**2. Generalized variational principle for dissipative hydrodynamics**

### **2.1 Hamilton's variational principle**

The non-dissipative case of Hamilton's variational principle can be formulated for a continuous medium in the form of the extreme condition for the action functional 0 δ*S* = :

$$S = \bigwedge\_{t\_1}^{t\_2} dt \bigwedge\_V d\vec{r} L \tag{1}$$

which is an integral over the time interval ( <sup>1</sup>*t* , <sup>2</sup>*t* ) and the initial volume *V* of a given mass of a continuum medium in terms of Lagrangian's coordinates. From the principles of particle mechanics the Lagrangian density *L* is represented as the difference between the kinetic *K* and potential *U* energies:

$$L(\dot{\vec{\boldsymbol{\mu}}}, \nabla \vec{\boldsymbol{\mu}}) = K(\dot{\vec{\boldsymbol{\mu}}}) - L(\nabla \vec{\boldsymbol{\mu}}) \,. \tag{2}$$

Expression (2) implies that the Lagrangian can be considered as a function of the velocities of the displacements *<sup>u</sup> u t* ∂ = ∂ and deformations ∇ = *u div u*( ) .

The motion equations derived from variational principles (1), (2) have the following form

$$
\frac{d}{dt}\frac{\partial L}{\partial \dot{\vec{u}}} + \nabla \frac{\partial L}{\partial \nabla \vec{u}} = 0 \; \text{}.\tag{3}
$$

In the simplest case, when the kinetic and potential energies are determined by the quadratic forms

$$2\operatorname{2K}(\dot{\vec{\mu}}^2) = \rho\_0 \dot{\vec{\mu}}^2, \quad 2\operatorname{2I} = \lambda \varepsilon\_{\text{ll}}^2 + 2\mu \varepsilon\_{\text{ik}}^2, \quad \varepsilon\_{\text{ik}} = \frac{1}{2} \left( \frac{\partial u\_i}{\partial \mathbf{x}\_k} + \frac{\partial u\_k}{\partial \mathbf{x}\_i} \right) \tag{4}$$

the well-known equation of motion for an elastic medium (Landau & Lifshitz, 1972) can be derived:

$$
\rho\_0 \frac{d}{dt} \dot{\vec{u}} - \mu \Delta \vec{u} - (\mathcal{k} + \mu) \nabla(\nabla \vec{u}) = 0 \tag{5}
$$

where ρ0 is the density of the medium, and λ and μare the Lamé's constants.

### **2.2 Onsager's variational principle**

36 Hydrodynamics – Advanced Topics

principle of symmetry of kinetic coefficients (Landau & Lifshitz, 1964). There is an established opinion that for a dissipative system it is impossible to formulate the variational principle analogously to the least action principle of Hamilton (Landau & Lifshitz, 1964). At the same time there are successful approaches (Onsager, 1931a, 1931b; Glensdorf & Prigogine, 1971; Biot, 1970; Gyarmati, 1970; Berdichevsky, 2009) in which the variational principles for heat conduction theory and for irreversible thermodynamics are applied to account explicitly for the dissipation processes. In spite of many attempts to formulate a variational principle for dissipative hydrodynamics or continuum mechanics (see for example (Onsager, 1931a, 1931b; Glensdorf & Prigogine, 1971; Biot, 1970; Gyarmati, 1970; Berdichevsky, 2009) and references inside) a consistent and predictive formulation is still absent. Therefore, there are good reasons to attempt to formulate the generalized Hamilton's variational principle for dissipative systems, which argues against its established opposition (Landau & Lifshitz, 1964). Thus the objective of the chapter is a new formulation of the generalized variational principle (GVP) for dissipative hydrodynamics (continuum mechanics) as a direct combination of Hamilton's and Osager's variational principles. The first part of the chapter is devoted to formulation of the GVP by itself with application to the well-known Navier-Stokes hydrodynamical system for heat conductive fluid. The second part of the chapter is devoted to the consistent introduction of viscous terms into the equation of fluid motion on the basis of the GVP. Two different approaches are considered. The first one dealt with iternal degree of freedom described in terms of some internal parameter in the framework of Mandelshtam – Leontovich approach (Mandelshtam & Leontovich, 1937). In the second approach the rotational degree of freedom as independent variable appears additionally to the mean mass displacement field. For the dissipationless case this approach leads to the well-known Cosserat continuum (Kunin, 1975; Novatsky, 1975; Erofeev, 1998). When dissipation prevails over angular inertion this approach describes local relaxation of angular momentum and corresponds to the sense of internal parameter. Finally, it is shown that the nature of viscosity phenomenon can be interpreted

as relaxation of angular momentum of material points on the kinetic level.

