid Δ*V*

∇ !

**7. Energy equation for moving fluid**

**63**

� *ρ V* ! ⊗ *V* � � !

the case of moving fluid, it can be written as follows:

*ρ du dt* ¼ �<sup>∇</sup> ! � *q* ! <sup>þ</sup> *<sup>σ</sup>* : <sup>∇</sup> ! *V* !

is zero, and then *E* ¼ *const* in the whole area of the flow.

� *ν*∇ ! � ∇ ! � *V* � � !

¼ 0 and then we have

� *ν*∇ ! � ∇ ! � *V* !

!

þ *V* ! � ∇ � � !

*ρU* � *μ*∇ ! � ∇ ! � *V* !

*<sup>ε</sup>* � � � <sup>∇</sup> ! � *V*

*V* !


þ ∇ ! *U*,

*:* (56)

<sup>2</sup> <sup>þ</sup> *<sup>p</sup> <sup>ρ</sup>* þ *gz*

¼ 0 the right part of Eq. (56)

*:* (57)

*:* (58)

*<sup>k</sup>*, we can

! *i e* ! *j e* !

¼ 0*:* (59)

� *S*. Finally, the divergence

þ *qv*, (61)

As a result, we could conclude that the rate of change of total mechanical energy of a flow is equal to the sum of the powers of the pressure forces and viscous friction.

Navier–Stokes equation for a steady flow of viscous incompressible fluid is

$$
\rho \left( \vec{\mathbf{V}} \cdot \vec{\nabla} \right) \vec{\mathbf{V}} = -\vec{\nabla}p + \mu \Delta \vec{\mathbf{V}} + \rho \vec{\mathbf{f}} \,. \tag{54}
$$

The Laplacian of velocity in the right part can be written in the form:

$$
\Delta \overrightarrow{V} = \overrightarrow{\nabla} \left( \overrightarrow{\nabla} \cdot \overrightarrow{V} \right) - \overrightarrow{\nabla} \times \left( \overrightarrow{\nabla} \times \overrightarrow{V} \right). \tag{55}
$$

It can be obtained by consideration of operation ∇ ! � ∇ ! � *V* ! :

$$\begin{split} \overrightarrow{\nabla} \times \left( \overrightarrow{\nabla} \times \overrightarrow{V} \right) &= \overrightarrow{e\_i} \frac{\partial}{\partial \mathbf{x}\_i} \times \left( \frac{\partial V\_j}{\partial \mathbf{x}\_k} \varepsilon\_{k\dot{\mathbf{y}}} \overrightarrow{e\_s} \right) = \\\\ &= \frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_k} \varepsilon\_{k\dot{\mathbf{y}}} \varepsilon\_{l\dot{\mathbf{x}}} \overrightarrow{e\_I} = \frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_k} \varepsilon\_{k\dot{\mathbf{y}}} \varepsilon\_{1\dot{\mathbf{x}}} \overrightarrow{e\_1} + \frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_k} \varepsilon\_{k\dot{\mathbf{y}}} \varepsilon\_{2\dot{\mathbf{x}}} \overrightarrow{e\_2} + \frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_k} \varepsilon\_{k\dot{\mathbf{y}}} \varepsilon\_{3\dot{\mathbf{y}}} \overrightarrow{e\_3}. \end{split}$$

The member with the basis vector *e* ! <sup>1</sup> is determined as

$$\frac{\partial^2 V\_j}{\partial \mathbf{x}\_i \partial \mathbf{x}\_k} \varepsilon\_{k\mathbf{j}\uparrow} \varepsilon\_{1\mathbf{x}} = \frac{\partial^2 V\_j}{\partial \mathbf{x}\_2 \partial \mathbf{x}\_k} \varepsilon\_{3\mathbf{j}j} - \frac{\partial^2 V\_j}{\partial \mathbf{x}\_3} \varepsilon\_{2\mathbf{j}} = \frac{\partial^2 V\_2}{\partial \mathbf{x}\_2 \partial \mathbf{x}\_1} - \frac{\partial^2 V\_1}{\partial \mathbf{x}\_3 \partial \mathbf{x}\_3} + \frac{\partial^2 V\_3}{\partial \mathbf{x}\_3 \partial \mathbf{x}\_1} =$$

$$= \frac{\partial}{\partial \mathbf{x}\_1} \left( \frac{\partial^2 V\_1}{\partial \mathbf{x}\_1} + \frac{\partial^2 V\_2}{\partial \mathbf{x}\_2} + \frac{\partial^2 V\_3}{\partial \mathbf{x}\_3} \right) - \frac{\partial^2 V\_1}{\partial \mathbf{x}\_1^2} - \frac{\partial^2 V\_1}{\partial \mathbf{x}\_2^2} - \frac{\partial^2 V\_1}{\partial \mathbf{x}\_3^2} = \frac{\partial}{\partial \mathbf{x}\_1} \left( \vec{\nabla} \cdot \vec{V} \right) - \Delta V\_1.$$

The same can be written for the members with basis vectors *e* ! <sup>2</sup> and *e* ! 3. As a result, we obtain

$$
\begin{split}
\overrightarrow{\nabla} \times \left(\overrightarrow{\nabla} \times \overrightarrow{\nabla}\right) &= \left[\frac{\partial}{\partial \mathbf{x}\_{1}} \left(\overrightarrow{\nabla} \cdot \overrightarrow{\nabla}\right) - \Delta V\_{1}\right] \overrightarrow{\overline{e}\_{1}} + \left[\frac{\partial}{\partial \mathbf{x}\_{2}} \left(\overrightarrow{\nabla} \cdot \overrightarrow{\overline{V}}\right) - \Delta V\_{2}\right] \overrightarrow{\overline{e}\_{2}} + \\ &+ \left[\frac{\partial}{\partial \mathbf{x}\_{3}} \left(\overrightarrow{\nabla} \cdot \overrightarrow{\overline{V}}\right) - \Delta V\_{3}\right] \overrightarrow{\overline{e}\_{3}} = \overrightarrow{\overline{\nabla}} \left(\overrightarrow{\nabla} \cdot \overrightarrow{\overline{V}}\right) - \Delta \overrightarrow{\overline{V}}.
\end{split}
$$

And therefore formula Eq. (55) is valid. One more useful expression based on Eq. (55) is

$$
\overrightarrow{\nabla} \cdot \underline{\mathfrak{L}} = \Delta \overrightarrow{V} + \overrightarrow{\nabla} \left( \overrightarrow{\nabla} \cdot \overrightarrow{V} \right) = 2 \overrightarrow{\nabla} \left( \overrightarrow{\nabla} \cdot \overrightarrow{V} \right) - \overrightarrow{\nabla} \times \overrightarrow{\nabla} \times \overrightarrow{V}.
$$

Using Eqs. (34) and (55) and considering the mass force in field of gravity with the help of potential *U*, *U* ¼ �*gz* (*z* axis is positive upwards), we obtain instead Eq. (54):

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

When we substitute these terms in Eq. (52) and then in Eq. (51), the equation for

*p* þ ∇ ! *V* ! � *τ* 

*p* þ *μ*Δ*V* ! þ *ρ f* !

> ! � ∇ ! � *V* !

*∂*2 *Vj ∂xi∂xk*

� ∇ ! � ∇ ! � *V* !

*εskjε*1*is e* ! <sup>1</sup> þ

*V*<sup>2</sup> *∂x*2*∂x*<sup>1</sup>

> � *∂*2 *V*<sup>1</sup> *∂x*<sup>2</sup> 1

� Δ*V*<sup>1</sup>

� Δ*V*<sup>3</sup>

¼ 2∇ ! ∇ ! � *V* !

Using Eqs. (34) and (55) and considering the mass force in field of gravity with the help of potential *U*, *U* ¼ �*gz* (*z* axis is positive upwards), we obtain instead

*e* ! <sup>1</sup> þ

> *e* ! <sup>3</sup> ¼ ∇ ! ∇ ! � *V* !

<sup>1</sup> is determined as

� *∂*2 *V*1 *∂x*3*∂x*<sup>3</sup> þ *∂*2 *V*<sup>3</sup> *∂x*3*∂x*<sup>1</sup>

> � *∂*2 *V*<sup>1</sup> *∂x*<sup>2</sup> 2

> > *∂ ∂x*<sup>2</sup> ∇ ! � *V* !

> > > � ∇ ! � ∇ ! � *V* ! *:*

� *∂*2 *V*1 *∂x*<sup>2</sup> 3

� *τ* : ∇ ! *V* !

*:* (53)

*:* (54)

*:* (55)

*∂*2 *Vj ∂xi∂xk*

*εskjε*3*is e* ! 3*:*

� Δ*V*1*:*

:

*εskjε*2*is e* ! <sup>2</sup> þ

¼

¼ *∂ ∂x*<sup>1</sup> ∇ ! � *V* !

> ! <sup>2</sup> and *e* ! 3. As a

� Δ*V*<sup>2</sup>

� Δ*V* ! *:*

*e* ! <sup>2</sup>þ

¼ � *V* ! � ∇ !

> *V* !

As a result, we could conclude that the rate of change of total mechanical energy of a flow is equal to the sum of the powers of the pressure forces and viscous

Navier–Stokes equation for a steady flow of viscous incompressible fluid is

¼ �∇ !

The Laplacian of velocity in the right part can be written in the form:

¼

!

*<sup>ε</sup>*2*kj* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

The same can be written for the members with basis vectors *e*

mechanical energy of a fluid flow will take the form:

*ρ V* ! � ∇ !

Δ*V* ! ¼ ∇ ! ∇ ! � *V* !

It can be obtained by consideration of operation ∇

*∂Vj ∂xk εskj e* ! *s*

*εskjεlis e* ! *<sup>l</sup>* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup> *Vj ∂xi∂xk*

*<sup>ε</sup>*3*kj* � *<sup>∂</sup>*<sup>2</sup>

<sup>¼</sup> *<sup>∂</sup> ∂x*<sup>1</sup> ∇ ! � *V* !

þ

One more useful expression based on Eq. (55) is

And therefore formula Eq. (55) is valid.

� *S* ¼ Δ*V* ! þ ∇ ! ∇ ! � *V* !

∇ !

*∂ ∂x*<sup>3</sup> ∇ ! � *V* !

*Vj ∂x*<sup>3</sup>

*V*2 <sup>2</sup> <sup>þ</sup> *<sup>U</sup>* 

*Advances on Tensor Analysis and Their Applications*

*ρ d dt*

¼ *e* ! *i ∂ ∂xi* �

¼ *∂*2 *Vj ∂xi∂xk*

*<sup>ε</sup>skjε*1*is* <sup>¼</sup> *<sup>∂</sup>*<sup>2</sup>

result, we obtain

∇ ! � ∇ ! � *V* !

Eq. (54):

**62**

¼ *∂ ∂x*<sup>1</sup>

The member with the basis vector *e*

*∂*2 *V*<sup>1</sup> *∂x*<sup>1</sup> þ *∂*2 *V*<sup>2</sup> *∂x*<sup>2</sup> þ *∂*2 *V*<sup>3</sup> *∂x*<sup>3</sup>

*Vj ∂x*2*∂xk*

friction.

∇ ! � ∇ ! � *V* !

*∂*2 *Vj ∂xi∂xk*

$$
\vec{\nabla}\frac{\vec{V}^2}{2} - \vec{\nabla} \times \left(\vec{\nabla} \times \vec{\nabla}\right) = -\frac{1}{\rho}\vec{\nabla}p + \nu \vec{\nabla} \left(\vec{\nabla} \cdot \vec{\nabla}\right) - \nu \vec{\nabla} \times \left(\vec{\nabla} \times \vec{\nabla}\right) + \vec{\nabla}U,
$$

where *<sup>ν</sup>* <sup>¼</sup> *<sup>μ</sup> <sup>ρ</sup>* is the kinematic (momentum) viscosity of fluid. In incompressible fluid *ρ* ¼ *const*, ∇ ! � *V* ! ¼ 0 and then we have

$$\vec{\nabla} \left( \frac{\mathbf{V}^2}{2} + \frac{\mathbf{p}}{\rho} + \mathbf{g} \mathbf{z} \right) = \underbrace{\vec{\mathbf{V}} \times \left( \vec{\nabla} \times \vec{\mathbf{V}} \right) - \nu \vec{\nabla} \times \vec{\nabla} \times \vec{\mathbf{V}}}\_{\left( \vec{\nabla} - \nu \vec{\nabla} \right) \times \vec{\nabla} \times \vec{\nabla}}. \tag{56}$$

The gradient of total mechanical energy of a fluid particle *<sup>E</sup>* <sup>¼</sup> *<sup>V</sup>*<sup>2</sup> <sup>2</sup> <sup>þ</sup> *<sup>p</sup> <sup>ρ</sup>* þ *gz* depends on vortex structure of the flow. When ∇ � *V* ! ¼ 0 the right part of Eq. (56) is zero, and then *E* ¼ *const* in the whole area of the flow.

It is possible to obtain the divergence form of Eq. (54) for incompressible fluid considering ∇ ! � *V* ! ¼ 0 and using Eq. (55) and the relation:

$$
\overrightarrow{\nabla} \cdot \left( \overrightarrow{V} \otimes \overrightarrow{V} \right) = \overrightarrow{V} \left( \overrightarrow{\nabla} \cdot \overrightarrow{V} \right) + \left( \overrightarrow{V} \cdot \overrightarrow{\nabla} \right) \overrightarrow{V}.\tag{57}
$$

Then Eq. (54) will take the form:

$$\overrightarrow{\nabla} \cdot \left(\rho \overrightarrow{V} \otimes \overrightarrow{V}\right) = -\overrightarrow{\nabla}p + \overrightarrow{\nabla}\rho U - \mu \overrightarrow{\nabla} \times \overrightarrow{\nabla} \times \overrightarrow{V}.\tag{58}$$

Using concepts of identity tensor *<sup>I</sup>* and Levi-Civita tensor <sup>3</sup>*<sup>ε</sup>* <sup>¼</sup> *<sup>ε</sup>ijk <sup>e</sup>* ! *i e* ! *j e* ! *<sup>k</sup>*, we can write members of the right part in the divergence form, and as a result, the whole Navier–Stokes equation for steady flow of incompressible fluid can be written as

$$\overrightarrow{\nabla} \cdot \left[ \rho \overrightarrow{V} \otimes \overrightarrow{V} + p\underline{I} - \rho U \underline{I} + \rho \nu \left(^3 \varepsilon \right) \cdot \left( \overrightarrow{\nabla} \times \overrightarrow{V} \right) \right] = \mathbf{0}.\tag{59}$$

The last term of Eq. (59) can be considered in a more simple form due to relation for incompressible fluid Δ*V* ! ¼ ∇ ! � ∇ ! *V* ! ¼ ∇ ! � *S*. Finally, the divergence form of the Navier–Stokes equation for steady flow of an incompressible fluid is

$$\overrightarrow{\nabla} \cdot \left[ \rho \left( \overrightarrow{\mathbf{V}} \otimes \overrightarrow{\mathbf{V}} \right) + (p + \rho \mathbf{g} \mathbf{z}) \mathbf{\underline{I}} - \mu \underline{\mathbf{S}} \right] = \mathbf{0}.\tag{60}$$

#### **7. Energy equation for moving fluid**

The first law of thermodynamics connects internal energy, heat, and work. In the case of moving fluid, it can be written as follows:

$$
\rho \frac{du}{dt} = -\overrightarrow{\nabla} \cdot \overrightarrow{q} + \underline{\sigma} : \overrightarrow{\nabla} \overrightarrow{V} + q\_v,\tag{61}
$$

where *t* is the time; *u* is the specific internal energy; *q* ! is the heat flux density vector due to thermal conductivity; *σ* is the stress tensor; and *qv* is the value of heat entering into the particle volume from action of external or internal sources per unit time. In this expression, the colon denotes double scalar product of tensors; in this case these are the stress tensor and velocity gradient tensor.

The physical meaning of this equation is that the rate of change of internal energy per unit volume is equal to rate of energy supply due to heat conduction, due to dissipation of mechanical energy of the flow, and due to heat from external or internal sources. Since stress tensor *σ* can be written as *σ* ¼ �*pE* þ *τ*, where *τ* is the shear stress tensor, taking into account material derivative definition, we can rewrite Eq. (61) in this form:

$$
\rho \left[ \frac{\partial u}{\partial t} + \left( \overrightarrow{\mathbf{V}} \cdot \overrightarrow{\mathbf{V}} \right) u \right] = -\overrightarrow{\mathbf{V}} \cdot \overrightarrow{q} - p \overrightarrow{\mathbf{V}} \cdot \overrightarrow{\mathbf{V}} + \underline{\tau} : \overrightarrow{\mathbf{V}} \overrightarrow{\mathbf{V}} + q\_v. \tag{62}
$$

This is the energy equation in terms of transfer of specific internal energy *u*. Vector *q* ! in the energy equation is determined by Fourier's law:

$$
\overrightarrow{q} = -\lambda \overrightarrow{\nabla} T,\tag{63}
$$

*Ф* ¼ *τ* : ∇ ! *V* !

> ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � !

> ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � !

> ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � !

¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � ! *∂Vi*

þ*μ*

*∂Vj ∂xi*

or in usual notations

3

*∂Vx ∂y* þ *∂Vy ∂x* � �<sup>2</sup>

*Ф* ¼ *τ* : ∇ ! *V* ! ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � ! <sup>2</sup>

temperature transport in the form:

� �

*<sup>Ф</sup>* <sup>¼</sup> *<sup>μ</sup>* � <sup>2</sup>

þ

*ρcv ∂T ∂t* þ *V* ! � ∇ � � !

*<sup>h</sup>* <sup>¼</sup> *<sup>u</sup>* <sup>þ</sup> *<sup>p</sup>*

**65**

*∂Vi ∂xj* þ *∂Vi ∂xj*

¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � !

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

> *E* : ∇ ! *V* ! þ 2*μ* � 1 <sup>2</sup> <sup>∇</sup> ! *V* ! þ ∇ ! *V* !*<sup>T</sup>* � � : <sup>∇</sup>

*δij e* ! *i e* ! *j* : *∂Vs ∂xk e* ! *k e* !

*δijδjkδis*

*∂xi* þ *μ*

� �

*Ф* ¼ *τ* : ∇ ! *V* ! ¼ � <sup>2</sup> 3 *μ ∂Vi ∂xi* � �<sup>2</sup>

*∂Vx ∂x* þ *∂Vy ∂y* þ *∂Vz ∂z*

� �<sup>2</sup>

þ

Eq. (62) with the help of expression for vector *q*

*T*

parts of the equation. Then we obtain in the left part *ρ dh*

shall get this term in transformed form shown as follows:

*p ρ* � �

*ρ d dt* ¼ ∇ ! � *λ*∇ ! *T* � � � *p V*!

*<sup>ρ</sup>*. With this purpose we need to add the term *ρ <sup>d</sup>*

<sup>¼</sup> *dp dt* � *<sup>p</sup> ρ dρ dt* <sup>¼</sup> *dp*

*∂Vx ∂z* þ *∂Vz ∂x* � �<sup>2</sup>

This function can also be written in the componentless form:

*∂Vs ∂xk* þ *μ*

*∂Vi ∂xj*

*∂Vj ∂xi*

*∂Vs ∂xk*

¼ � <sup>2</sup> 3 *μ ∂Vi ∂xi* � �<sup>2</sup>

Thus, in component form Rayleigh function *Ф* can be written as

*E* þ 2*μS*

! *V* ! ¼

*<sup>s</sup>* þ *μ* ∇ ! *V* ! : ∇ ! *V* ! þ ∇ ! *V* !*<sup>T</sup>* : ∇ ! *V*

*δjkδis* þ

þ *μ*

<sup>þ</sup> <sup>2</sup> *<sup>∂</sup>Vx ∂x* � �<sup>2</sup>

þ

� �<sup>2</sup> ( " #

þ *μ* ∇ ! *V* ! : ∇ ! *V* ! þ ∇ ! *V* ! : ∇ ! *V* !*<sup>T</sup>* � �*:* (68)

For perfect gases [6, 7] internal energy is connected with temperature by the relation *du* ¼ *cvdT*, where *cv* is the isochore thermal capacity. Then, instead of

The energy equation (Eq. (62)) can be also written in terms of enthalpy

*∂Vi ∂xj*

*∂Vs ∂xk <sup>δ</sup>jkδis* � � ¼ � <sup>2</sup>

> *∂Vj ∂xi*

*∂Vi ∂xj* þ *∂Vi ∂xj*

� �*:*

þ *μ*

*∂Vj ∂xi*

*∂Vi ∂xj* þ *∂Vi ∂xj* : *∂Vi ∂xj* � �, (66)

þ

� ∇ � � !

*dt* <sup>þ</sup> *<sup>ρ</sup>* <sup>∇</sup> ! � *V* � � !

*∂Vy ∂z* þ *∂Vz ∂y* � �<sup>2</sup>

*∂Vy ∂y* � �<sup>2</sup>

þ

)

!, we can write equation for

þ *τ* : ∇ ! *V* !

> *dt p ρ*

*∂Vz ∂z*

*:* (67)

þ *qv:* (69)

� � to the left and right

*dt*, but in the right part, we

*:* (70)

*∂Vj ∂xi e* ! *i e* ! *j* : *∂Vs ∂xk e* ! *k e* ! *s* þ *∂Vi ∂xj e* ! *i e* ! *j* : *∂Vs ∂xk e* ! *k e* ! *s*

! *V* ! ¼

! � � <sup>¼</sup>

� �

3 *μ* ∇ ! � *V* � � ! <sup>2</sup>

> *∂Vi ∂xj*

¼

þ

� � : <sup>∇</sup>

where *T* is the temperature and *λ* is the coefficient of thermal conductivity.

Fourier's law of thermal conductivity can also be written in terms of enthalpy, which for an ideal gas is related to temperature by the formula *h* ¼ *cpT*, where *ср* is the isobar heat capacity. Then considering *λ* ¼ *ρcpa*, where a is the thermal diffusivity, heat flux density vector can be written in the form:

$$
\overrightarrow{q} = \lambda \overrightarrow{\nabla} T = \rho c\_p a \overrightarrow{\nabla} T = \rho \frac{\nu}{a\_{\prime\prime}^{\prime}} \overrightarrow{\nabla} c\_p T = \rho \frac{\nu}{\text{Pr}} \overrightarrow{\nabla} h = \frac{\mu}{\text{Pr}} \overrightarrow{\nabla} h,
$$

where Pr ¼ *a=ν* is the Prandtl number.

In Cartesian coordinates Eq. (62) can be written as follows:

$$
\rho \left[ \frac{\partial u}{\partial t} + V\_j \frac{\partial u}{\partial \mathbf{x}\_j} \right] = \frac{\partial}{\partial \mathbf{x}\_j} \lambda \frac{\partial T}{\partial \mathbf{x}\_j} - p \frac{\partial V\_j}{\partial \mathbf{x}\_j} + \tau\_{ij} \frac{\partial V\_i}{\partial \mathbf{x}\_j} + q\_v. \tag{64}
$$

The terms *p*∇ ! � *V* ! and *τ* : ∇ ! *V* ! show us in a moving fluid heating or cooling can occur. The term *p*∇ ! � *V* ! may cause significant change of temperature, when gas expands (compresses) rapidly. The term *τ* : ∇ ! *V* ! is always positive; it characterizes dissipation of mechanical energy and its transformation to heat energy. This scalar quantity is usually named as Rayleigh dissipation function [6] and denoted as *τ* : ∇ ! *V* ! ¼ *Ф*. Let us write this function in Cartesian coordinates for Newtonian viscous fluid, when rheological relation has the form:

$$
\underline{\pi} = -\frac{2}{3}\mu \overline{(\vec{\nabla} \cdot \vec{V})} \underline{E} + 2\mu \underline{\mathbb{S}},\tag{65}
$$

where *μ* is the fluid shear viscosity and *S* is the strain rate tensor.

Now we could write the dissipative term *τ* : ∇ ! *V* ! in Eq. (62) by simple transformations:

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

where *t* is the time; *u* is the specific internal energy; *q*

*Advances on Tensor Analysis and Their Applications*

case these are the stress tensor and velocity gradient tensor.

*u*

diffusivity, heat flux density vector can be written in the form:

! *T* ¼ *ρ*

In Cartesian coordinates Eq. (62) can be written as follows:

*<sup>τ</sup>* ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* !

where *μ* is the fluid shear viscosity and *S* is the strain rate tensor.

¼ *∂ ∂xj λ ∂T ∂xj* � *p ∂Vj ∂xj* þ *τij*

*∂u ∂xj*

and *τ* : ∇ ! *V* !

expands (compresses) rapidly. The term *τ* : ∇

fluid, when rheological relation has the form:

Now we could write the dissipative term *τ* : ∇

*T* ¼ *ρcpa*∇

where Pr ¼ *a=ν* is the Prandtl number.

*ρ ∂u ∂t* þ *Vj*

! � *V* !

! � *V* !

The terms *p*∇

occur. The term *p*∇

transformations:

∇ ! *V* !

**64**

¼ �∇ ! � *q* ! � *<sup>p</sup>*<sup>∇</sup> ! � *V* !

! in the energy equation is determined by Fourier's law:

*q* ! ¼ �*λ*<sup>∇</sup> !

This is the energy equation in terms of transfer of specific internal energy *u*.

where *T* is the temperature and *λ* is the coefficient of thermal conductivity. Fourier's law of thermal conductivity can also be written in terms of enthalpy, which for an ideal gas is related to temperature by the formula *h* ¼ *cpT*, where *ср* is the isobar heat capacity. Then considering *λ* ¼ *ρcpa*, where a is the thermal

> *ν a=ν* ∇ !

*cpT* ¼ *ρ*

*ν* Pr <sup>∇</sup> ! *<sup>h</sup>* <sup>¼</sup> *<sup>μ</sup>* Pr <sup>∇</sup> ! *h*,

> *∂Vi ∂xj*

show us in a moving fluid heating or cooling can

may cause significant change of temperature, when gas

! *V* !

dissipation of mechanical energy and its transformation to heat energy. This scalar quantity is usually named as Rayleigh dissipation function [6] and denoted as *τ* :

¼ *Ф*. Let us write this function in Cartesian coordinates for Newtonian viscous

! *V* !

rewrite Eq. (61) in this form:

Vector *q*

*ρ ∂u ∂t* þ *V* ! � ∇ !

*q* ! <sup>¼</sup> *<sup>λ</sup>*<sup>∇</sup> !

vector due to thermal conductivity; *σ* is the stress tensor; and *qv* is the value of heat entering into the particle volume from action of external or internal sources per unit time. In this expression, the colon denotes double scalar product of tensors; in this

The physical meaning of this equation is that the rate of change of internal energy per unit volume is equal to rate of energy supply due to heat conduction, due to dissipation of mechanical energy of the flow, and due to heat from external or internal sources. Since stress tensor *σ* can be written as *σ* ¼ �*pE* þ *τ*, where *τ* is the shear stress tensor, taking into account material derivative definition, we can

! is the heat flux density

þ *qv:* (62)

þ *qv:* (64)

is always positive; it characterizes

*E* þ 2*μS*, (65)

in Eq. (62) by simple

þ *τ* : ∇ ! *V* !

*T*, (63)

*Ф* ¼ *τ* : ∇ ! *V* ! ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � ! *E* þ 2*μS* � � : <sup>∇</sup> ! *V* ! ¼ ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � ! *E* : ∇ ! *V* ! þ 2*μ* � 1 <sup>2</sup> <sup>∇</sup> ! *V* ! þ ∇ ! *V* !*<sup>T</sup>* � � : <sup>∇</sup> ! *V* ! ¼ ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � ! *δij e* ! *i e* ! *j* : *∂Vs ∂xk e* ! *k e* ! *<sup>s</sup>* þ *μ* ∇ ! *V* ! : ∇ ! *V* ! þ ∇ ! *V* !*<sup>T</sup>* : ∇ ! *V* ! � � <sup>¼</sup> ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � ! *δijδjkδis ∂Vs ∂xk* þ *μ ∂Vj ∂xi e* ! *i e* ! *j* : *∂Vs ∂xk e* ! *k e* ! *s* þ *∂Vi ∂xj e* ! *i e* ! *j* : *∂Vs ∂xk e* ! *k e* ! *s* � � ¼ ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � ! *∂Vi ∂xi* þ *μ ∂Vj ∂xi ∂Vs ∂xk δjkδis* þ *∂Vi ∂xj ∂Vs ∂xk <sup>δ</sup>jkδis* � � ¼ � <sup>2</sup> 3 *μ* ∇ ! � *V* � � ! <sup>2</sup> þ þ*μ ∂Vj ∂xi ∂Vi ∂xj* þ *∂Vi ∂xj ∂Vi ∂xj* � � ¼ � <sup>2</sup> 3 *μ ∂Vi ∂xi* � �<sup>2</sup> þ *μ ∂Vj ∂xi ∂Vi ∂xj* þ *∂Vi ∂xj ∂Vi ∂xj* � �*:*

Thus, in component form Rayleigh function *Ф* can be written as

$$\Phi = \underline{\tau} : \overrightarrow{\nabla V} = -\frac{2}{3}\mu \left(\frac{\partial V\_i}{\partial \mathbf{x}\_i}\right)^2 + \mu \left[\frac{\partial V\_j}{\partial \mathbf{x}\_i}\frac{\partial V\_i}{\partial \mathbf{x}\_j} + \frac{\partial V\_i}{\partial \mathbf{x}\_j} : \frac{\partial V\_i}{\partial \mathbf{x}\_j}\right],\tag{66}$$

or in usual notations

$$\begin{split} \Phi = \mu \left\{ -\frac{2}{3} \left( \frac{\partial V\_x}{\partial \mathbf{x}} + \frac{\partial V\_y}{\partial \mathbf{y}} + \frac{\partial V\_x}{\partial \mathbf{z}} \right)^2 + 2 \left[ \left( \frac{\partial V\_x}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial V\_y}{\partial \mathbf{y}} \right)^2 + \left( \frac{\partial V\_x}{\partial \mathbf{z}} \right)^2 \right] \\ \quad + \left( \frac{\partial V\_x}{\partial \mathbf{y}} + \frac{\partial V\_y}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial V\_x}{\partial \mathbf{z}} + \frac{\partial V\_x}{\partial \mathbf{x}} \right)^2 + \left( \frac{\partial V\_y}{\partial \mathbf{z}} + \frac{\partial V\_x}{\partial \mathbf{y}} \right)^2 \right\}. \end{split} \tag{67}$$

This function can also be written in the componentless form:

$$\Phi = \underline{\underline{r}} : \overrightarrow{\nabla} \vec{V} = -\frac{2}{3}\mu \left( \overrightarrow{\nabla} \cdot \overrightarrow{\underline{V}} \right)^{2} + \mu \left[ \overrightarrow{\nabla} \vec{V} : \overrightarrow{\nabla} \vec{V} + \overrightarrow{\nabla} \vec{V} : \overrightarrow{\nabla} \vec{V}^{T} \right]. \tag{68}$$

For perfect gases [6, 7] internal energy is connected with temperature by the relation *du* ¼ *cvdT*, where *cv* is the isochore thermal capacity. Then, instead of Eq. (62) with the help of expression for vector *q* !, we can write equation for temperature transport in the form:

$$
\rho c\_v \left[ \frac{\partial T}{\partial t} + \left( \vec{V} \cdot \vec{\nabla} \right) T \right] = \vec{\nabla} \cdot \left( \vec{\lambda} \vec{\nabla} T \right) - p \left( \vec{V} \cdot \vec{\nabla} \right) + \underline{\varepsilon} : \vec{\nabla} \vec{V} + q\_v. \tag{69}
$$

The energy equation (Eq. (62)) can be also written in terms of enthalpy *<sup>h</sup>* <sup>¼</sup> *<sup>u</sup>* <sup>þ</sup> *<sup>p</sup> <sup>ρ</sup>*. With this purpose we need to add the term *ρ <sup>d</sup> dt p ρ* � � to the left and right parts of the equation. Then we obtain in the left part *ρ dh dt*, but in the right part, we shall get this term in transformed form shown as follows:

$$
\rho \frac{d}{dt} \left( \frac{p}{\rho} \right) = \frac{dp}{dt} - \frac{p}{\rho} \frac{d\rho}{dt} = \frac{dp}{dt} + \rho \left( \vec{\nabla} \cdot \vec{V} \right). \tag{70}
$$

Here we also used continuity equation (Eq. (2)). Finally, we can obtain energy equation in the form of enthalpy transport as

$$
\rho \frac{dh}{dt} = -\overrightarrow{\nabla} \cdot \overrightarrow{q} + \frac{dp}{dt} + \underline{\tau} : \overrightarrow{\nabla} \overrightarrow{V} + q\_v. \tag{71}
$$

In usual axis designations, the first term is

*∂ ∂xk*

> *∂ ∂x*<sup>1</sup>

> > *∂ ∂x*<sup>1</sup>

> > *∂ ∂x*<sup>1</sup>

þ *Vy ∂ ∂y μ* ∇ ! � *V* !

> þ *∂ ∂y* 2*μ ∂Vy ∂y*

þ *∂ ∂y μ ∂Vz ∂y* þ *∂Vy ∂z* 

transport. According to the fundamental thermodynamic relation,

*<sup>T</sup> ds dt* <sup>¼</sup> *dh dt* � <sup>1</sup> *ρ dp dt* ,

*dt* ¼ �<sup>∇</sup> ! � *q* ! <sup>þ</sup> *<sup>τ</sup>* : <sup>∇</sup> ! *V* !

*<sup>ρ</sup><sup>T</sup> ds*

þ *∂ ∂y μ ∂Vx ∂y* þ *∂Vy ∂x*

*∂ ∂z*

ð Þþ 2*μSk*<sup>1</sup> *V*<sup>2</sup>

ð Þþ 2*μS*<sup>11</sup>

ð Þþ 2*μS*<sup>12</sup>

ð Þþ 2*μS*<sup>13</sup>

Finally function Ψ in Cartesian coordinates can be written as follows:

!

The fourth form of the energy equation can be written in terms of entropy *s*

*Tds* <sup>¼</sup> *dh* � <sup>1</sup>

*ρ*

All forms of the equation energy (in terms of internal energy, enthalpy, stagna-

*dt* for arbitrary gases and liquids must be specified using known

Equation for temperature field of an arbitrary gas in the form of equation of transport of temperature *T* can be obtained from Eq. (62) or Eq. (71). In these cases

� 2 3 *μ ∂Vx ∂x* þ *∂Vy ∂y* þ *∂Vz ∂z*

> *∂ ∂xk*

*∂ ∂x*<sup>2</sup>

> *∂ ∂x*<sup>2</sup>

*∂ ∂x*<sup>2</sup>

,

ð Þþ 2*μSk*<sup>2</sup> *V*<sup>3</sup>

*∂ ∂x*<sup>3</sup>

> *∂ ∂x*<sup>3</sup>

*∂ ∂x*<sup>3</sup>

ð Þþ 2*μS*<sup>21</sup>

ð Þ <sup>2</sup>*μS*<sup>32</sup> <sup>þ</sup>

ð Þ <sup>2</sup>*μS*<sup>33</sup> *:*

þ *Vz*

þ *∂ ∂z μ ∂Vz ∂x* þ *∂Vx ∂z*

þ *∂ ∂z μ ∂Vz ∂y* þ *∂Vy ∂z*

*:* (76)

ð Þþ 2*μS*<sup>22</sup>

ð Þþ 2*μS*<sup>23</sup>

*∂ ∂z μ* ∇ ! � *V*

> þ *∂ ∂z* 2*μ ∂Vz ∂z*

*dt* from Eq. (77) to Eq. (71), we obtain

*dp:* (77)

þ *qv:* (78)

ð Þ <sup>2</sup>*μS*<sup>31</sup> <sup>þ</sup>

*∂ ∂xk*

ð Þ¼ 2*μSk*<sup>3</sup>

*Vx ∂ ∂x* þ *Vy ∂ ∂y* þ *Vz*

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

<sup>2</sup>*μSkj* <sup>¼</sup> *<sup>V</sup>*<sup>1</sup>

¼ *V*<sup>1</sup>

þ*V*<sup>2</sup>

þ*V*<sup>3</sup>

while second term is

*Vj ∂ ∂xk*

<sup>Ψ</sup> ¼ � <sup>2</sup> 3 *Vx ∂ ∂x μ* ∇ ! � *V* !