**2.1 Hamilton's variational principle** 

kinetic *K* and potential *U* energies:

*u*

*t* ∂ = ∂ 

of the displacements *<sup>u</sup>*

**2. Generalized variational principle for dissipative hydrodynamics** 

The non-dissipative case of Hamilton's variational principle can be formulated for a continuous medium in the form of the extreme condition for the action functional 0

2

*t*

1

*t V*

which is an integral over the time interval ( <sup>1</sup>*t* , <sup>2</sup>*t* ) and the initial volume *V* of a given mass of a continuum medium in terms of Lagrangian's coordinates. From the principles of particle mechanics the Lagrangian density *L* is represented as the difference between the

Expression (2) implies that the Lagrangian can be considered as a function of the velocities

and deformations ∇ = *u div u*( ) .

δ*S* = :

*S dt drL* <sup>=</sup> , (1)

*Lu u Ku U u* (, ) () ( ) ∇= − ∇ . (2)

If we consider quasi-equilibrium systems, then the Onsager's variational principle for least energy dissipation can be formulated (Onsager, 1931a, 1931b). This principle is based on the symmetry of the kinetic coefficients and can be formulated as the extreme condition for the functional constructed as the difference between the rate of increase of entropy, *s* , and the dissipation function, *D* . Here the entropy *s* is considered as a function of some thermodynamic relaxation process α , and the dissipation function *D* as a function of the rate of change of α, i.e.

$$\left. \delta\_{\dot{\alpha}} \left[ \dot{s}(\alpha) - D(\dot{\alpha}) \right] \right| = 0 \tag{6}$$

The kinetic equation can then be derived from variational principle (6) to describe the relaxation of a thermodynamic system to its equilibrium state, i.e.:

$$\frac{d}{dt}s(\alpha) = 2D(\dot{\alpha})\,. \tag{7}$$

The above equation satisfies strictly the symmetry principle for the kinetic coefficients (Landau & Lifshitz, 1986).

#### **2.3 Variational principle for mechanical systems with dissipation**

As was mentioned above, the generalization of the equation of motion (3) in the presence of dissipation is obtained by introducing the derivative of the dissipation function with respect to the velocities into the right hand side of the equation (3). Therefore, in accordance with Onsager's symmetry principle for the kinetic coefficients (Landau & Lifshitz, 1964) we have

$$\frac{d}{dt}\frac{\partial L}{\partial \dot{\vec{u}}} + \nabla \frac{\partial L}{\partial \nabla \vec{u}} = -\frac{\partial D}{\partial \dot{\vec{u}}}\tag{8}$$

Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity

considered as a basis for variation rule of integral term in Lagrangian.

η

velocities, i.e.:

4 3 ς ′ ′ + η

entropy , , , *v Ps* ρ

field *u*

variables.

case.

density *P s const* (, ) ρ

Navier–Stokes equation:

**2.4 Independent variables** 

*d*

*dt*

respectively, from the constants in (12).

 λμ

hydrodinamics equations and look at variables for their description.

Velocity by definition is a time derivative from displacement *v u* <sup>=</sup>

 ρ

ρ

where the shear and volume viscosities,

through divergence of the displacement field

implying in the following set of variables: , , , , *v PsT*

ρ

continuity equation leads to relation

density and temperature *P Ts T* ( , ), ( , )

of displacements and temperatures: *u T*,

determined by the divergence of the field *uT*

from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics 39

not singular in the point *t t* ′ = , then its contribution can be neglected in this point in comparison with singular contribution from delta-function. The presented arguments can be

In particular, if the dissipation function is considered as a quadratic form of the deformation

2( ) *ik l*

*uu u D u*

∂∂ ∂ ∇= + + ′ ′ ∂∂ ∂

then the derived equation of motion with account of (4) corresponds to the linearized

<sup>0</sup> ( ) () ( ) <sup>3</sup>

When GVP is formulated in the form (9) we need to determine variables in which terms the Lagrange's function has to be expressed. To answer on this question let's return to the

In absence of dissipation, as it easy to see, these variables are velocity, density, pressure and

material point, hence a pressure can be considered, for example, as a function of solely

its volume. Hence variation of density can be expressed in terms of variation of volume or