þ *Vx*

þ *Vy*

þ *Vz*

Hence we have

quantities *du*

**67**

*∂ ∂x* 2*μ ∂Vx ∂x* 

*∂ ∂x μ ∂Vx ∂y* þ *∂Vy ∂x*

*∂ ∂x μ ∂Vz ∂x* þ *∂Vx ∂z* 

and then if we substitute quantity *dh*

tion enthalpy, and entropy) are equivalent.

formulas, which follow from Maxwell's relations:

*dt* and *dh*

This is the second form of energy equation for the perfect gas in which *dh* ¼ *cpdT*, where *cp* is the isobar heat capacity, which leads to the following transport equation for temperature:

$$
\rho c\_p \frac{dT}{dt} = \overrightarrow{\nabla} \cdot \left( \lambda \overrightarrow{\nabla} T \right) + \frac{dp}{dt} + \underline{\mathfrak{z}} : \overrightarrow{\nabla} \overrightarrow{\mathcal{V}} + q\_v. \tag{72}
$$

One more form of the energy equation can be written if we introduce stagnation enthalpy *<sup>h</sup>* <sup>þ</sup> *<sup>V</sup>*<sup>2</sup> <sup>2</sup> . To do it we need to add equations for mechanical energy (Eq. (48) to Eq. (71)); as a result we obtain the equation:

$$
\rho \frac{d}{dt} \left( h + \frac{V^2}{2} \right) = -\overrightarrow{\nabla} \cdot \overrightarrow{q} + \frac{\partial p}{\partial t} + \overrightarrow{\nabla} \cdot \left( \underline{\mathbf{z}} \cdot \overrightarrow{\mathbf{V}} \right) + \rho \overrightarrow{\mathbf{V}} \cdot \overrightarrow{f} + q\_v. \tag{73}
$$

Here we used the relation:

$$
\overrightarrow{\nabla} \cdot \left( \underline{\tau} \cdot \overrightarrow{\nabla} \right) = \overrightarrow{\nabla} \cdot \left( \overrightarrow{\nabla} \cdot \underline{\tau} \right) + \underline{\tau} : \overrightarrow{\nabla} \overrightarrow{\nabla}, \tag{74}
$$

which is easy to be proven if we write it down in the component form considering symmetry of stress tensor *τ*.

As a result, the dissipative term in Eq. (73) can be written as follows:

$$
\overrightarrow{\nabla} \cdot \left( \underline{\mathbf{z}} \cdot \overrightarrow{\mathbf{V}} \right) = \Psi + \Phi,\tag{75}
$$

where *Ф* is the Rayleigh dissipation function and Ψ ¼ *V* ! � ∇ ! � *τ* � is the scalar quantity, which can be named as additional dissipation function. This additional dissipation function in Cartesian coordinates for Newtonian Stokes liquid can be written as

Ψ ¼ *V* ! � ∇ ! � *τ* <sup>¼</sup> *<sup>V</sup>* ! � ∇ ! � � <sup>2</sup> 3 *μ* ∇ ! � *V* ! *E* þ 2*μS* ¼ ¼ *V* ! � *e* ! *k ∂ ∂xk* � � <sup>2</sup> 3 *μ* ∇ ! � *V* ! *δij e* ! *i e* ! *<sup>j</sup>* þ 2*μSij e* ! *i e* ! *j* ¼ ¼ *V* ! � *∂ ∂xk* � � <sup>2</sup> 3 *μ* ∇ ! � *V* ! *δijδki e* ! *<sup>j</sup>* þ 2*μSijδki e* ! *j* ¼ ¼ *V* ! � *∂ ∂xk* � � <sup>2</sup> 3 *μ* ∇ ! � *V* ! *e* ! *<sup>k</sup>* þ 2*μSkj e* ! *j* ¼ ¼ *Vs e* ! *<sup>s</sup>* � *<sup>∂</sup> ∂xk* � 2 3 *μ* ∇ ! � *V* ! *e* ! *<sup>k</sup>* þ 2*μSkj e* ! *j* ¼ ¼ *V* ! � ∇ ! � 2 3 *μ* ∇ ! � *V* ! <sup>þ</sup> *Vj ∂ ∂xk* 2*μSkj :*

In usual axis designations, the first term is

$$
\left(V\_x \frac{\partial}{\partial \mathbf{x}} + V\_y \frac{\partial}{\partial \mathbf{y}} + V\_x \frac{\partial}{\partial \mathbf{z}}\right) \left(-\frac{2}{3}\mu \left(\frac{\partial V\_x}{\partial \mathbf{x}} + \frac{\partial V\_y}{\partial \mathbf{y}} + \frac{\partial V\_x}{\partial \mathbf{z}}\right)\right), \dots
$$

while second term is

Here we also used continuity equation (Eq. (2)). Finally, we can obtain energy

This is the second form of energy equation for the perfect gas in which *dh* ¼ *cpdT*, where *cp* is the isobar heat capacity, which leads to the following transport

<sup>þ</sup> *dp*

One more form of the energy equation can be written if we introduce stagnation

which is easy to be proven if we write it down in the component form consider-

*dt* <sup>þ</sup> *<sup>τ</sup>* : <sup>∇</sup> ! *V* !

<sup>2</sup> . To do it we need to add equations for mechanical energy (Eq. (48)

� *τ* � *V* !

> þ *τ* : ∇ ! *V* !

þ *ρV* ! � *f* !

¼ Ψ þ *Ф*, (75)

! � ∇ ! � *τ* 

*E* þ 2*μS*

! *j*

¼

¼

*<sup>j</sup>* þ 2*μSij e* ! *i e* ! *j*

*<sup>j</sup>* þ 2*μSijδki e*

2*μSkj :* ¼

¼

¼

*δij e* ! *i e* !

*<sup>k</sup>* þ 2*μSkj e* ! *j*

> *<sup>k</sup>* þ 2*μSkj e* ! *j*

> > *∂ ∂xk*

*δijδki e* !

*e* !

*e* !

þ *Vj*

*dt* <sup>þ</sup> *<sup>τ</sup>* : <sup>∇</sup> ! *V* !

þ *qv:* (71)

þ *qv:* (72)

þ *qv:* (73)

� is the scalar

, (74)

equation in the form of enthalpy transport as

*Advances on Tensor Analysis and Their Applications*

*ρcp dT dt* <sup>¼</sup> <sup>∇</sup> ! � *λ*∇ ! *T* 

to Eq. (71)); as a result we obtain the equation:

*V*2 2 

> ∇ !

¼ �∇ ! � *q* ! <sup>þ</sup> *∂p ∂t* þ ∇ !

� *τ* � *V* !

> ∇ !

where *Ф* is the Rayleigh dissipation function and Ψ ¼ *V*

¼ *V* ! � ∇ ! � � <sup>2</sup> 3 *μ* ∇ ! � *V* !

� � <sup>2</sup> 3 *μ* ∇ ! � *V* !

� � <sup>2</sup> 3 *μ* ∇ ! � *V* !

� � <sup>2</sup> 3 *μ* ∇ ! � *V* !

� 2 3 *μ* ∇ ! � *V* !

� 2 3 *μ* ∇ ! � *V* !

¼ *V* ! � ∇ ! � *τ* 

As a result, the dissipative term in Eq. (73) can be written as follows:

� *τ* � *V* !

quantity, which can be named as additional dissipation function. This additional dissipation function in Cartesian coordinates for Newtonian Stokes liquid can be

equation for temperature:

*ρ d dt <sup>h</sup>* <sup>þ</sup>

Here we used the relation:

ing symmetry of stress tensor *τ*.

Ψ ¼ *V* ! � ∇ ! � *τ* 

> ¼ *V* ! � *e* ! *k ∂ ∂xk*

¼ *V* ! � *∂ ∂xk*

¼ *V* ! � *∂ ∂xk*

¼ *Vs e* ! *<sup>s</sup>* � *<sup>∂</sup> ∂xk*

¼ *V* ! � ∇ !

enthalpy *<sup>h</sup>* <sup>þ</sup> *<sup>V</sup>*<sup>2</sup>

written as

**66**

*ρ dh dt* ¼ �<sup>∇</sup> ! � *q* ! <sup>þ</sup> *dp*

$$\begin{split} V\_{j}\frac{\partial}{\partial \mathbf{x}\_{k}} \left( 2\mu \mathbf{S}\_{kj} \right) &= V\_{1} \frac{\partial}{\partial \mathbf{x}\_{k}} \left( 2\mu \mathbf{S}\_{k1} \right) + V\_{2} \frac{\partial}{\partial \mathbf{x}\_{k}} \left( 2\mu \mathbf{S}\_{k2} \right) + V\_{3} \frac{\partial}{\partial \mathbf{x}\_{k}} \left( 2\mu \mathbf{S}\_{kj} \right) = 0 \\ &= V\_{1} \left[ \frac{\partial}{\partial \mathbf{x}\_{1}} \left( 2\mu \mathbf{S}\_{11} \right) + \frac{\partial}{\partial \mathbf{x}\_{2}} \left( 2\mu \mathbf{S}\_{21} \right) + \frac{\partial}{\partial \mathbf{x}\_{3}} \left( 2\mu \mathbf{S}\_{31} \right) \right] + \\ &\quad + V\_{2} \left[ \frac{\partial}{\partial \mathbf{x}\_{1}} \left( 2\mu \mathbf{S}\_{12} \right) + \frac{\partial}{\partial \mathbf{x}\_{2}} \left( 2\mu \mathbf{S}\_{22} \right) + \frac{\partial}{\partial \mathbf{x}\_{3}} \left( 2\mu \mathbf{S}\_{32} \right) \right] + \\ &\quad + V\_{3} \left[ \frac{\partial}{\partial \mathbf{x}\_{1}} \left( 2\mu \mathbf{S}\_{13} \right) + \frac{\partial}{\partial \mathbf{x}\_{2}} \left( 2\mu \mathbf{S}\_{23} \right) + \frac{\partial}{\partial \mathbf{x}\_{3}} \left( 2\mu \mathbf{S}\_{33} \right) \right]. \end{split}$$

Finally function Ψ in Cartesian coordinates can be written as follows:

<sup>Ψ</sup> ¼ � <sup>2</sup> 3 *Vx ∂ ∂x μ* ∇ ! � *V* ! þ *Vy ∂ ∂y μ* ∇ ! � *V* ! þ *Vz ∂ ∂z μ* ∇ ! � *V* ! þ *Vx ∂ ∂x* 2*μ ∂Vx ∂x* þ *∂ ∂y μ ∂Vx ∂y* þ *∂Vy ∂x* þ *∂ ∂z μ ∂Vz ∂x* þ *∂Vx ∂z* þ *Vy ∂ ∂x μ ∂Vx ∂y* þ *∂Vy ∂x* þ *∂ ∂y* 2*μ ∂Vy ∂y* þ *∂ ∂z μ ∂Vz ∂y* þ *∂Vy ∂z* þ *Vz ∂ ∂x μ ∂Vz ∂x* þ *∂Vx ∂z* þ *∂ ∂y μ ∂Vz ∂y* þ *∂Vy ∂z* þ *∂ ∂z* 2*μ ∂Vz ∂z :* (76)

The fourth form of the energy equation can be written in terms of entropy *s* transport. According to the fundamental thermodynamic relation,

$$Tds = dh - \frac{1}{\rho}dp.\tag{77}$$

Hence we have

$$T\frac{ds}{dt} = \frac{dh}{dt} - \frac{1}{\rho}\frac{dp}{dt},$$

and then if we substitute quantity *dh dt* from Eq. (77) to Eq. (71), we obtain

$$
\rho \, T \frac{d\mathbf{s}}{dt} = -\overrightarrow{\nabla} \cdot \overrightarrow{q} + \underline{\mathbf{z}} : \overrightarrow{\nabla} \overrightarrow{V} + q\_v. \tag{78}
$$

All forms of the equation energy (in terms of internal energy, enthalpy, stagnation enthalpy, and entropy) are equivalent.

Equation for temperature field of an arbitrary gas in the form of equation of transport of temperature *T* can be obtained from Eq. (62) or Eq. (71). In these cases quantities *du dt* and *dh dt* for arbitrary gases and liquids must be specified using known formulas, which follow from Maxwell's relations:

*Advances on Tensor Analysis and Their Applications*

$$du = c\_v dT - \frac{1}{\rho^2} \left[ T \left( \frac{\partial p}{\partial T} \right)\_\rho - p \right] d\rho,\tag{79}$$

We could prove this equality if we write the left and right parts in component

*T* ¼ *ρ*

*ρVjT* ¼

*∂ρ ∂xj* þ *ρT*

Since the expression in parentheses is zero (due to continuity equation), the

! ⊗ *V* !

We could prove this equality if we write the left and right parts in the compo-

*V* ! ¼ *ρ dVi dt <sup>e</sup>* ! *<sup>i</sup>* þ *ρVj*

*ρVjV e*!

*ρVjVi e* ! *<sup>i</sup>* ¼ *ρ*

*<sup>k</sup>* � *e* ! *j* � � *<sup>e</sup>* ! *<sup>i</sup>* ¼

> *∂Vj ∂xj Vi e* ! *<sup>i</sup>* þ *Vj*

!

*∂T ∂t*

> *∂Vj ∂xj*

þ *ρ ∂T ∂t*

þ *ρVj*

þ *ρVj*

*∂T ∂xj* ¼

*∂T ∂xj* ¼ *ρ ∂T ∂t*

. In this case its material derivative

*:* (85)

(momentum flow tensor); the sign

*∂Vi ∂xj e* ! *i:*

> *∂ρ ∂xj Vi e* ! *<sup>i</sup>* þ *ρVj*

*∂Vi ∂xj e* ! *i:*

*∂Vi ∂t e* ! *i:* þ *ρVj*

*∂T ∂xj :*

þ *ρVj*

*∂T ∂xj :*

þ *ρ V* ! � ∇ � � !

þ *TVj*

*∂ρ ∂xj* þ *ρ ∂Vj ∂xj*


� �

form. For the left part, we have

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

For the right part, we have

*∂ ∂t* *ρT* þ ∇ ! � *ρV* ! *T* � � <sup>¼</sup> *<sup>T</sup> <sup>∂</sup><sup>ρ</sup>*

*ρ dT dt* <sup>¼</sup> *<sup>∂</sup><sup>T</sup> ∂t*

> *∂t* þ *ρ ∂T ∂t* þ *∂ ∂xj*

<sup>¼</sup> *<sup>T</sup> <sup>∂</sup><sup>ρ</sup> ∂t* þ *ρ ∂T ∂t*

<sup>¼</sup> *<sup>T</sup> <sup>∂</sup><sup>ρ</sup> ∂t* þ *Vj*

equality of the left and right parts is obvious.

Let us assume that quantity Θ is velocity *V*

Here in the last term, we see tensor *ρV*

nent form and use continuity equation. For the left part, we have

> *ρ dV* ! *dt* <sup>¼</sup> *<sup>ρ</sup>*

The first term of the right part is

The second term of the right part is

*∂ ∂xk*

¼ *e* ! *k ∂ ∂xk*

¼ *δkj*

∇ ! � *ρV* ! *V* � � !

**69**

*∂ ∂t ρV* ! ¼ *∂ ∂t ρVi e* ! *<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup><sup>ρ</sup> ∂t Vi e* ! *<sup>i</sup>* þ *ρ*

� *ρVjVi e* ! *j e* ! *<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup> ∂xk*

*ρVjVi e* ! *<sup>i</sup>* <sup>¼</sup> *<sup>∂</sup> ∂xj*

*ρ dV* ! *dt* <sup>¼</sup> *<sup>∂</sup> ∂t ρV* ! þ ∇ ! � *ρV* ! *V* � � !

of tensor multiplication ⊗ is omitted for ease of recording.

*∂V ∂t*

þ *ρ V* ! � ∇ � � !

**8.2 Quantity Θ is a vector**

can be written in the form:

*∂t* þ*V* ! �∇ ! *ρ*þ*ρ*∇ ! �*V* ! ¼0

$$dh = c\_p dT + \frac{1}{\rho^2} \left[ \rho + T \left( \frac{\partial \rho}{\partial T} \right)\_p \right] dp. \tag{80}$$

The subscripts in derivatives here fix the parameters, with the constancy of which the derivatives are calculated. From these formulas the expressions for derivatives can be obtained:

$$\frac{du}{dt} = c\_v \frac{dT}{dt} - \frac{1}{\rho^2} \left[ T \left( \frac{\partial p}{\partial T} \right)\_\rho - p \right] \frac{d\rho}{dt},\tag{81}$$

$$\frac{dh}{dt} = c\_p \frac{dT}{dt} + \frac{1}{\rho^2} \left[ \rho + T \left( \frac{\partial \rho}{\partial T} \right)\_p \right] \frac{dp}{dt}. \tag{82}$$

If we substitute them into Eq. (62) and Eq. (71), we obtain two forms of the equation energy in terms of temperature transport.

In heat transfer problems, boundary conditions are specified in three different kinds—the first, second, and third kind:


#### **8. Divergence form of transport equations**

Material derivative of any physical quantity Θ multiplied by density *ρ* always can be written in the "divergent" form as

$$
\rho \frac{d\Theta}{dt} = \frac{\partial}{\partial t} \rho \Theta + \vec{\nabla} \cdot \left(\rho \vec{V} \Theta\right). \tag{83}
$$

This directly follows from the continuity equation [Eq. (2)].

Let us consider in detail the following cases for three ranks of a certain physical quantity Θ.

#### **8.1 Quantity Θ is a scalar**

Let us assume that quantity Θ is temperature *T:*

$$
\rho \frac{dT}{dt} = \frac{\partial}{\partial t} \rho T + \vec{\nabla} \cdot \left(\rho \vec{V} T\right). \tag{84}
$$

We could prove this equality if we write the left and right parts in component form. For the left part, we have

$$
\rho \frac{dT}{dt} = \frac{\partial T}{\partial t} + \rho \left(\vec{V} \cdot \vec{\nabla}\right) T = \rho \frac{\partial T}{\partial t} + \rho V\_j \frac{\partial T}{\partial \mathbf{x}\_j}.
$$

For the right part, we have

*du* <sup>¼</sup> *cvdT* � <sup>1</sup>

*dh* ¼ *cpdT* þ

*dT dt* � <sup>1</sup>

*dT dt* þ

*du dt* <sup>¼</sup> *cv*

*Advances on Tensor Analysis and Their Applications*

*dh dt* <sup>¼</sup> *cp*

equation energy in terms of temperature transport.

the heat flux density q on the surface of the body.

**8. Divergence form of transport equations**

*ρ d*Θ *dt* <sup>¼</sup> *<sup>∂</sup> ∂t*

Let us assume that quantity Θ is temperature *T:*

*ρ dT dt* <sup>¼</sup> *<sup>∂</sup> ∂t*

This directly follows from the continuity equation [Eq. (2)].

can be written in the "divergent" form as

kinds—the first, second, and third kind:

the surface of the body.

surface.

quantity Θ.

**68**

**8.1 Quantity Θ is a scalar**

derivatives can be obtained:

*<sup>ρ</sup>*<sup>2</sup> *<sup>T</sup> <sup>∂</sup><sup>p</sup> ∂T* � �

*<sup>ρ</sup>*<sup>2</sup> *<sup>ρ</sup>* <sup>þ</sup> *<sup>T</sup> <sup>∂</sup><sup>ρ</sup>*

1

The subscripts in derivatives here fix the parameters, with the constancy of which the derivatives are calculated. From these formulas the expressions for

> *<sup>ρ</sup>*<sup>2</sup> *<sup>T</sup> <sup>∂</sup><sup>p</sup> ∂T* � �

If we substitute them into Eq. (62) and Eq. (71), we obtain two forms of the

In heat transfer problems, boundary conditions are specified in three different

1.The boundary conditions of the first kind consist in setting the temperature on

2.The boundary conditions of the second kind are setting of the distribution of

3.The boundary conditions of the third kind consist in setting the temperature of the flow over the surface of the body and the heat transfer conditions on its

Material derivative of any physical quantity Θ multiplied by density *ρ* always

*ρ*Θ þ ∇ ! � *ρV* ! Θ � �

Let us consider in detail the following cases for three ranks of a certain physical

*ρT* þ ∇ ! � *ρV* ! *T* � �

*<sup>ρ</sup>*<sup>2</sup> *<sup>ρ</sup>* <sup>þ</sup> *<sup>T</sup> <sup>∂</sup><sup>ρ</sup>*

1

*ρ* � *p*

*∂T* � �

> *ρ* � *p*

*∂T* � �

*p*

" #

" #

*p*

*dρ*

*dp*

*dρ*, (79)

*dp:* (80)

*dt* , (81)

*dt :* (82)

*:* (83)

*:* (84)

" #

" #

$$\begin{split} \frac{\partial}{\partial t} \rho T + \vec{\nabla} \cdot \left( \rho \vec{V} T \right) &= T \frac{\partial \rho}{\partial t} + \rho \frac{\partial T}{\partial t} + \frac{\partial}{\partial \mathbf{x}\_j} \rho V\_j T = \\ &= T \frac{\partial \rho}{\partial t} + \rho \frac{\partial T}{\partial t} + T V\_j \frac{\partial \rho}{\partial \mathbf{x}\_j} + \rho T \frac{\partial V\_j}{\partial \mathbf{x}\_j} + \rho V\_j \frac{\partial T}{\partial \mathbf{x}\_j} = \\ &= T \underbrace{\left( \frac{\partial \rho}{\partial t} + V\_j \frac{\partial \rho}{\partial \mathbf{x}\_j} + \rho \frac{\partial V\_j}{\partial \mathbf{x}\_j} \right)}\_{\frac{\partial}{\partial t} + \vec{V} \cdot \vec{\nabla} \rho + \rho \vec{V} \cdot \vec{V} = 0} + \rho \frac{\partial T}{\partial \mathbf{x}\_j} + \rho V\_j \frac{\partial T}{\partial \mathbf{x}\_j} = \rho \frac{\partial T}{\partial t} + \rho V\_j \frac{\partial T}{\partial \mathbf{x}\_j}. \end{split}$$

Since the expression in parentheses is zero (due to continuity equation), the equality of the left and right parts is obvious.

#### **8.2 Quantity Θ is a vector**

Let us assume that quantity Θ is velocity *V* ! . In this case its material derivative can be written in the form:

$$
\rho \frac{d\vec{V}}{dt} = \frac{\partial}{\partial t} \rho \vec{V} + \vec{\nabla} \cdot \left(\rho \vec{V} \vec{V}\right). \tag{85}
$$

Here in the last term, we see tensor *ρV* ! ⊗ *V* ! (momentum flow tensor); the sign of tensor multiplication ⊗ is omitted for ease of recording.

We could prove this equality if we write the left and right parts in the component form and use continuity equation.

For the left part, we have

$$
\rho \frac{d\vec{V}}{dt} = \rho \frac{\partial V}{\partial t} + \rho \left(\vec{V} \cdot \vec{\nabla}\right) \vec{V} = \rho \frac{dV\_i}{dt} \vec{e\_i} + \rho V\_j \frac{\partial V\_i}{\partial \mathbf{x\_j}} \vec{e\_i}.
$$

The first term of the right part is

$$\frac{\partial}{\partial t}\rho \overrightarrow{V} = \frac{\partial}{\partial t}\rho V\_i \overrightarrow{e\_i} = \frac{\partial \rho}{\partial t}V\_i \overrightarrow{e\_i} + \rho \frac{\partial V\_i}{\partial t} \overrightarrow{e\_i}.$$

The second term of the right part is

$$\begin{split} \overrightarrow{\nabla} \cdot \left( \rho \overrightarrow{V} \overrightarrow{V} \right) &= \overrightarrow{\varepsilon}\_{k} \frac{\partial}{\partial \mathbf{x}\_{k}} \cdot \rho V\_{j} V\_{i} \overrightarrow{\varepsilon}\_{j} \overrightarrow{\varepsilon}\_{i} = \frac{\partial}{\partial \mathbf{x}\_{k}} \rho V\_{j} V \left( \overrightarrow{\varepsilon}\_{k} \cdot \overrightarrow{\varepsilon}\_{j} \right) \overrightarrow{\varepsilon}\_{i} = \\ &= \delta\_{kj} \frac{\partial}{\partial \mathbf{x}\_{k}} \rho V\_{j} V\_{i} \overrightarrow{\varepsilon}\_{i} = \frac{\partial}{\partial \mathbf{x}\_{j}} \rho V\_{j} V\_{i} \overrightarrow{\varepsilon}\_{i} = \rho \frac{\partial V\_{j}}{\partial \mathbf{x}\_{j}} V\_{i} \overrightarrow{\varepsilon}\_{i} + V\_{j} \frac{\partial \rho}{\partial \mathbf{x}\_{j}} V\_{i} \overrightarrow{\varepsilon}\_{i} + \rho V\_{j} \frac{\partial V\_{i}}{\partial \mathbf{x}\_{j}} \overrightarrow{\varepsilon}\_{i}. \end{split}$$

The right part as a whole is

$$
\rho \frac{\partial V\_i}{\partial t} \overrightarrow{e\_i} + \rho V\_j \frac{\partial V\_i}{\partial \mathbf{x}\_j} \overrightarrow{e\_i} + V\_i \overrightarrow{e\_i} \underbrace{\left(\frac{\partial \rho}{\partial t} + V\_j \frac{\partial \rho}{\partial \mathbf{x}\_j} + \rho \frac{\partial V\_j}{\partial \mathbf{x}\_j}\right)}\_{\frac{\partial \rho}{\partial t} + \vec{V} \cdot \vec{\nabla}\rho + \rho \vec{\nabla} \cdot \vec{V} = 0}.
$$

Therefore, the expression (Eq. (83)) is valid in case Θ is a vector.

#### **8.3 Quantity Θ is a tensor**

Let us assume that Θ is a tensor, for instance, stress tensor *σ*. Stress tensor is a second-rank symmetric tensor, which in Cartesian coordinates can be written in the form *σ* ¼ *σij e* ! *i e* ! *<sup>j</sup>*. Let us prove the equality:

$$
\rho \frac{d\underline{\sigma}}{dt} = \frac{\partial}{\partial t} \rho \underline{\sigma} + \vec{\nabla} \cdot \left(\rho \vec{V} \underline{\sigma}\right). \tag{86}
$$

Jaumann G.: *<sup>A</sup>* � <sup>Ω</sup> <sup>þ</sup> ð Þ *<sup>A</sup>* � <sup>Ω</sup> *<sup>T</sup>*, <sup>Ω</sup>—antisymmetric spin tensor.

by Jaumann, for an arbitrary tensor of the second rank, has the form

Material derivative in the form by Rivlin is written as follows:

þ *tr*∇ ! *V* !

At present, the question of which derivative is more appropriate to use when constructing rheological equations is unclear. The most common is the rotational derivative by Gustav Jaumann. The corresponding material derivative in the form

> *A* þ *A* � ∇ ! *V* !

It is easy to see that Rivlin's derivative differs from Jaumann's one by the addi-

In this chapter, some applications of tensor calculus in fluid dynamics and heat transfer are presented. Typical transformations of equations and governing relations are discussed. Main conservation equations are given and analyzed. The

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

Rotational derivative of a symmetric tensor is also a symmetric tensor. As an example, let us consider the rotational derivative of strain-rate tensor and spin tensor:

> ! *V* ! � ∇ ! *V* !*<sup>T</sup>* <sup>þ</sup> <sup>∇</sup>

As a result, we have obtained the symmetrical second-rank tensor.

*A*.

*<sup>A</sup>* <sup>þ</sup> *<sup>A</sup>* � <sup>Ω</sup> <sup>þ</sup> ð Þ *<sup>A</sup>* � <sup>Ω</sup> *<sup>T</sup>:* (87)

*:* (88)

! *V* !*<sup>T</sup>* � ∇ ! *V*

þ *A* � ∇ ! *V* ! *<sup>T</sup>*

! *V* !*<sup>T</sup>* þ ∇ ! *V* ! � <sup>∇</sup>

! <sup>¼</sup>

! *V* ! *<sup>T</sup>*

� *A* þ *A* � ∇

Oldroyd J., Sedov L.I., etc. [8, 10, 11]

*DA Dt* <sup>¼</sup> *<sup>∂</sup><sup>A</sup> ∂t* þ *V* ! � ∇ !

*DA Dt* <sup>¼</sup> *<sup>∂</sup><sup>A</sup> ∂t* þ *V* ! � ∇ !

4

¼ 1 2 ∇ ! *V* !*<sup>T</sup>* � ∇ ! *V* ! � ∇ ! *V* ! � ∇ ! *V*

∇ ! *V* ! þ ∇ ! *V* !*<sup>T</sup>* � <sup>∇</sup>

Dmitry Nikushchenko\* and Valery Pavlovsky

\*Address all correspondence to: ndmitry@list.ru

provided the original work is properly cited.

State Marine Technical University, St. Petersburg, Russia

tional term *<sup>A</sup>* � *<sup>S</sup>* <sup>þ</sup> ð Þ *<sup>A</sup>* � *<sup>S</sup> <sup>T</sup>*, which is neutral by itself.

!*<sup>T</sup> :*

governing equations of fluid motion and energy were obtained.

Rivlin R.: *A* � ∇

Truesdell C.: ∇

*<sup>S</sup>* � <sup>Ω</sup> <sup>þ</sup> ð Þ *<sup>S</sup>* � <sup>Ω</sup> *<sup>T</sup>* <sup>¼</sup> <sup>1</sup>

**9. Conclusions**

**Author details**

**71**

! *V* ! þ ∇ ! *V* !*<sup>T</sup>* � *A*.

*Fluid Motion Equations in Tensor Form DOI: http://dx.doi.org/10.5772/intechopen.91284*

> ! *V* !

The left part of this equation in the component form can be written as follows:

$$
\rho \frac{d\underline{\sigma}}{dt} = \rho \frac{d}{dt} \sigma\_{\vec{\imath}\vec{\imath}} \overrightarrow{\overline{e}\_{i}} \overrightarrow{e}\_{\vec{\jmath}} = \rho \frac{\partial}{\partial t} \sigma\_{\vec{\imath}\vec{\imath}} \overrightarrow{e}\_{i} \overrightarrow{e}\_{\vec{\jmath}} + \rho \left(\overrightarrow{V} \cdot \overrightarrow{\nabla}\right) \sigma\_{\vec{\imath}\vec{\imath}} \overrightarrow{e}\_{i} \overrightarrow{e}\_{\vec{\jmath}} = 0
$$

$$
= \rho \frac{\partial \sigma\_{\vec{\imath}\vec{\imath}}}{\partial t} \overrightarrow{e}\_{i} \overrightarrow{e}\_{\vec{\jmath}} + \rho V\_{k} \frac{\partial \sigma\_{\vec{\imath}\vec{\jmath}}}{\partial \textbf{x}\_{k}} \overrightarrow{e}\_{i} \overrightarrow{e}\_{\vec{\jmath}}.
$$

The right part is

*∂ ∂t ρσ* þ ∇ ! � *ρV* ! *σ* � � <sup>¼</sup> *<sup>∂</sup> ∂t ρσij e* ! *i e* ! *<sup>j</sup>* þ *e* ! *k ∂ ∂xk* � *ρVsσij e* ! *s e* ! *i e* ! *<sup>j</sup>* ¼ ¼ *σij ∂ρ ∂t e* ! *i e* ! *<sup>j</sup>* þ *ρ ∂σij <sup>∂</sup><sup>t</sup> <sup>e</sup>* ! *i e* ! *<sup>j</sup>* þ *δks ∂ ∂xk ρVsσij e* ! *i e* ! *<sup>j</sup>* ¼ ¼ *σij ∂ρ ∂t e* ! *i e* ! *<sup>j</sup>* þ *ρ ∂σij <sup>∂</sup><sup>t</sup> <sup>e</sup>* ! *i e* ! *<sup>j</sup>* þ *∂ ∂xk ρVkσij e* ! *i e* ! *<sup>j</sup>* ¼ ¼ *σij ∂ρ ∂t e* ! *i e* ! *<sup>j</sup>* þ *ρ ∂σij <sup>∂</sup><sup>t</sup> <sup>e</sup>* ! *i e* ! *<sup>j</sup>* þ *Vkσij ∂ρ ∂xk e* ! *i e* ! *<sup>j</sup>* þ *ρσij ∂Vk ∂xk e* ! *i e* ! *<sup>j</sup>* þ *ρVk ∂σij ∂xk e* ! *i e* ! *<sup>j</sup>* ¼ ¼ *ρ ∂σij <sup>∂</sup><sup>t</sup> <sup>e</sup>* ! *i e* ! *<sup>j</sup>* þ *ρVk ∂σij ∂xk e* ! *i e* ! *<sup>j</sup>* þ *σij e* ! *i e* ! *j ∂ρ ∂t* þ *Vk ∂ρ ∂xk* þ *ρ ∂Vk ∂xk* � � |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} <sup>¼</sup><sup>0</sup>

Therefore, the expression (Eq. (83)) is also valid in case Θ is a tensor.

It is necessary to note the derivative *<sup>d</sup> dt σ*, which contains local and convective parts *<sup>d</sup> dt* <sup>¼</sup> *<sup>∂</sup> <sup>∂</sup><sup>t</sup>* þ *V* ! � ∇ � � ! , can, at first glance, be used in fluid models when writing the defining equation in the form of a differential transport Equation [2, 5]. However, careful analysis shows that the material derivative for a second-rank tensor is not an invariant quantity [5, 8, 9]. By this reason, instead of derivative *<sup>d</sup> dt*, derivative *<sup>D</sup> Dt* is usually used as the material derivative for a second-rank tensor, which contains also rotational part (deviatoric stress rate), which provides symmetry relative to rotations. The rotational part cannot be written in divergent form.

There are different forms of deviatoric stress rate for an arbitrary second-rank tensor A, for instance:

The right part as a whole is

*Advances on Tensor Analysis and Their Applications*

*ρ ∂Vi ∂t e* ! *<sup>i</sup>* þ *ρVj*

**8.3 Quantity Θ is a tensor**

! *i e* !

The right part is

*∂ ∂t ρσ* þ ∇ ! � *ρV* ! *σ* � �

parts *<sup>d</sup>*

**70**

*dt* <sup>¼</sup> *<sup>∂</sup>*

*<sup>∂</sup><sup>t</sup>* þ *V* ! � ∇ � � !

tensor A, for instance:

*ρ dσ dt* <sup>¼</sup> *<sup>ρ</sup>*

form *σ* ¼ *σij e*

*∂Vi ∂xj e* ! *<sup>i</sup>* þ *Vi e* ! *i ∂ρ ∂t* þ *Vj*

*<sup>j</sup>*. Let us prove the equality:

*ρ dσ dt* <sup>¼</sup> *<sup>∂</sup> ∂t*

*d dt <sup>σ</sup>ij <sup>e</sup>* ! *i e* ! *<sup>j</sup>* ¼ *ρ ∂ ∂t σij e* ! *i e* ! *<sup>j</sup>* þ *ρ V* ! � ∇ � � !

¼ *ρ ∂σij ∂t e* ! *i e* ! *<sup>j</sup>* þ *ρVk*

¼ *∂ ∂t ρσij e* ! *i e* ! *<sup>j</sup>* þ *e* ! *k ∂ ∂xk*

¼ *σij ∂ρ ∂t e* ! *i e* ! *<sup>j</sup>* þ *ρ ∂σij <sup>∂</sup><sup>t</sup> <sup>e</sup>* ! *i e* ! *<sup>j</sup>* þ *δks*

¼ *σij ∂ρ ∂t e* ! *i e* ! *<sup>j</sup>* þ *ρ ∂σij <sup>∂</sup><sup>t</sup> <sup>e</sup>* ! *i e* ! *<sup>j</sup>* þ *∂ ∂xk*

¼ *σij ∂ρ ∂t e* ! *i e* ! *<sup>j</sup>* þ *ρ ∂σij <sup>∂</sup><sup>t</sup> <sup>e</sup>* ! *i e* ! *<sup>j</sup>* þ *Vkσij*

¼ *ρ ∂σij <sup>∂</sup><sup>t</sup> <sup>e</sup>* ! *i e* ! *<sup>j</sup>* þ *ρVk*

It is necessary to note the derivative *<sup>d</sup>*

Therefore, the expression (Eq. (83)) is valid in case Θ is a vector.