0

ρ ρ ρ ρ

can be considered as the principal hydrodinamical variable for the dissipationless

In the presence of dissipation, the hydrodynamic equations also involve the temperature *T* ,

Further, we will adopt the idea of Biot (Biot, 1970), and introduce some vector field *uT*

(some vector potential), called the heat displacement, as independent variable instead temperature, so that the relative deviation of temperature *T* from its equilibrium state *T*0 is

> 0(1 ) *T T divu* = −θ*T*

ρ

. For the dissipationless case the entropy holds to be constant for given

= . The density of the given mass of continuum is expressed in terms of

 λ

> η and ς

*u u uu u*

− + Δ − ∇∇ = Δ + + ∇∇

 η

2 2

, (12)

η

respectively are given by

<sup>=</sup> ( ) *divu* . In particular, linearization of the

. Thus, the displacement

= − (1 ) *divu* (14)

. If pressure and entropy depend on

(15a)

in accordance to the state equation, then the fields

can be considered as the principal hydrodynamical

. Namely in analogy with (14)

η

′ / 2 and

, (13)

 ς

*ki l*

 ς

*xx x*

Now it is possible to show, that the equation of motion can be derived in the form of equation (8) if Hamilton's variational principle is adapted with the following form of the Lagrangian function:

$$L(\dot{\vec{\mu}}, \nabla \vec{\mu}) = K(\dot{\vec{\mu}}) - L(\nabla \vec{\mu}) - \int\_0^t D(\dot{\vec{\mu}}) dt' \,\tag{9}$$

where the time integral of the dissipation function is introduced into equation (2). The initial time in integral (9) denoted for simplicity equal to 0 corresponds to the time 1*t* in functional (1). It needs, however, to pay attention that at variation of dissipative term in such approach an additional item appears, which has to be neglected by hands. Indeed, variation of the last term in (9) leads us to result

$$\delta \oint\_{0} D(\dot{\vec{u}}) dt' = \int\_{0}^{t} \frac{\partial D(\dot{\vec{u}})}{\partial \dot{\vec{u}}} \delta \dot{\vec{u}} \, dt' = \int\_{0}^{t} \frac{d}{dt'} \left( \frac{\partial D(\dot{\vec{u}})}{\partial \vec{u}} \delta \vec{u} \right) dt' - \int\_{0}^{t} \frac{d}{dt'} \left( \frac{\partial D(\dot{\vec{u}})}{\partial \dot{\vec{u}}} \right) \delta \vec{u} \, dt' \tag{10a}$$

If to neglect by the last item in this expression

$$\delta \int\_0^\dagger D(\dot{\vec{u}}(t'))dt' = \frac{\partial D(\dot{\vec{u}})}{\partial \dot{\vec{u}}} \delta \vec{u}(t) - \int\_0^\dagger \frac{d}{dt'} \left(\frac{\partial D(\dot{\vec{u}})}{\partial \dot{\vec{u}}}\right) \delta \vec{u} dt' = \frac{\partial D(\dot{\vec{u}})}{\partial \dot{\vec{u}}} \delta \vec{u}(t) \,, \tag{10b}$$

then the result gives us the same term *D u*( ) *u* ∂ ∂ , which we need artificially introduce in the

motion equation (8) for account of dissipation. From the one hand this approach can be considered as some rule at variation of integral term, because it leads us to the required form of the motion equation (8). From the other hand the following supporting basement can be proposed. Variation of action containing all terms in Lagrangian (9) with account of initial and boundary conditions can be written in the form

$$\int\_{t\_1}^{t\_2} dt \int dV \left[ \left( -\frac{d}{dt} \frac{\partial K(\dot{\vec{u}})}{\partial \dot{\vec{u}}} + \nabla \frac{\partial L(\nabla \vec{u})}{\partial \nabla \vec{u}} - \frac{\partial D(\dot{\vec{u}})}{\partial \dot{\vec{u}}} \right) \delta \vec{u} + \int\_0^t \frac{d}{dt'} \left( \frac{\partial D(\dot{\vec{u}})}{\partial \dot{\vec{u}}} \right) \delta \vec{u} dt' \right] = \tag{11a}$$