Let us assume that Θ is a tensor, for instance, stress tensor *σ*. Stress tensor is a second-rank symmetric tensor, which in Cartesian coordinates can be written in the

> *ρσ* þ ∇ ! � *ρV* ! *σ* � �

The left part of this equation in the component form can be written as follows:

*∂σij ∂xk e* ! *i e* ! *j:*

� *ρVsσij e* ! *s e* ! *i e* ! *<sup>j</sup>* ¼

*∂σij ∂xk e* ! *i e* ! *<sup>j</sup>* þ *σij e* ! *i e* ! *j ∂ρ ∂t* þ *Vk ∂ρ ∂xk* þ *ρ ∂Vk ∂xk*

Therefore, the expression (Eq. (83)) is also valid in case Θ is a tensor.

invariant quantity [5, 8, 9]. By this reason, instead of derivative *<sup>d</sup>*

tions. The rotational part cannot be written in divergent form.

defining equation in the form of a differential transport Equation [2, 5]. However, careful analysis shows that the material derivative for a second-rank tensor is not an

usually used as the material derivative for a second-rank tensor, which contains also rotational part (deviatoric stress rate), which provides symmetry relative to rota-

There are different forms of deviatoric stress rate for an arbitrary second-rank

*∂ ∂xk*

*ρVsσij e* ! *i e* ! *<sup>j</sup>* ¼

*ρVkσij e* ! *i e* ! *<sup>j</sup>* ¼

> *∂ρ ∂xk e* ! *i e* ! *<sup>j</sup>* þ *ρσij*

*∂ρ ∂xj* þ *ρ ∂Vj ∂xj*

*:*

*:* (86)

*σij e* ! *i e* ! *<sup>j</sup>* ¼

> *∂Vk ∂xk e* ! *i e* ! *<sup>j</sup>* þ *ρVk*

� �


*dt σ*, which contains local and convective

, can, at first glance, be used in fluid models when writing the

*∂σij ∂xk e* ! *i e* ! *<sup>j</sup>* ¼

*dt*, derivative *<sup>D</sup>*

*Dt* is


*∂t* þ*V* ! �∇ ! *ρ*þ*ρ*∇ ! �*V* ! ¼0

� �

Jaumann G.: *<sup>A</sup>* � <sup>Ω</sup> <sup>þ</sup> ð Þ *<sup>A</sup>* � <sup>Ω</sup> *<sup>T</sup>*, <sup>Ω</sup>—antisymmetric spin tensor. Rivlin R.: *A* � ∇ ! *V* ! þ ∇ ! *V* !*<sup>T</sup>* � *A*. Truesdell C.: ∇ ! *V* ! � *A* þ *A* � ∇ ! *V* ! *<sup>T</sup>* þ *tr*∇ ! *V* ! *A*. Oldroyd J., Sedov L.I., etc. [8, 10, 11]

At present, the question of which derivative is more appropriate to use when constructing rheological equations is unclear. The most common is the rotational derivative by Gustav Jaumann. The corresponding material derivative in the form by Jaumann, for an arbitrary tensor of the second rank, has the form

$$\frac{D\underline{A}}{Dt} = \frac{\partial \underline{A}}{\partial t} + \left(\vec{V} \cdot \vec{\nabla}\right) \underline{A} + \underline{A} \cdot \underline{\Omega} + \left(\underline{A} \cdot \underline{\Omega}\right)^{T} . \tag{87}$$

Material derivative in the form by Rivlin is written as follows:

$$\frac{D\underline{A}}{Dt} = \frac{\partial \underline{A}}{\partial t} + \left(\vec{V} \cdot \vec{\nabla}\right) \underline{A} + \underline{A} \cdot \vec{\nabla} \vec{V} + \left(\underline{A} \cdot \vec{\nabla} \vec{V}\right)^{T}.\tag{88}$$

It is easy to see that Rivlin's derivative differs from Jaumann's one by the additional term *<sup>A</sup>* � *<sup>S</sup>* <sup>þ</sup> ð Þ *<sup>A</sup>* � *<sup>S</sup> <sup>T</sup>*, which is neutral by itself.

Rotational derivative of a symmetric tensor is also a symmetric tensor. As an example, let us consider the rotational derivative of strain-rate tensor and spin tensor:

$$\begin{split} \frac{1}{2}\boldsymbol{\mathfrak{S}} \cdot \boldsymbol{\mathfrak{Q}} + (\boldsymbol{\mathfrak{S}} \cdot \boldsymbol{\mathfrak{Q}})^{T} &= \frac{1}{4} \Big[ \Big( \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\boldsymbol{\tilde{V}}} \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\boldsymbol{\tilde{V}}} + \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\boldsymbol{\tilde{V}}} \Big) \cdot \Big( \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{\tilde{V}}}{\boldsymbol{V}}} - \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}}} \Big) + \Big( \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}}} + \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}}} \Big) \cdot \Big( \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}}} - \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}}} \Big) \\ &= \frac{1}{2} \Big[ \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}} \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}} \cdot \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}} - \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}} \overset{\scriptstyle \boldsymbol{\tilde{V}}}{\overset{\scriptstyle \boldsymbol{V}}} \Big] . \end{split}$$

As a result, we have obtained the symmetrical second-rank tensor.

#### **9. Conclusions**

In this chapter, some applications of tensor calculus in fluid dynamics and heat transfer are presented. Typical transformations of equations and governing relations are discussed. Main conservation equations are given and analyzed. The governing equations of fluid motion and energy were obtained.

#### **Author details**

Dmitry Nikushchenko\* and Valery Pavlovsky State Marine Technical University, St. Petersburg, Russia

\*Address all correspondence to: ndmitry@list.ru

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Chapter 5**

**Abstract**

Differential Geometry and

Nonequilibrium Process

*Claudia B. Ruscitti, Laura B. Langoni*

*and Augusto A. Melgarejo*

instabilities in diffusive processes.

**1. Introduction**

**73**

manifold, *α*-connections, macroscopic potential

Macroscopic Descriptions in

The method of Riemannian geometry is fruitful in equilibrium thermodynamics. From the theory of fluctuations it has been possible to construct a metric for the space of thermodynamic equilibrium states. Inspired by these geometric elements, we will discuss the geometric-differential approach of nonequilibrium systems. In particular we will study the geometric aspects from the knowledge of the macroscopic potential associated with the Uhlenbeck-Ornstein (UO) nonequilibrium process. Assuming the geodesic curve as an optimal path and using the affine connection, known as *α*-connection, we will study the conditions under which a diffusive process can be considered optimal. We will also analyze the impact of this

behavior on the entropy of the system, relating these results with studies of

**Keywords:** nonequilibrium processes, Uhlenbeck-Ornstein process, statistical

The use of Riemannian geometry associated with the space of thermodynamic equilibrium states has been successful. In this framework the geometric elements are constructed from the knowledge of the thermodynamic potential. In this sense, one of the most interesting ideas is associated with the study of the phase transitions

visualized by means of the singularities of the scalar curvature [1, 2]. As a geometric-differential approach of nonequilibrium systems, we consider in our study the geometric properties of a statistical manifold associated with trajectorydependent entropy [3]. In statistics mechanics it is known that any statistical system has an associated metric-affine manifold having a special affine connection whether it is in equilibrium or not. The affine connection, called the *α*-connection [4], is a generalization of the Levi-Civita connection in Riemannian geometry, and in the case *α* ¼ 0 the metric-affine manifold reduces to the so-called Riemannian manifold. A first element that appears for nonequilibrium systems is the visualization of the phase transitions through the curvature tensor [5]. A second issue is the study of temporal evolution of a statistical system in order to study the optimal evolution.

#### **References**

[1] Kochin N, Kibel I, Rose N. Theoretical Hydrodynamic. 6nd Ed. Vol. 1. Moscow: PhysMathLit; 1963. p. 583. (in Russian)

[2] Nikushchenko D, Pavlovsky V. Computational Hydrodynamics. Theoretical Basis. St. Petersburg: Lan; 2018. p. 364. ISBN: 978-5-8114-2924-0 (in Russian)

[3] Batchelor G. An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press; 2000. p. 615

[4] Sedov L. Continuum Mechanics. Moscow: Nauka; 1970. p. 492. (in Russian)

[5] Pavlovsky V. A Short Course in Continuum Mechanics. SPbGTURP: St. Petersburg; 1993. p. 212. (in Russian)

[6] Udaev B. Heat Transfer. Vysshaya shkola: Moscow; 1973. p. 360. (in Russian)

[7] Kutateladze S. Fundamentals of Heat Transfer Theory. Moscow: Atomizdat; 1979. p. 416. (in Russian)

[8] Astarita G, Marrucci G. Principles of Non-Newtonian Fluid Mechanics. Maidenhead–Berkshire: McGraw-Hill; 1974. p. 296

[9] Ehrhard P, editor. Fuhrer Durch Die Stromungslehre: Grundlagen Und Phanomene. Wiesbaden: Springer Fachmedien; 2014. p. 548

[10] Truesdell C, Noll W. The Non-Linear Field Theories of Mechanics. Heidelberg: Springer; 2004. p. 602. ISBN: 978-3-662-10388-3

[11] Oldroyd JG. On the formulation of rheological equations of state. Proceedings of the Royal Society. 1950; **A200**:523

#### **Chapter 5**

**References**

(in Russian)

Russian)

Russian)

1974. p. 296

**A200**:523

**72**

p. 583. (in Russian)

[1] Kochin N, Kibel I, Rose N. Theoretical Hydrodynamic. 6nd Ed. Vol. 1. Moscow: PhysMathLit; 1963.

*Advances on Tensor Analysis and Their Applications*

[2] Nikushchenko D, Pavlovsky V. Computational Hydrodynamics. Theoretical Basis. St. Petersburg: Lan; 2018. p. 364. ISBN: 978-5-8114-2924-0

[3] Batchelor G. An Introduction to Fluid Dynamics. Cambridge: Cambridge University Press; 2000. p. 615

[4] Sedov L. Continuum Mechanics. Moscow: Nauka; 1970. p. 492. (in

[5] Pavlovsky V. A Short Course in Continuum Mechanics. SPbGTURP: St. Petersburg; 1993. p. 212. (in Russian)

[6] Udaev B. Heat Transfer. Vysshaya shkola: Moscow; 1973. p. 360. (in

[7] Kutateladze S. Fundamentals of Heat Transfer Theory. Moscow: Atomizdat;

[8] Astarita G, Marrucci G. Principles of Non-Newtonian Fluid Mechanics. Maidenhead–Berkshire: McGraw-Hill;

[9] Ehrhard P, editor. Fuhrer Durch Die Stromungslehre: Grundlagen Und Phanomene. Wiesbaden: Springer

[10] Truesdell C, Noll W. The Non-Linear Field Theories of Mechanics. Heidelberg: Springer; 2004. p. 602.

[11] Oldroyd JG. On the formulation of rheological equations of state. Proceedings of the Royal Society. 1950;

1979. p. 416. (in Russian)

Fachmedien; 2014. p. 548

ISBN: 978-3-662-10388-3

## Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process

*Claudia B. Ruscitti, Laura B. Langoni and Augusto A. Melgarejo*

#### **Abstract**

The method of Riemannian geometry is fruitful in equilibrium thermodynamics. From the theory of fluctuations it has been possible to construct a metric for the space of thermodynamic equilibrium states. Inspired by these geometric elements, we will discuss the geometric-differential approach of nonequilibrium systems. In particular we will study the geometric aspects from the knowledge of the macroscopic potential associated with the Uhlenbeck-Ornstein (UO) nonequilibrium process. Assuming the geodesic curve as an optimal path and using the affine connection, known as *α*-connection, we will study the conditions under which a diffusive process can be considered optimal. We will also analyze the impact of this behavior on the entropy of the system, relating these results with studies of instabilities in diffusive processes.

**Keywords:** nonequilibrium processes, Uhlenbeck-Ornstein process, statistical manifold, *α*-connections, macroscopic potential

#### **1. Introduction**

The use of Riemannian geometry associated with the space of thermodynamic equilibrium states has been successful. In this framework the geometric elements are constructed from the knowledge of the thermodynamic potential. In this sense, one of the most interesting ideas is associated with the study of the phase transitions visualized by means of the singularities of the scalar curvature [1, 2]. As a geometric-differential approach of nonequilibrium systems, we consider in our study the geometric properties of a statistical manifold associated with trajectorydependent entropy [3]. In statistics mechanics it is known that any statistical system has an associated metric-affine manifold having a special affine connection whether it is in equilibrium or not. The affine connection, called the *α*-connection [4], is a generalization of the Levi-Civita connection in Riemannian geometry, and in the case *α* ¼ 0 the metric-affine manifold reduces to the so-called Riemannian manifold. A first element that appears for nonequilibrium systems is the visualization of the phase transitions through the curvature tensor [5]. A second issue is the study of temporal evolution of a statistical system in order to study the optimal evolution.

In other words, it is the analysis of the dynamic behavior of the system and the study of the conditions that are optimal.

We now restrict our attention to a special family of probability density function,

*i*

where *C x*ð Þ and *Fi*ð Þ *x* are arbitrary functions of *x* and *ψ θ*ð Þ is a function of *θ<sup>i</sup>*

Particularly for a probability density function belonging to the exponential family, from Eqs. (2) and (3) the covariant coefficients and metric tensor are

2

*ψ θ*ð Þ *∂θi∂θ <sup>j</sup>*

*ijk* ð Þ¼� *<sup>θ</sup>* ð Þ <sup>1</sup> � *<sup>α</sup>*

Γ*<sup>k</sup>*ð Þ *<sup>α</sup>*

where *<sup>g</sup>km* � � is the inverse matrix of the metric and *gij* � � and <sup>Γ</sup>*<sup>k</sup>*ð Þ *<sup>α</sup>*

From Eq. (3) the curvature tensor for an *α*-connection is written as

*jk* � *<sup>∂</sup> <sup>j</sup>*Γ*<sup>s</sup>*ð Þ *<sup>α</sup> ik* � �*gsm* <sup>þ</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>*

struction associated with the probability density function (Eq. (4)).

In the case *α* ¼ 0, the coefficients reduce to the Levi-Civita's connection:

*gij*ð Þ¼ *<sup>θ</sup> <sup>∂</sup>*<sup>2</sup>

These coefficients, for *α* ∈ ℝ, determine a one-parameter family of affine connections, and each element of this family is called an *α*-connection. An affine connection allows one to compare vectors in nearby tangent spaces [9]. Moreover,

*ij* <sup>¼</sup> *<sup>g</sup>km*Γð Þ *<sup>α</sup>*

*<sup>θ</sup>iFi*ð Þ� *<sup>x</sup> ψ θ*ð Þ !, (4)

*<sup>∂</sup>i<sup>∂</sup> <sup>j</sup>∂kψ θ*ð Þ, (5)

*:* (6)

*ijm*, (7)

<sup>2</sup> *<sup>∂</sup>igjk* <sup>þ</sup> *<sup>∂</sup> jgik* � *<sup>∂</sup>kgij* � �*:* (8)

*jk* � <sup>Γ</sup>ð Þ *<sup>α</sup>*

*jrm*Γ*<sup>r</sup>*ð Þ *<sup>α</sup> ik* � �*:* (9)

*irm*Γ*<sup>r</sup>*ð Þ *<sup>α</sup>*

*ijk* ð Þ*θ* vanishes identically for *α* ¼ 1, any exponential family of PDF constitutes an uncurved space when the *α* ¼ 1 connection is used. The one connection is therefore called the exponential connection as we mentioned above [11]. The relationships (Eqs. (5) and (6)) lead to the simplification of the geometric con-

For systems in thermodynamic equilibrium, the parameters *θ<sup>i</sup>* may include inverse temperature, chemical potential, pressure, magnetic field, and so on. Also, *ψ θ*ð Þ represents the thermodynamic potential of the system [8]. It is worth noting that, if we consider the equilibrium density functions and *α* ¼ 0, this formalism reproduces the geometric structure found by Ruppeiner for the space of equilibrium states. In the Ruppeiner formalism, the metric is constructed using the theory of fluctuations, and the geometric elements are obtained as the second derivatives of the corresponding thermodynamic potential [2]. In this paper we have chosen the approach of statistical manifold because it allows us to address nonequilibrium

*ij* are the

called an exponential family, which is described by [8].

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

Γð Þ *<sup>α</sup>*

the covariant coefficients satisfy the following relation:

Γð Þ <sup>0</sup> *ijk* <sup>¼</sup> <sup>1</sup>

coordinates.

written as [9, 10].

contravariant coefficients.

*R*ð Þ *<sup>α</sup>*

Since Γð Þ *<sup>α</sup>*

problems.

**75**

*ijkm* <sup>¼</sup> *<sup>∂</sup>i*Γ*<sup>s</sup>*ð Þ *<sup>α</sup>*

*p x*ð Þ¼ , *<sup>θ</sup>* exp *C x*ð ÞþX*<sup>m</sup>*

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

In the previous context, this chapter will focus on the analysis of the optimal evolution. In particular we consider the Uhlenbeck-Ornstein (UO) nonequilibrium process described by the probability density function (PDF) solution of the Fokker-Planck Equation [6]. From this probability density function, we build a twodimensional metric-affine manifold in the coordinates ð Þ *μ*, *σ* , where *μ* is the mean and *σ* the standard deviation. In this coordinates and for the connections *α* ¼ 0 and *α* ¼ �1, the system evolves on a geodesic curve [7].

However, due to simplicity in geometric construction, we are interested in studying the behavior of the system in coordinates ð Þ *<sup>θ</sup>*1, *<sup>θ</sup>*<sup>2</sup> , where *<sup>θ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* <sup>2</sup>*σ*<sup>2</sup> ð Þ and *<sup>θ</sup>*<sup>2</sup> ¼ �*μ=σ*2. In these coordinates the probability density function belongs to the exponential family, and using this formal expression and analogously with the equilibrium probability density function, we build a macroscopic potential *ψ θ*ð Þ 1, *θ*<sup>2</sup> for the UO process. From the geometry constructed using the *ψ θ*ð Þ 1, *θ*<sup>2</sup> potential, we show that for *α* ¼ 3 and *α* ¼ 2, the system evolves on a geodesic curve. In particular for *α* ¼ 3, we show that the manifold is flat and for the steady state the macroscopic potential and entropy have the same functional dependence. Thinking the geodesic curve as an optimal trajectory, our results allow us to conjecture that the entropy describes the steady state of an optimal evolution.

In the second section of this chapter, we summarize the most relevant aspects of the theory of the statistical manifold. The geometric development associated with the fundamental solution of the Fokker-Planck equation of UO process is found in the third section. The fourth section is devoted to the construction of the potential. In the fifth section, we analyze the geometric relationship between macroscopic potential and entropy. In the sixth section, we present our conclusions and perspectives.

#### **2. Elements of statistical manifold**

In this section we briefly review the information of geometrical theory [4] that is used to analyze geometrically a family of probability density functions (PDF) and its application to thermodynamics. Let *p x*ð Þ , *θ* be a PDF described by a random variable *x* and parameters *θ* ¼ ð Þ *θ*1, *θ*2, … , *θ<sup>m</sup>* that characterize a system. The set of PDFs

$$M = \{ p(\mathbf{x}, \theta) : \theta \in \Omega \subset \mathbb{R}^m \} \tag{1}$$

becomes an *m*-dimensional statistical manifold having *θ<sup>i</sup>* coordinates. According to the information geometrical theory, we can make a metric tensor *gij*

$$\mathbf{g}\_{i\bar{\jmath}}(\theta) = E\left[\partial\_i l(\varkappa, \theta)\partial\_j l(\varkappa, \theta)\right] = -E\left[\partial\_i \partial\_j l(\varkappa, \theta)\right],\tag{2}$$

where *l x*ð Þ¼ , *θ* ln ½ � *p x*ð Þ , *θ* and *E*½ �*:* means the expectation operation with respect to *p x*ð Þ , *θ* . The last expression is obtained by the use of the normalization condition *<sup>E</sup> <sup>∂</sup><sup>i</sup>* ½ �¼ *l x*ð Þ , *<sup>θ</sup>* 0. This metric tensor is the Fisher information matrix in information theory.

In the statistical manifold *M*, we can introduce a natural derivative of the vector field *B* toward the tangent vector *A*, denoted by ∇*<sup>α</sup> AB*. It is obtained through the covariant coefficients [7]:

$$\Gamma^{(a)}\_{ijk}(\theta) = E\left[ \left( \partial\_i \partial\_j l(\mathbf{x}, \theta) + \frac{1-a}{2} \partial\_i l(\mathbf{x}, \theta) \partial\_j l(\mathbf{x}, \theta) \right) \partial\_k l(\mathbf{x}, \theta) \right]. \tag{3}$$

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

We now restrict our attention to a special family of probability density function, called an exponential family, which is described by [8].

$$p(\mathbf{x}, \theta) = \exp\left(\mathbf{C}(\mathbf{x}) + \sum\_{i}^{m} \theta\_{i} F\_{i}(\mathbf{x}) - \boldsymbol{\psi}(\theta)\right),\tag{4}$$

where *C x*ð Þ and *Fi*ð Þ *x* are arbitrary functions of *x* and *ψ θ*ð Þ is a function of *θ<sup>i</sup>* coordinates.

Particularly for a probability density function belonging to the exponential family, from Eqs. (2) and (3) the covariant coefficients and metric tensor are written as [9, 10].

$$\Gamma\_{ijk}^{(a)}(\theta) = -\frac{(1-a)}{2} \partial\_i \partial\_j \partial\_k \varphi(\theta),\tag{5}$$

$$\mathcal{g}\_{\vec{\eta}}(\theta) = \frac{\partial^2 \boldsymbol{\mu}(\theta)}{\partial \theta\_i \partial \theta\_j}. \tag{6}$$

These coefficients, for *α* ∈ ℝ, determine a one-parameter family of affine connections, and each element of this family is called an *α*-connection. An affine connection allows one to compare vectors in nearby tangent spaces [9]. Moreover, the covariant coefficients satisfy the following relation:

$$
\Gamma\_{ij}^{k(a)} = \mathbf{g}^{km} \Gamma\_{ijm}^{(a)},\tag{7}
$$

where *<sup>g</sup>km* � � is the inverse matrix of the metric and *gij* � � and <sup>Γ</sup>*<sup>k</sup>*ð Þ *<sup>α</sup> ij* are the contravariant coefficients.

In the case *α* ¼ 0, the coefficients reduce to the Levi-Civita's connection:

$$
\Gamma^{(0)}\_{ijk} = \frac{1}{2} \left( \partial\_i \mathbf{g}\_{jk} + \partial\_j \mathbf{g}\_{ik} - \partial\_k \mathbf{g}\_{ij} \right). \tag{8}
$$

From Eq. (3) the curvature tensor for an *α*-connection is written as

$$R^{(a)}\_{ijkm} = \left(\partial\_i \Gamma^{s(a)}\_{jk} - \partial\_j \Gamma^{s(a)}\_{ik}\right) \mathbf{g}\_{im} + \left(\Gamma^{(a)}\_{im} \Gamma^{r(a)}\_{jk} - \Gamma^{(a)}\_{jrm} \Gamma^{r(a)}\_{ik}\right). \tag{9}$$

Since Γð Þ *<sup>α</sup> ijk* ð Þ*θ* vanishes identically for *α* ¼ 1, any exponential family of PDF constitutes an uncurved space when the *α* ¼ 1 connection is used. The one connection is therefore called the exponential connection as we mentioned above [11]. The relationships (Eqs. (5) and (6)) lead to the simplification of the geometric construction associated with the probability density function (Eq. (4)).

For systems in thermodynamic equilibrium, the parameters *θ<sup>i</sup>* may include inverse temperature, chemical potential, pressure, magnetic field, and so on. Also, *ψ θ*ð Þ represents the thermodynamic potential of the system [8]. It is worth noting that, if we consider the equilibrium density functions and *α* ¼ 0, this formalism reproduces the geometric structure found by Ruppeiner for the space of equilibrium states. In the Ruppeiner formalism, the metric is constructed using the theory of fluctuations, and the geometric elements are obtained as the second derivatives of the corresponding thermodynamic potential [2]. In this paper we have chosen the approach of statistical manifold because it allows us to address nonequilibrium problems.

In other words, it is the analysis of the dynamic behavior of the system and the

In the previous context, this chapter will focus on the analysis of the optimal evolution. In particular we consider the Uhlenbeck-Ornstein (UO) nonequilibrium process described by the probability density function (PDF) solution of the Fokker-Planck Equation [6]. From this probability density function, we build a twodimensional metric-affine manifold in the coordinates ð Þ *μ*, *σ* , where *μ* is the mean and *σ* the standard deviation. In this coordinates and for the connections *α* ¼ 0 and

However, due to simplicity in geometric construction, we are interested in studying the behavior of the system in coordinates ð Þ *<sup>θ</sup>*1, *<sup>θ</sup>*<sup>2</sup> , where *<sup>θ</sup>*<sup>1</sup> <sup>¼</sup> <sup>1</sup>*<sup>=</sup>* <sup>2</sup>*σ*<sup>2</sup> ð Þ and *<sup>θ</sup>*<sup>2</sup> ¼ �*μ=σ*2. In these coordinates the probability density function belongs to the exponential family, and using this formal expression and analogously with the equilibrium probability density function, we build a macroscopic potential *ψ θ*ð Þ 1, *θ*<sup>2</sup> for the UO process. From the geometry constructed using the *ψ θ*ð Þ 1, *θ*<sup>2</sup> potential, we show that for *α* ¼ 3 and *α* ¼ 2, the system evolves on a geodesic curve. In particular for *α* ¼ 3, we show that the manifold is flat and for the steady state the macroscopic potential and entropy have the same functional dependence. Thinking the geodesic curve as an optimal trajectory, our results allow us to conjecture that the entropy

In the second section of this chapter, we summarize the most relevant aspects of the theory of the statistical manifold. The geometric development associated with the fundamental solution of the Fokker-Planck equation of UO process is found in the third section. The fourth section is devoted to the construction of the potential. In the fifth section, we analyze the geometric relationship between macroscopic potential and entropy. In the sixth section, we present our conclusions and perspectives.

In this section we briefly review the information of geometrical theory [4] that is used to analyze geometrically a family of probability density functions (PDF) and its application to thermodynamics. Let *p x*ð Þ , *θ* be a PDF described by a random variable *x* and parameters *θ* ¼ ð Þ *θ*1, *θ*2, … , *θ<sup>m</sup>* that characterize a system. The set of PDFs

becomes an *m*-dimensional statistical manifold having *θ<sup>i</sup>* coordinates. According

In the statistical manifold *M*, we can introduce a natural derivative of the vector

1 � *α*

where *l x*ð Þ¼ , *θ* ln ½ � *p x*ð Þ , *θ* and *E*½ �*:* means the expectation operation with respect to *p x*ð Þ , *θ* . The last expression is obtained by the use of the normalization condition *<sup>E</sup> <sup>∂</sup><sup>i</sup>* ½ �¼ *l x*ð Þ , *<sup>θ</sup>* 0. This metric tensor is the Fisher information matrix in

to the information geometrical theory, we can make a metric tensor *gij*

field *B* toward the tangent vector *A*, denoted by ∇*<sup>α</sup>*

*ijk* ð Þ¼ *<sup>θ</sup> <sup>E</sup> <sup>∂</sup>i<sup>∂</sup> jl x*ð Þþ , *<sup>θ</sup>*

*<sup>M</sup>* <sup>¼</sup> *p x*ð Þ , *<sup>θ</sup>* : *<sup>θ</sup>* <sup>∈</sup> <sup>Ω</sup> <sup>⊂</sup> <sup>ℝ</sup>*<sup>m</sup>* f g (1)

*AB*. It is obtained through the

*<sup>∂</sup>kl x*ð Þ , *<sup>θ</sup>*

*:* (3)

*gij*ð Þ¼ *<sup>θ</sup> <sup>E</sup> <sup>∂</sup>il x*ð Þ , *<sup>θ</sup> <sup>∂</sup> jl x*ð �¼� , *<sup>θ</sup>*<sup>Þ</sup> *<sup>E</sup> <sup>∂</sup>i<sup>∂</sup> jl x*ð Þ , *<sup>θ</sup>* , (2)

<sup>2</sup> *<sup>∂</sup>il*ð*x*, *<sup>θ</sup>*Þ*<sup>∂</sup> jl*ð*x*, *<sup>θ</sup>*<sup>Þ</sup>

study of the conditions that are optimal.

*Advances on Tensor Analysis and Their Applications*

*α* ¼ �1, the system evolves on a geodesic curve [7].

describes the steady state of an optimal evolution.

**2. Elements of statistical manifold**

information theory.

**74**

covariant coefficients [7]:

Γð Þ *<sup>α</sup>*

#### **3. Fokker-Planck equation and macroscopic potential**

A diffusion process can be thought of as a process of Uhlenbeck-Ornstein. The Uhlenbeck-Ornstein process is a stochastic process that, roughly speaking, describes the velocity of a massive Brownian particle under the influence of friction. The probability density function *P x*ð Þ , *t* of the Uhlenbeck-Ornstein process satisfies the Fokker-Planck equation [6]:

$$\frac{\partial P}{\partial t} = \frac{1}{\tau} \frac{\partial}{\partial \mathbf{x}} \left[ (\mathbf{x} - \boldsymbol{\varkappa}\_0) P \right] + D \frac{\partial^2 P}{\partial \mathbf{x}^2}. \tag{10}$$

By analogy with the probability density functions for systems in equilibrium, we

1 *θ*3 1 þ 3*θ*<sup>2</sup> 2 2*θ*<sup>4</sup> 1 � �, <sup>Γ</sup>ð Þ *<sup>α</sup>*

<sup>211</sup> <sup>¼</sup> ð Þ *<sup>α</sup>* � <sup>1</sup> 2

<sup>212</sup> ¼ � ð Þ *<sup>α</sup>* � <sup>1</sup> 4*θ*<sup>2</sup> 1

Using Eqs. (18) and (19), we calculate the curvature tensor component in the

From Eq. (19) we observe that there are two *α* values for which the manifold is

In a manifold with a connection, we can generalize the straight line of Euclidean geometry. The generalized straight line is called geodesic, and it is defined by the characteristic that its tangent vector does not change its direction. It satisfies the

> *dθ <sup>j</sup> du* ¼ � *<sup>d</sup>*<sup>2</sup>

with an arbitrary parameter *u*. The special parameter *s* is called the affine parameter. We note that if *u* is a linear transformation of *s*, the right-hand side vanishes. Let us now prove that the curve defined by Eq. (15) of the UO process is a geodesic. We choose the Newtonian time *t* for the arbitrary parameter *u*. From Eq. (20) for the study of geodetic curves, we consider that the temporal dependence

> 1 <sup>2</sup>*D<sup>τ</sup>* <sup>1</sup> � *<sup>e</sup>*�2*t=<sup>τ</sup>* ð Þ,

> > *<sup>x</sup>*<sup>0</sup> <sup>1</sup> � *<sup>e</sup>*�*t=<sup>τ</sup>* � � *<sup>D</sup><sup>τ</sup>* <sup>1</sup> � *<sup>e</sup>*�2*t=<sup>τ</sup>* ð Þ*:*

*dt*<sup>2</sup> <sup>¼</sup> cothð Þ *<sup>t</sup>=<sup>τ</sup>* csch2

ð Þ *<sup>t</sup>=*ð Þ <sup>2</sup>*<sup>τ</sup>* csch<sup>3</sup>

ð Þ t*=τ* <sup>2</sup>*Dτ*<sup>2</sup> (22)

ð Þ t*=τ <sup>D</sup>τ*<sup>3</sup> *:* (23)

*θ*1ðÞ¼ *t*

8 >>>><

>>>>:

*θ*2ðÞ¼� *t*

ð Þ *t=τ* <sup>4</sup>*Dτ*<sup>2</sup> , *<sup>d</sup>*<sup>2</sup>

*θ*2

The tangent vector coordinate *dθi=dt* and the acceleration coordinate

*θ*1

*dt*<sup>2</sup> <sup>¼</sup> <sup>2</sup>*x*<sup>0</sup> sinh <sup>4</sup>

<sup>222</sup> ¼ 0

(18)

*:* (19)

*du* (20)

(21)

*θ*2 2*θ*<sup>3</sup> 1

ð Þ *α* � 3 ð Þ *α* � 1 8*θ*<sup>3</sup> 1

> *u=ds*<sup>2</sup> ð Þ *du=ds* <sup>2</sup>

*dθ<sup>i</sup>*

will call the relationship (Eq. (17)) "nonequilibrium potential" [8].

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

<sup>111</sup> ¼ � ð Þ *<sup>α</sup>* � <sup>1</sup> 2

Γð Þ *<sup>α</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

Γð Þ *<sup>α</sup>* <sup>112</sup> <sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>*

Γð Þ *<sup>α</sup>* <sup>122</sup> <sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>*

*R*ð Þ *<sup>α</sup>*

*d*2 *θi du*<sup>2</sup> <sup>þ</sup> <sup>Γ</sup>*<sup>i</sup> jk dθ<sup>k</sup> du*

of the coordinates ð Þ *θ*1, *θ*<sup>2</sup> is given by

*dθ*<sup>1</sup>

*dt* ¼ � *<sup>x</sup>*0sech<sup>2</sup>

*dt* ¼ � cosh <sup>2</sup>

ð Þ *t=*ð Þ 2*τ* <sup>4</sup>*Dτ*<sup>2</sup> , *<sup>d</sup>*<sup>2</sup>

coordinates ð Þ *θ*1, *θ*<sup>2</sup>

Eq. (11)

*d*2

**77**

*θi=dt*<sup>2</sup> are

and

*dθ*<sup>2</sup>

Using the potential (Eq. (17)) we calculate the coefficients (Eq. (5))

<sup>121</sup> <sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>*

<sup>221</sup> <sup>¼</sup> <sup>Γ</sup>ð Þ *<sup>α</sup>*

<sup>1212</sup>ð Þ¼� *θ*1, *θ*<sup>2</sup>

flat, *α* ¼ 3 and *α* ¼ 1. The case *α* ¼ 1 is a direct consequence of Eq. (5).

The fundamental solution of this linear parabolic partial differential equation, and the initial condition consisting of a unit point mass at location *y*, is:

$$P(\mathbf{x},t) = \sqrt{\frac{1}{2\pi D\tau(1 - e^{-2t/\tau})}} \exp\left\{-\frac{\mathbf{1}}{2D\tau} \left[\frac{\left(\mathbf{x} - \mathbf{x}\_0 - (\mathbf{y} - \mathbf{x}\_0)e^{-t/\tau}\right)^2}{\mathbf{1} - e^{-2t/\tau}}\right] \right\} \tag{11}$$

which is the Gaussian density function with mean

$$
\mu = \varkappa\_0 + (\mathcal{y} - \varkappa\_0)e^{-t/\tau} \tag{12}
$$

and variance

$$
\sigma^2 = D\tau \left( 1 - e^{-2t/\tau} \right) \tag{13}
$$

where *x*<sup>0</sup> represents the average length of the displacement, *D* is the diffusion coefficient, and *τ* is a characteristic time. Without loss of generality, in the rest of the work, we consider *y* ¼ 0.

Considering Eqs. (12) and (13), we think the function (Eq. (11)) as a probability density function dependent on two parameters *μ* and *σ*, formally:

$$p(\mathbf{x}, \mu, \sigma) = \frac{\mathbf{1}}{\sqrt{2\pi\sigma^2}} e^{-(\mathbf{x}-\mu)^2/2\sigma^2}. \tag{14}$$

Considering these parameters as the coordinates ð Þ *μ*, *σ* of the manifold *M* and taking into account Eqs. (2) and (4), it can be seen that for *α* ¼ 0 and *α* ¼ �1, the system evolves on a geodesic curve [7].

Inspired by the simplicity of relations (Eqs. (5) and (6)), we use an alternative description of the UO process through the coordinates:

$$f(\theta\_1, \theta\_2) = \left(\frac{1}{2\sigma^2}, -\frac{\mu}{\sigma^2}\right). \tag{15}$$

In these coordinates the probability density function (Eq. (14)) belongs to the exponential family and is written as

$$p(\mathbf{x}, \theta\_1, \theta\_2) = \exp\left[-\theta\_1 \mathbf{x}^2 - \theta\_2 \mathbf{x} - \varphi(\theta\_1, \theta\_2)\right] \tag{16}$$

where

$$
\psi(\theta\_1, \theta\_2) = \frac{\theta\_2^2}{4\theta\_1} - \ln\left[\sqrt{\frac{\theta\_1}{\pi}}\right].\tag{17}
$$

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

By analogy with the probability density functions for systems in equilibrium, we will call the relationship (Eq. (17)) "nonequilibrium potential" [8].