It is seen from (11a) that the required form of the motion equation with dissipation arises due to zero value of coefficient at arbitrary variation of the displacement field δ*u* . The last additional item, containing variation δ*u* under integral, prevent to the strict conclusion in the given case. Nevertheless, if to rewrite the first term in (11a) in the same integral form as the last term

$$=\int\_{t\_1}^{t\_2} dt \int\_0 dV \int\_0 dt' \left\{ \mathcal{S}(t - t') \left( -\frac{d}{dt'} \frac{\partial \mathcal{K}(\dot{\vec{u}})}{\partial \dot{\vec{u}}} + \nabla \frac{\partial \mathcal{L}(\nabla \vec{u})}{\partial \nabla \vec{u}} - \frac{\partial \mathcal{D}(\dot{\vec{u}})}{\partial \dot{\vec{u}}} \right) + \frac{d}{dt'} \left( \frac{\partial D(\dot{\vec{u}})}{\partial \dot{\vec{u}}} \right) \right\} \delta \vec{u} \tag{11b}$$

then due to the same reason of arbitrary variation δ*u* the multiplier in brackets at this variation has to be equal to zero. It is possible to see now, that, if the function *d Du*( ) *dt u* ∂ ′ <sup>∂</sup> is not singular in the point *t t* ′ = , then its contribution can be neglected in this point in comparison with singular contribution from delta-function. The presented arguments can be considered as a basis for variation rule of integral term in Lagrangian.

In particular, if the dissipation function is considered as a quadratic form of the deformation velocities, i.e.:

$$2D(\nabla \dot{\overline{u}}) = \eta' \left(\frac{\partial \dot{u}\_i}{\partial x\_k} + \frac{\partial \dot{u}\_k}{\partial x\_i}\right)^2 + \zeta' \left(\frac{\partial \dot{u}\_l}{\partial x\_l}\right)^2,\tag{12}$$

then the derived equation of motion with account of (4) corresponds to the linearized Navier–Stokes equation:

$$
\rho\_0 \frac{d}{dt} \dot{\vec{u}} - (\dot{\mathcal{A}} + \mu) \Delta \vec{u} - \mathcal{A} \nabla(\nabla \vec{u}) = \eta \Delta \dot{\vec{u}} + \left(\boldsymbol{\mathcal{L}} + \frac{\eta}{3}\right) \nabla(\nabla \dot{\vec{u}})\,,\tag{13}
$$

where the shear and volume viscosities, η and ς respectively are given by η′ / 2 and 4 3 ς ′ ′ + ηrespectively, from the constants in (12).

### **2.4 Independent variables**

38 Hydrodynamics – Advanced Topics

Now it is possible to show, that the equation of motion can be derived in the form of equation (8) if Hamilton's variational principle is adapted with the following form of the

(, ) () ( ) ()

*L u u K u U u D u dt* ∇= − ∇− ′

where the time integral of the dissipation function is introduced into equation (2). The initial time in integral (9) denoted for simplicity equal to 0 corresponds to the time 1*t* in functional (1). It needs, however, to pay attention that at variation of dissipative term in such approach an additional item appears, which has to be neglected by hands. Indeed, variation of the last

> ( ) ( ) ( ) ( ) *tt t <sup>t</sup> Du d Du d Du D u dt udt u dt udt uu u dt dt*

> ∂∂ ∂ ′′ ′ ′ == − ∂∂ ∂ ′ ′

( ) () () ( ( )) ( ) ( ) *t t Du d Du Du D u t dt u t udt u t*

 ∂ ∂∂ ′ ′ =− ≈′ ∂ ∂∂ ′ 

> *u* ∂ ∂

motion equation (8) for account of dissipation. From the one hand this approach can be considered as some rule at variation of integral term, because it leads us to the required form of the motion equation (8). From the other hand the following supporting basement can be proposed. Variation of action containing all terms in Lagrangian (9) with account of

 

*u uu dt*

δ

(11a)

under integral, prevent to the strict conclusion in

(11b)

00 0 0

 δ

<sup>1</sup> 0

() ( ) () ( ) *<sup>t</sup> <sup>t</sup>*

due to zero value of coefficient at arbitrary variation of the displacement field

δ*u*

*d Ku U u Du d Du dt dV <sup>u</sup> udt dt u uu u dt*

∂ ∂∇ ∂ <sup>∂</sup> − +∇ − + ′ <sup>=</sup> ∂ ∂∂ ∂∇ ′

It is seen from (11a) that the required form of the motion equation with dissipation arises

the given case. Nevertheless, if to rewrite the first term in (11a) in the same integral form as