Using the potential (Eq. (17)) we calculate the coefficients (Eq. (5))

$$\begin{aligned} \Gamma\_{111}^{(a)} &= -\frac{(a-1)}{2} \left( \frac{1}{\theta\_1^3} + \frac{3\theta\_2^2}{2\theta\_1^4} \right), & \Gamma\_{222}^{(a)} &= \mathbf{0} \\\\ \Gamma\_{12}^{(a)} &= \Gamma\_{121}^{(a)} = \Gamma\_{211}^{(a)} = \frac{(a-1)}{2} \frac{\theta\_2}{2\theta\_1^3} \\\\ \Gamma\_{122}^{(a)} &= \Gamma\_{221}^{(a)} = \Gamma\_{212}^{(a)} = -\frac{(a-1)}{4\theta\_1^2} \end{aligned} \tag{18}$$

Using Eqs. (18) and (19), we calculate the curvature tensor component in the coordinates ð Þ *θ*1, *θ*<sup>2</sup>

$$R\_{1212}^{(a)}(\\\theta\_1,\\\\\theta\_2) = -\frac{(a-\mathfrak{Z})(a-\mathfrak{1})}{8\theta\_1^3}.\tag{19}$$

From Eq. (19) we observe that there are two *α* values for which the manifold is flat, *α* ¼ 3 and *α* ¼ 1. The case *α* ¼ 1 is a direct consequence of Eq. (5).

In a manifold with a connection, we can generalize the straight line of Euclidean geometry. The generalized straight line is called geodesic, and it is defined by the characteristic that its tangent vector does not change its direction. It satisfies the Eq. (11)

$$\frac{d^2\theta\_i}{du^2} + \Gamma^i\_{jk}\frac{d\theta\_k}{du}\frac{d\theta\_j}{du} = -\frac{d^2u/ds^2}{(du/ds)^2}\frac{d\theta\_i}{du} \tag{20}$$

with an arbitrary parameter *u*. The special parameter *s* is called the affine parameter. We note that if *u* is a linear transformation of *s*, the right-hand side vanishes. Let us now prove that the curve defined by Eq. (15) of the UO process is a geodesic. We choose the Newtonian time *t* for the arbitrary parameter *u*. From Eq. (20) for the study of geodetic curves, we consider that the temporal dependence of the coordinates ð Þ *θ*1, *θ*<sup>2</sup> is given by

$$\begin{cases} \theta\_1(t) = \frac{1}{2D\tau(1 - e^{-2t/\tau})}, \\\\ \theta\_2(t) = -\frac{\varkappa\_0 \left(1 - e^{-t/\tau}\right)}{D\tau(1 - e^{-2t/\tau})}. \end{cases} \tag{21}$$

The tangent vector coordinate *dθi=dt* and the acceleration coordinate *d*2 *θi=dt*<sup>2</sup> are

$$\frac{d\theta\_1}{dt} = -\frac{\cosh^2(t/\tau)}{4D\tau^2}, \quad \frac{d^2\theta\_1}{dt^2} = \frac{\coth(t/\tau)\text{csch}^2(t/\tau)}{2D\tau^2} \tag{22}$$

and

$$\frac{d\theta\_2}{dt} = -\frac{\varkappa\_0 \text{sech}^2(t/(2\tau))}{4D\tau^2}, \quad \frac{d^2\theta\_2}{dt^2} = \frac{2\varkappa\_0 \sinh^4(t/(2\tau))\text{csch}^3(t/\tau)}{D\tau^3}.\tag{23}$$

**3. Fokker-Planck equation and macroscopic potential**

*∂P <sup>∂</sup><sup>t</sup>* <sup>¼</sup> <sup>1</sup> *τ ∂*

which is the Gaussian density function with mean

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2*πDτ* 1 � *e*�2*t=<sup>τ</sup>* ð Þ

*Advances on Tensor Analysis and Their Applications*

Fokker-Planck equation [6]:

s

the work, we consider *y* ¼ 0.

system evolves on a geodesic curve [7].

exponential family and is written as

where

**76**

description of the UO process through the coordinates:

*P x*ð Þ¼ , *t*

and variance

A diffusion process can be thought of as a process of Uhlenbeck-Ornstein. The Uhlenbeck-Ornstein process is a stochastic process that, roughly speaking, describes the velocity of a massive Brownian particle under the influence of friction. The probability density function *P x*ð Þ , *t* of the Uhlenbeck-Ornstein process satisfies the

*<sup>∂</sup><sup>x</sup>* ½ �þ ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> *<sup>P</sup> <sup>D</sup> <sup>∂</sup>*<sup>2</sup>

The fundamental solution of this linear parabolic partial differential equation,

2*Dτ*

*μ* ¼ *x*<sup>0</sup> þ ð Þ *y* � *x*<sup>0</sup> *e*

*<sup>σ</sup>*<sup>2</sup> <sup>¼</sup> *<sup>D</sup><sup>τ</sup>* <sup>1</sup> � *<sup>e</sup>*

density function dependent on two parameters *μ* and *σ*, formally:

*p x*ð Þ¼ , *<sup>μ</sup>*, *<sup>σ</sup>* <sup>1</sup>

ð Þ¼ *θ*1, *θ*<sup>2</sup>

*ψ θ*ð Þ¼ 1, *θ*<sup>2</sup>

where *x*<sup>0</sup> represents the average length of the displacement, *D* is the diffusion coefficient, and *τ* is a characteristic time. Without loss of generality, in the rest of

Considering Eqs. (12) and (13), we think the function (Eq. (11)) as a probability

ffiffiffiffiffiffiffiffiffi <sup>2</sup>*πσ*<sup>2</sup> <sup>p</sup> *<sup>e</sup>*

Considering these parameters as the coordinates ð Þ *μ*, *σ* of the manifold *M* and taking into account Eqs. (2) and (4), it can be seen that for *α* ¼ 0 and *α* ¼ �1, the

Inspired by the simplicity of relations (Eqs. (5) and (6)), we use an alternative

1 <sup>2</sup>*σ*<sup>2</sup> , � *<sup>μ</sup> σ*2

In these coordinates the probability density function (Eq. (14)) belongs to the

*p x*ð Þ¼ , *<sup>θ</sup>*1, *<sup>θ</sup>*<sup>2</sup> exp �*θ*1*x*<sup>2</sup> � *<sup>θ</sup>*2*<sup>x</sup>* � *ψ θ*ð Þ 1, *<sup>θ</sup>*<sup>2</sup>

*θ*2 2 4*θ*<sup>1</sup>

� ln

ffiffiffiffi *θ*1 *π* " # r

� �

�ð Þ *<sup>x</sup>*�*<sup>μ</sup>* <sup>2</sup> *=*2*σ*<sup>2</sup>

�2*t=<sup>τ</sup>* � �

and the initial condition consisting of a unit point mass at location *y*, is:

exp � <sup>1</sup>

*P*

*<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> � ð Þ *<sup>y</sup>* � *<sup>x</sup>*<sup>0</sup> *<sup>e</sup>*�*t=<sup>τ</sup>* � �<sup>2</sup> 1 � *e*�2*t=<sup>τ</sup>*

( ) " #

*<sup>∂</sup>x*<sup>2</sup> *:* (10)

�*t=<sup>τ</sup>* (12)

*:* (14)

*:* (15)

*:* (17)

� � (16)

(11)

(13)

Substituting Eqs. (22) and (23) in Eq. (20) for the coordinate *θ*1, we have

$$\frac{1}{2D\tau^2} \left\{ 2D\tau[(a-1)-(a-3)\coth(t/\tau)] - (a-1)\mathbf{x}\_0^2 \right\} = \frac{d^2\mathbf{t}/d\mathbf{s}^2}{\left(dt/d\mathbf{s}\right)^2},\tag{24}$$

and for the coordinate *θ*<sup>2</sup>

$$\begin{aligned} \frac{\text{csch}(t/\tau)}{2D\tau^2} \{2D\tau[-1-(a-2)\cosh\left(t/\tau\right)+(a-1)\sinh\left(t/\tau\right)]\}+\\ \frac{\text{csch}(t/\tau)}{2D\tau^2} \left[\left(1+e^{-t/\tau}\right)(a-1)\chi\_0^2\right] = \frac{d^2t/ds^2}{(dt/ds)^2}. \end{aligned} \tag{25}$$

The results (Eqs. (24) and (25)) are compatible if we choose *α* as

$$a = 1 + 2\frac{D\tau}{x\_0^2}.\tag{26}$$

In this case the characteristic time *τ* represents the tumbling time, and *ν* represents

*t as a function of s for two values of τ. τ* ¼ 10 *in the solid line and τ* ¼ 1 *in the dashed line.*

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

Thinking the geodesic curve as an optimal trajectory, from Eq. (21) we will analyze the optimal evolution of the system in terms of the behavior of the macroscopic potential (Eq. (17)) and the entropy function. To begin with this discussion, we first notice that the common definition of a nonequilibrium Gibbs entropy

suggests to define a trajectory-dependent entropy for the particle (or "system") *h x*ð Þ¼� , *θ* ln ½ � *p x*ð Þ , *θ* [3]. The trajectory-dependent entropy is related to the function *l x*ð Þ , *θ* used in the construction of the metric of the statistical manifold *M* by means of the relation *h x*ð Þ¼� , *θ l x*ð Þ , *θ* . In our analysis we are interested in the

Moreover, given that the probability density function (Eq. (16)) belongs to the exponential family, we can calculate the entropy by means of the relation [13].

> *∂ψ ∂θ*1 � *θ*<sup>2</sup> *∂ψ ∂θ*2

> > ffiffiffiffi *θ*1 *π* " # r

*H*ð Þ¼ *θ ψ* � *θ*<sup>1</sup>

*H*ð Þ¼ *θ*

1 <sup>2</sup> � ln

In order to study the temporal evolution of potential *ψ* and entropy *H*, we use the relations Eqs. (12) and (13) in Eq. (21). In particular, we focus on investigating the asymptotic behavior, that is, the behavior for *t* ! ∞. In this limit and using Eq. (21), we have that *θ*<sup>1</sup> ¼ 1*=*ð Þ 2*Dτ* and *θ*<sup>2</sup> ¼ �*x*0*=*ð Þ *Dτ* . Therefore, the asymptotic

*p x*ð Þ , *θ* ln ½ � *p x*ð Þ , *θ dx* ¼ h i *h x*ð Þ , *θ* (30)

*:* (31)

*:* (32)

the average speed of bacteria [12].

**Figure 1.**

**4. Optimal trajectory and entropy**

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

*H*ð Þ¼� *θ*

In any case, using Eqs. (30) or (31), we obtain

behavior of the potential and entropy is written as

**79**

entropy averaged over trajectories *H*ð Þ*θ* .

∞ð

�∞

Taking as reference the geometric study in the coordinates ð Þ *μ*, *σ* [7], we consider two possible relationships between the characteristic time *τ* and the other two parameters, *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>0</sup>*=<sup>D</sup>* or *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>0</sup>*=*ð Þ 2*D* . The first case leads to a flat geometry, that is, if *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>0</sup>*=D*, we have that *<sup>α</sup>* <sup>¼</sup> 3 and for Eq. (19) *<sup>R</sup>*ð Þ<sup>3</sup> <sup>1212</sup>ð Þ¼ *θ*1, *θ*<sup>2</sup> 0. This case is equivalent to the connection *α* ¼ �1 in the coordinates ð Þ *μ*, *σ* , where the manifold is also flat and the affine time *s* and Newtonian time *t* are related by the differential equation

$$\frac{1}{\tau} = \frac{d^2 t / ds^2}{\left(dt / ds\right)^2} \tag{27}$$

whose solution has the form

$$t = -\tau \ln\left(s + a\tau\right) + b.\tag{28}$$

On the other hand if *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>0</sup>*=*ð Þ 2*D* , we have that *α* ¼ 2 and the affine and Newtonian time are now related by

$$\frac{\coth(t/\tau)}{\tau} = \frac{d^2t/ds^2}{\left(dt/ds\right)^2}.\tag{29}$$

Due to its nature, the equation (Eq. (29)) has been solved numerically. For each *τ* we find solutions that make sense in the context of our problem. In **Figure 1** we show two examples of these solutions.

In the coordinates ð Þ *μ*, *σ* , Eqs. (27) and (29) correspond to the choices *α* ¼ �1 and *α* ¼ 0, respectively [7]. On the other hand in the context of geometric construction from the potential (Eq. (17)), the main interpretation of the equation (Eq. (26)) is that the temporal evolution described by Eq. (21) coincides with a geodesic curve of space ð Þ *<sup>θ</sup>*1, *<sup>θ</sup>*<sup>2</sup> . In this sense we find two relationships *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>0</sup>*=D* and *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>0</sup>*=*ð Þ 2*D* between the parameters *x*0, *D*, and *τ* for which the evolution is optimized.

It is interesting to note that if we assume that *v* ¼ *x*0*=τ*, the relationship *τ* ¼ *x*2 <sup>0</sup>*=<sup>D</sup>* leads us to *<sup>D</sup>* <sup>¼</sup> *<sup>v</sup>*<sup>2</sup>*τ*. This is a well-known and a widely used relationship, for example, in the understanding of bacterial mobility through a diffusive model.

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

**Figure 1.** *t as a function of s for two values of τ. τ* ¼ 10 *in the solid line and τ* ¼ 1 *in the dashed line.*

In this case the characteristic time *τ* represents the tumbling time, and *ν* represents the average speed of bacteria [12].

#### **4. Optimal trajectory and entropy**

Thinking the geodesic curve as an optimal trajectory, from Eq. (21) we will analyze the optimal evolution of the system in terms of the behavior of the macroscopic potential (Eq. (17)) and the entropy function. To begin with this discussion, we first notice that the common definition of a nonequilibrium Gibbs entropy

$$H(\theta) = -\int\_{-\infty}^{\infty} p(\mathbf{x}, \theta) \ln \left[ p(\mathbf{x}, \theta) \right] d\mathbf{x} = \langle h(\mathbf{x}, \theta) \rangle \tag{30}$$

suggests to define a trajectory-dependent entropy for the particle (or "system") *h x*ð Þ¼� , *θ* ln ½ � *p x*ð Þ , *θ* [3]. The trajectory-dependent entropy is related to the function *l x*ð Þ , *θ* used in the construction of the metric of the statistical manifold *M* by means of the relation *h x*ð Þ¼� , *θ l x*ð Þ , *θ* . In our analysis we are interested in the entropy averaged over trajectories *H*ð Þ*θ* .

Moreover, given that the probability density function (Eq. (16)) belongs to the exponential family, we can calculate the entropy by means of the relation [13].

$$H(\theta) = \psi - \theta\_1 \frac{\partial \psi}{\partial \theta\_1} - \theta\_2 \frac{\partial \psi}{\partial \theta\_2}. \tag{31}$$

In any case, using Eqs. (30) or (31), we obtain

$$H(\theta) = \frac{1}{2} - \ln\left[\sqrt{\frac{\theta\_1}{\pi}}\right].\tag{32}$$

In order to study the temporal evolution of potential *ψ* and entropy *H*, we use the relations Eqs. (12) and (13) in Eq. (21). In particular, we focus on investigating the asymptotic behavior, that is, the behavior for *t* ! ∞. In this limit and using Eq. (21), we have that *θ*<sup>1</sup> ¼ 1*=*ð Þ 2*Dτ* and *θ*<sup>2</sup> ¼ �*x*0*=*ð Þ *Dτ* . Therefore, the asymptotic behavior of the potential and entropy is written as

Substituting Eqs. (22) and (23) in Eq. (20) for the coordinate *θ*1, we have

<sup>2</sup>*Dτ*<sup>2</sup> <sup>f</sup>2*Dτ*½ � �<sup>1</sup> � ð Þ *<sup>α</sup>* � <sup>2</sup> cosh ð Þþ *<sup>t</sup>=<sup>τ</sup>* ð Þ *<sup>α</sup>* � <sup>1</sup> sinh ð Þ *<sup>t</sup>=<sup>τ</sup>* gþ

*α* ¼ 1 þ 2

Taking as reference the geometric study in the coordinates ð Þ *μ*, *σ* [7], we consider two possible relationships between the characteristic time *τ* and the other two

equivalent to the connection *α* ¼ �1 in the coordinates ð Þ *μ*, *σ* , where the manifold is also flat and the affine time *s* and Newtonian time *t* are related by the differential

*t=ds*<sup>2</sup>

1 *<sup>τ</sup>* <sup>¼</sup> *<sup>d</sup>*<sup>2</sup>

cothð Þ *t=τ*

*<sup>τ</sup>* <sup>¼</sup> *<sup>d</sup>*<sup>2</sup>

Due to its nature, the equation (Eq. (29)) has been solved numerically. For each *τ* we find solutions that make sense in the context of our problem. In **Figure 1** we

In the coordinates ð Þ *μ*, *σ* , Eqs. (27) and (29) correspond to the choices *α* ¼ �1 and *α* ¼ 0, respectively [7]. On the other hand in the context of geometric construction from the potential (Eq. (17)), the main interpretation of the equation (Eq. (26)) is that the temporal evolution described by Eq. (21) coincides with a geodesic curve of space ð Þ *<sup>θ</sup>*1, *<sup>θ</sup>*<sup>2</sup> . In this sense we find two relationships *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

<sup>0</sup>*=*ð Þ 2*D* between the parameters *x*0, *D*, and *τ* for which the evolution is

It is interesting to note that if we assume that *v* ¼ *x*0*=τ*, the relationship *τ* ¼

<sup>0</sup>*=<sup>D</sup>* leads us to *<sup>D</sup>* <sup>¼</sup> *<sup>v</sup>*<sup>2</sup>*τ*. This is a well-known and a widely used relationship, for example, in the understanding of bacterial mobility through a diffusive model.