*d Ku U u Du d Du dt dV dt t t <sup>u</sup>*

∂ ∂∇ ∂ ∂ <sup>=</sup> ′ ′ − − +∇ − + ′ ′ ∂ ∂∂ ∂∇

variation has to be equal to zero. It is possible to see now, that, if the function *d Du*( )

() ( ) () () ( )

*dt u uu u dt*

δ*u*

δδδ

0 0

initial and boundary conditions can be written in the form

0

*t*

, (9)

(10a)

δδ

, which we need artificially introduce in the

 δ

> δ*u*

 δ

*dt u* ∂ ′ ∂

 is

the multiplier in brackets at this

. The last

, (10b)

 δ

Lagrangian function:

term in (9) leads us to result

If to neglect by the last item in this expression

then the result gives us the same term *D u*( )

δ

2

*t*

2

*t*

<sup>1</sup> 0

*t t*

the last term

additional item, containing variation

δ

then due to the same reason of arbitrary variation

When GVP is formulated in the form (9) we need to determine variables in which terms the Lagrange's function has to be expressed. To answer on this question let's return to the hydrodinamics equations and look at variables for their description.

In absence of dissipation, as it easy to see, these variables are velocity, density, pressure and entropy , , , *v Ps* ρ . For the dissipationless case the entropy holds to be constant for given material point, hence a pressure can be considered, for example, as a function of solely density *P s const* (, ) ρ = . The density of the given mass of continuum is expressed in terms of its volume. Hence variation of density can be expressed in terms of variation of volume or through divergence of the displacement field ρ ρ <sup>=</sup> ( ) *divu* . In particular, linearization of the continuity equation leads to relation

$$
\rho = \rho\_0 (1 - div\vec{u}) \tag{14}
$$

Velocity by definition is a time derivative from displacement *v u* <sup>=</sup> . Thus, the displacement field *u* can be considered as the principal hydrodinamical variable for the dissipationless case.

In the presence of dissipation, the hydrodynamic equations also involve the temperature *T* , implying in the following set of variables: , , , , *v PsT* ρ . If pressure and entropy depend on density and temperature *P Ts T* ( , ), ( , ) ρ ρ in accordance to the state equation, then the fields of displacements and temperatures: *u T*, can be considered as the principal hydrodynamical variables.

Further, we will adopt the idea of Biot (Biot, 1970), and introduce some vector field *uT* (some vector potential), called the heat displacement, as independent variable instead temperature, so that the relative deviation of temperature *T* from its equilibrium state *T*0 is determined by the divergence of the field *uT* . Namely in analogy with (14)

$$T = T\_0 \left(1 - \theta \, d\bar{v} \, \vec{u}\_T\right) \tag{15a}$$

Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity

με

κ , α and β

= 0 ) to the linearized traditional system of hydrodynamics equations:

 λμα

> θ

or with substitution of expression (13) instead of the temperature terms:

με

1972):

displacements

equations.

μ

The meanings of the coefficients

*dt* ρ

2

*dt*

ρ

heat conductivity coefficient, and

ς

viscosity

η= 0 and

 μ

β

**2.6 Comparison with the system of hydrodynamics equations** 

2

 ρ

ρ

from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics 41

Taking into account that the kinetic energy is given by quadratic form (4), the free energy is given by its usual expression for thermo-elasticity in quadratic form (Landau & Lifshitz,

θ

− − ∇ = ++ +

() () 2 2 <sup>2</sup> 2( , ) 2 <sup>2</sup> *Fuu* ∇∇ = + + ∇ + ∇ *T ik ll T*

The dissipation function is the square of the difference between the mean mass and the heat

<sup>2</sup> 2 (, ) ( ) *Duu u u T T* = − β

the next section by comparison with the classical Navier-Stokes hydrodynamical system of

In this case the motion equations for the mean displacement field and for the temperature field derived on the basis of the generalized variational principle are just equivalent (at

> 0 0 ( ) ( ) ( )/( ) *<sup>d</sup> u u u TT*

> > 0 0

Coefficients of the quadratic forms in equations (19) and (20) can be determined by comparison between the system of equation (21) and (22) and the linearized system of

hydrodynamics equations (Landau & Lifshitz, 1986) considering the variables *u T*,

ρ ρ= −∇ (1 ) *u*

0

*d u cu T u <sup>u</sup>*

0 0 <sup>0</sup> 0 *<sup>V</sup> dT C Tu T*

the structure of equations (21), (22) nearly coincides with the second (24) and the third (25) equations in the system of hydrodynamics equations (Landau & Lifshitz, 1986). The only