¼ *d*2

*t=ds*<sup>2</sup> ð Þ *dt=ds* <sup>2</sup> *:*

> *Dτ x*2 0

� � <sup>¼</sup> *<sup>d</sup>*<sup>2</sup>

0

*t=ds*<sup>2</sup>

*:* (26)

<sup>1212</sup>ð Þ¼ *θ*1, *θ*<sup>2</sup> 0. This case is

ð Þ *dt=ds* <sup>2</sup> (27)

ð Þ *dt=ds* <sup>2</sup> *:* (29)

<sup>0</sup>*=D*

*t* ¼ �*τ* ln ð Þþ *s* þ *aτ b:* (28)

<sup>0</sup>*=*ð Þ 2*D* , we have that *α* ¼ 2 and the affine and

*t=ds*<sup>2</sup>

<sup>0</sup>*=*ð Þ 2*D* . The first case leads to a flat geometry, that is,

ð Þ *dt=ds* <sup>2</sup> , (24)

(25)

<sup>2</sup>*Dτ*<sup>2</sup> <sup>2</sup>*Dτ α*½ð Þ� � <sup>1</sup> ð Þ *<sup>α</sup>* � <sup>3</sup> cothð Þ *<sup>t</sup>=<sup>τ</sup>* � � ð Þ *<sup>α</sup>* � <sup>1</sup> *<sup>x</sup>*<sup>2</sup>

ð Þ *<sup>α</sup>* � <sup>1</sup> *<sup>x</sup>*<sup>2</sup> 0

The results (Eqs. (24) and (25)) are compatible if we choose *α* as

1

and for the coordinate *θ*<sup>2</sup>

<sup>2</sup>*Dτ*<sup>2</sup> <sup>1</sup> <sup>þ</sup> *<sup>e</sup>*

whose solution has the form

On the other hand if *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

Newtonian time are now related by

show two examples of these solutions.

�*t=<sup>τ</sup>* � �

*Advances on Tensor Analysis and Their Applications*

<sup>0</sup>*=<sup>D</sup>* or *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

<sup>0</sup>*=D*, we have that *<sup>α</sup>* <sup>¼</sup> 3 and for Eq. (19) *<sup>R</sup>*ð Þ<sup>3</sup>

h i

cschð Þ *t=τ*

cschð Þ *t=τ*

parameters, *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

if *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

equation

and *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

optimized.

*x*2

**78**

*Advances on Tensor Analysis and Their Applications*

$$\psi = \frac{\varkappa\_0^2}{2D\tau} + \frac{1}{2}\ln\left[2\pi D\tau\right] \tag{33}$$

**5. Discussion and perspectives**

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

steady state of an optimal evolution in a flat manifold.

(UNLP). AM, LL, and CR are professors at the UNLP.

*R*ð Þ <sup>0</sup>

**81**

nonequilibrium systems.

**Acknowledgements**

In this chapter we have studied the PDF (Eq. (14)) that is the fundamental solution of the Fokker-Planck equation associated with the UO process. Our main interest was relating the geometric aspects of the process with the steady-state behavior. In our analysis we used the theoretical framework of the statistical manifold *M* with *α*-connections for two different coordinates ð Þ *μ*, *σ* and ð Þ *θ*1, *θ*<sup>2</sup> [4, 7]. In the first case, there exist two interesting values of *α*, namely, *α* ¼ �1 and *α* ¼ 0, for which the process evolves on a geodesic of space ð Þ *μ*, *σ* with different values of *τ*. However, in the search for a simpler geometric construction, we find that for the coordinates ð Þ *θ*1, *θ*<sup>2</sup> , we can define a macroscopic potential *ψ θ*ð Þ 1, *θ*<sup>2</sup> , and the values of *α* that lead to the system evolving on a geodesic curve are *α* ¼ 3 and *α* ¼ 2. In our study we show that the connection *α* ¼ �1 for the coordinates ð Þ *μ*, *σ* corresponds to the connection *α* ¼ 3 for the coordinates ð Þ *θ*1, *θ*<sup>2</sup> in the sense that both connections lead to a flat curvature and the same relationship between the parameters of the system. An important consequence of this behavior is that when the system evolves over geodesics, the macroscopic potential *ψ* and entropy *H* have the same functional dependence in the steady state. If we think of the geodesic curve as an optimal trajectory, our results allow us to conjecture that the entropy describes the

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

Additionally and poorly developed in this chapter is the use of geometric aspects in the study of instabilities in nonequilibrium system. In equilibrium thermodynamics this information is contained in the scalar curvature of the manifold of equilibrium states [2]. For nonequilibrium problems, the instabilities are associated with the singularities or discontinuities of the curvature tensor [5, 15]. In the case of diffusive problems, the instabilities can be found by studying the singularities of

<sup>1212</sup>ð Þ *μ*, *σ* . In terms of diffusivity, we see that as it decreases, the system has different macroscopic behavior. An example of this behavior is the formation of density patterns in the populations of self-propelled bacteria whose mobility can be investigated in terms of the diffusion coefficient *D* [12]. From the perspective of the macroscopic potential *ψ*, the instabilities can be associated to singularities or discontinuities of *ψ*. In this sense, from a wider point of view, we consider that the potential *ψ* represents an alternative way to study the phase transitions in

This work was supported by the Universidad Nacional de La Plata, Argentina

and

$$H = \frac{1}{2} + \frac{1}{2} \ln\left[2\pi D\tau\right].\tag{34}$$

If we think that in the coordinates ð Þ *θ*1, *θ*<sup>2</sup> , with *α* ¼ 3, the system evolves on a geodesic of a flat manifold (*R*ð Þ<sup>3</sup> <sup>1212</sup>ð Þ¼ *<sup>θ</sup>*1, *<sup>θ</sup>*<sup>2</sup> 0), we have *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> <sup>0</sup>*=D*, and both the potential *ψ* and the entropy *H* tend to the same asymptotic value. In other words the entropy describes the asymptotic behavior of an optimal evolution:

$$
\Delta \varphi = H = \frac{1}{2} + \ln \sqrt{2 \pi D \tau}. \tag{35}
$$

In this last result, it can be seen that the asymptotic behavior depends fundamentally on the transport coefficient *D*. In this sense, the relevant variable in the description of the process is the diffusion coefficient *D*. It is interesting to note that the relations (Eq. (21)) are not invariant when we change *t* by �*t*. In this context, the evolution of the system is irreversible.

While in the coordinates ð Þ *θ*1, *θ*<sup>2</sup> are two *α*-connections which leads to the system on a geodesic curve (*α* ¼ 3 and *α* ¼ 2), only *α* ¼ 3 makes that the macroscopic potential and entropy have the same functional dependence. It is interesting to note that for this choice of *α*, the manifold in the coordinates ð Þ *θ*1, *θ*<sup>2</sup> is flat.

In the study of processes, the speed at which they occur is important. In this sense we will study the rate of change of potential:

$$\begin{split} \dot{\boldsymbol{\nu}} &= \frac{\mathbf{x}\_0^2}{4D\tau^2} \text{sech}\left(\frac{t}{2\tau}\right) + \frac{1}{2\tau} \left[\coth\left(\frac{t}{\tau}\right) - \mathbf{1}\right] = \\ &= \frac{1}{2\tau} \text{sech}\left(\frac{t}{2\tau}\right) + \frac{1}{2\tau} \left[\coth\left(\frac{t}{\tau}\right) - \mathbf{1}\right] \ge \mathbf{0}, \end{split} \tag{36}$$

and the entropy

$$\dot{H} = \frac{1}{2\pi} \left[ \coth\left(\frac{t}{\pi}\right) - 1 \right] \ge 0. \tag{37}$$

The equal sign in Eqs. (36) and (37) corresponds to the steady state [14]. In this regard, we associate the asymptotic behavior with the steady state, and we can indicate that the entropy describes the steady state of an optimal evolution.

From the perspective of the probability density function (PDF), when the diffusion coefficient takes small values, the distribution (Eq. (14)) diffuses from a uniform state to a sharp concentrated state, that is, from a stable state to an unstable state [15]. Although in this chapter we have not studied the behavior of the curvature tensor in the ð Þ *μ*, *σ* coordinates, these instabilities observed from the perspective of the PDF can be expressed in the singularity of the curvature tensor *R*ð Þ <sup>0</sup> <sup>1212</sup>ð Þ *μ*, *σ* . An example of this behavior is the formation of density patterns in the populations of self-propelled bacteria whose mobility can be investigated in terms of the diffusion coefficient *D* [5]. We also observe from Eq. (35) that for *D* ! 0, the entropy and the macroscopic potential have a singular behavior. The behavior that we observe both the tensor *R*ð Þ <sup>0</sup> <sup>1212</sup>ð Þ *μ*, *σ* and the potential *ψ* allows us to conjecture that the instabilities of the system can also be observed in the singularities or discontinuities of the macroscopic potential.

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

#### **5. Discussion and perspectives**

*<sup>ψ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> 0 2*Dτ* þ 1 2

*<sup>H</sup>* <sup>¼</sup> <sup>1</sup> 2 þ 1 2

entropy describes the asymptotic behavior of an optimal evolution:

*<sup>ψ</sup>* <sup>¼</sup> *<sup>H</sup>* <sup>¼</sup> <sup>1</sup>

If we think that in the coordinates ð Þ *θ*1, *θ*<sup>2</sup> , with *α* ¼ 3, the system evolves on a

potential *ψ* and the entropy *H* tend to the same asymptotic value. In other words the

In this last result, it can be seen that the asymptotic behavior depends fundamentally on the transport coefficient *D*. In this sense, the relevant variable in the description of the process is the diffusion coefficient *D*. It is interesting to note that the relations (Eq. (21)) are not invariant when we change *t* by �*t*. In this context,

While in the coordinates ð Þ *θ*1, *θ*<sup>2</sup> are two *α*-connections which leads to the system on a geodesic curve (*α* ¼ 3 and *α* ¼ 2), only *α* ¼ 3 makes that the macroscopic potential and entropy have the same functional dependence. It is interesting to note that for this choice of *α*, the manifold in the coordinates ð Þ *θ*1, *θ*<sup>2</sup> is flat. In the study of processes, the speed at which they occur is important. In this

> 2*τ* � � þ 1 <sup>2</sup>*<sup>τ</sup>* coth *<sup>t</sup>*

<sup>2</sup>*<sup>τ</sup>* coth *<sup>t</sup>*

regard, we associate the asymptotic behavior with the steady state, and we can indicate that the entropy describes the steady state of an optimal evolution.

*τ* � � � 1 h i

The equal sign in Eqs. (36) and (37) corresponds to the steady state [14]. In this

From the perspective of the probability density function (PDF), when the diffusion coefficient takes small values, the distribution (Eq. (14)) diffuses from a uniform state to a sharp concentrated state, that is, from a stable state to an unstable state [15]. Although in this chapter we have not studied the behavior of the curvature tensor in the ð Þ *μ*, *σ* coordinates, these instabilities observed from the perspective of the PDF can be expressed in the singularity of the curvature tensor *R*ð Þ <sup>0</sup>

An example of this behavior is the formation of density patterns in the populations of self-propelled bacteria whose mobility can be investigated in terms of the diffusion coefficient *D* [5]. We also observe from Eq. (35) that for *D* ! 0, the entropy and the macroscopic potential have a singular behavior. The behavior that we observe both

of the system can also be observed in the singularities or discontinuities of the

<sup>1212</sup>ð Þ *μ*, *σ* and the potential *ψ* allows us to conjecture that the instabilities

sech *<sup>t</sup>* 2*τ* � � þ 1 <sup>2</sup>*<sup>τ</sup>* coth *<sup>t</sup>*

*<sup>H</sup>*\_ <sup>¼</sup> <sup>1</sup>

*τ* � � � 1 h i

*τ* � � � 1 h i ¼

≥0*:* (37)

(36)

<sup>1212</sup>ð Þ *μ*, *σ* .

≥0,

<sup>1212</sup>ð Þ¼ *<sup>θ</sup>*1, *<sup>θ</sup>*<sup>2</sup> 0), we have *<sup>τ</sup>* <sup>¼</sup> *<sup>x</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> ln ffiffiffiffiffiffiffiffiffiffiffi

and

geodesic of a flat manifold (*R*ð Þ<sup>3</sup>

*Advances on Tensor Analysis and Their Applications*

the evolution of the system is irreversible.

sense we will study the rate of change of potential:

*<sup>ψ</sup>*\_ <sup>¼</sup> *<sup>x</sup>*<sup>2</sup> 0 <sup>4</sup>*Dτ*<sup>2</sup> sech *<sup>t</sup>*

> ¼ 1 2*τ*

and the entropy

the tensor *R*ð Þ <sup>0</sup>

**80**

macroscopic potential.

ln 2½ � *πDτ* (33)

ln 2½ � *πDτ :* (34)

<sup>2</sup>*πD<sup>τ</sup>* <sup>p</sup> *:* (35)

<sup>0</sup>*=D*, and both the

In this chapter we have studied the PDF (Eq. (14)) that is the fundamental solution of the Fokker-Planck equation associated with the UO process. Our main interest was relating the geometric aspects of the process with the steady-state behavior. In our analysis we used the theoretical framework of the statistical manifold *M* with *α*-connections for two different coordinates ð Þ *μ*, *σ* and ð Þ *θ*1, *θ*<sup>2</sup> [4, 7]. In the first case, there exist two interesting values of *α*, namely, *α* ¼ �1 and *α* ¼ 0, for which the process evolves on a geodesic of space ð Þ *μ*, *σ* with different values of *τ*. However, in the search for a simpler geometric construction, we find that for the coordinates ð Þ *θ*1, *θ*<sup>2</sup> , we can define a macroscopic potential *ψ θ*ð Þ 1, *θ*<sup>2</sup> , and the values of *α* that lead to the system evolving on a geodesic curve are *α* ¼ 3 and *α* ¼ 2. In our study we show that the connection *α* ¼ �1 for the coordinates ð Þ *μ*, *σ* corresponds to the connection *α* ¼ 3 for the coordinates ð Þ *θ*1, *θ*<sup>2</sup> in the sense that both connections lead to a flat curvature and the same relationship between the parameters of the system. An important consequence of this behavior is that when the system evolves over geodesics, the macroscopic potential *ψ* and entropy *H* have the same functional dependence in the steady state. If we think of the geodesic curve as an optimal trajectory, our results allow us to conjecture that the entropy describes the steady state of an optimal evolution in a flat manifold.

Additionally and poorly developed in this chapter is the use of geometric aspects in the study of instabilities in nonequilibrium system. In equilibrium thermodynamics this information is contained in the scalar curvature of the manifold of equilibrium states [2]. For nonequilibrium problems, the instabilities are associated with the singularities or discontinuities of the curvature tensor [5, 15]. In the case of diffusive problems, the instabilities can be found by studying the singularities of *R*ð Þ <sup>0</sup> <sup>1212</sup>ð Þ *μ*, *σ* . In terms of diffusivity, we see that as it decreases, the system has different macroscopic behavior. An example of this behavior is the formation of density patterns in the populations of self-propelled bacteria whose mobility can be investigated in terms of the diffusion coefficient *D* [12]. From the perspective of the macroscopic potential *ψ*, the instabilities can be associated to singularities or discontinuities of *ψ*. In this sense, from a wider point of view, we consider that the potential *ψ* represents an alternative way to study the phase transitions in nonequilibrium systems.

#### **Acknowledgements**

This work was supported by the Universidad Nacional de La Plata, Argentina (UNLP). AM, LL, and CR are professors at the UNLP.

*Advances on Tensor Analysis and Their Applications*

**References**

5135-5145

[1] Ruppeiner G. Thermodynamics: A Riemannian geometric model. Physical

*DOI: http://dx.doi.org/10.5772/intechopen.92274*

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process*

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[15] Obata T, Hara H, Endo K. Differential geometry of non-equilibrium processes. Physical Review A. 1992;**45**:6997-7001

[12] Cates M, Marenduzzo D,

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[2] Ruppeiner G. Riemannian geometric approach to critical points: General theory. Physical Review E. 1998;**57**:

[3] Seifert U. Entropy production along a stochastic trajectory and an integral fluctuation theorem. Physical Review

[4] Amari S. Differential-geometrical methods in statistics. In: Lecture Notes in Statistics. Vol. 28. Heindelberg:

[5] Melgarejo A, Langoni L, Ruscitti C. Instability in bacterial populations and the curvature tensor. Physica A: Statistical Mechanics and its Applications. 2016;**458**:189-193

[6] Uhlenbeck G, Ornstein L. On the theory of the Brownian motion. Physics

[7] Ruscitti C, Langoni L, Melgarejo A.

Uhlenbeck–Ornstein process. Physica

[8] Brody D, Hook D. Information geometry in vapour-liquid equilibrium. Journal of Physics A: Mathematical and

[9] Amari S. Differential geometry of curved exponential families-curvatures

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#### **Author details**

Claudia B. Ruscitti1†, Laura B. Langoni2† and Augusto A. Melgarejo<sup>2</sup> \*†

1 Facultad de Ciencias Exactas, Departamento de Matemática, UNLP, La Plata, Argentina

2 Facultad de Ingeniería, Departamento de Ciencias Básicas, UNLP, La Plata, Argentina

\*Address all correspondence to: augusto.melgarejo@ing.unlp.edu.ar

† These authors are contributed equally.

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Differential Geometry and Macroscopic Descriptions in Nonequilibrium Process DOI: http://dx.doi.org/10.5772/intechopen.92274*

#### **References**

[1] Ruppeiner G. Thermodynamics: A Riemannian geometric model. Physical Review A. 1979;**20**:1608-1613

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[3] Seifert U. Entropy production along a stochastic trajectory and an integral fluctuation theorem. Physical Review Letters. 2005;**95**:040602

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**Author details**

Argentina

Argentina

**82**

Claudia B. Ruscitti1†, Laura B. Langoni2† and Augusto A. Melgarejo<sup>2</sup>

\*Address all correspondence to: augusto.melgarejo@ing.unlp.edu.ar

† These authors are contributed equally.

*Advances on Tensor Analysis and Their Applications*

provided the original work is properly cited.

1 Facultad de Ciencias Exactas, Departamento de Matemática, UNLP, La Plata,

2 Facultad de Ingeniería, Departamento de Ciencias Básicas, UNLP, La Plata,

© 2020 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

\*†

Section 4

Advanced Topics of

Tensor Analysis

**85**

Section 4