 η

 κ+ ∇−Δ =′

 − Δ =− ∇ + Δ + + ∇ ∇ 

0 00 0 <sup>2</sup> 3

 ρα

where 0*c* is the isothermal sound velocity, *CV* - the heat capacity at constant volume,

 ρα

*dt*

α

 κ  ακ

 αθ

 λε κ

2( , ) 2 2 *ik ll ll TT TT F uT*

 λε κ

2 2 2 0 0

0 0

 αε *u u ll T* , (19b)

αε

 θ

. (20)

 θ− Δ − + + ∇∇ = + ∇ (21)

( ) *TT u T T u* − ∇ − Δ = Δ∇ . (22)

in quadratic forms (19), (20) will be defined in

:

κthe

, (23)

, (24)

. (25)

( )

the thermal expansion coefficient. In the absence of

η

 ζ

= 0 , which was not taken into account in the dissipation function (20),

*T T*

, (19a)

where θ is some dimensionless constant which is specially introduced in definition (15a) for simplification of the expression for the dissipation function. Thus, the divergence of the heat displacement field *uT* determines temperature deviation from its equilibrium level

$$\frac{T - T\_0}{T\_0} = -\theta \nabla \vec{u}\_T \ . \tag{15b}$$

#### **2.5 Generalized variational principle (GVP) for dissipative hydrodynamics**

The above example (12), (13) of the derivation of the equation of motion for dissipative systems on the basis of Hamilton's variational principle with the Lagrange's function (9) suggests the possibility of formulating a generalized variational principle for dissipative hydrodynamical systems. This formulation can be obtained by a simple combination of Hamilton's variational principle (eqs. (1) and (2)) and Onsager's variational principle (eq. (6)), if the latter is integrated over time and multiplied by a temperature term (Maximov , 2008, 2010, originally formulated by Maximov , 2006). The Lagrangian density in this case can be written in the following form:

$$L = K - E + T\left[s - \bigwedge\limits\_{0}^{t} Ddt'\right] = K - F - T\left[\underset{0}{Ddt'}\right]Ddt'\tag{16}$$

where *E* and *F* are the internal energy (potential for the dissipationless case) and the free energy respectively. For the non-dissipative case, the Lagrangian depends on the time and spatial derivatives of the mean mass displacement field *u* , which is a basic independent variable in this formulation. For the dissipative case, the temperature should be considered as an additional independent variable for a complete description. Hence, a free energy and dissipation function should also depend on the temperature variations. But temperature by itself is not a convenient variable here. Instead it is more convenient to consider the heat displacements *uT* , introduced in previous section, of which the divergence will give us temperature.

In this case the generalized Lagrangian can be written in the following form:

$$L(\dot{\vec{\boldsymbol{\mu}}}, \nabla \vec{\boldsymbol{\mu}}, \nabla \vec{\boldsymbol{\mu}}\_{\Gamma}) = K(\dot{\vec{\boldsymbol{\mu}}}) - F(\nabla \vec{\boldsymbol{\mu}}, \nabla \vec{\boldsymbol{\mu}}\_{\Gamma}) - T\_0 \int\_0^t D(\dot{\vec{\boldsymbol{\mu}}}, \dot{\vec{\boldsymbol{\mu}}}\_{\Gamma}) dt' \,. \tag{17}$$

It is important to note here that the opportunity to formulate the variational principle for a dissipative system arises due to the energy conservation for two interacting fields: the mean mass displacement *u* and the heat displacement *uT* . The dissipation function only plays a role in the transformation rate between these fields.

In this way the motion equations derived by variation of action with the Lagrangian (17), can be expressed in the following forms

$$\begin{aligned} \frac{d}{dt} \frac{\partial \mathcal{K}}{\partial \vec{\bar{u}}} - \nabla \frac{\partial F}{\partial \nabla \vec{u}} &= -T\_0 \frac{\partial D}{\partial \vec{\bar{u}}} \; \prime \\\\ T\_0 \frac{\partial D}{\partial \vec{\bar{u}}\_T} - \nabla \frac{\partial F}{\partial \nabla \vec{\bar{u}}\_T} &= 0 \; \cdot \end{aligned} \tag{18}$$

Taking into account that the kinetic energy is given by quadratic form (4), the free energy is given by its usual expression for thermo-elasticity in quadratic form (Landau & Lifshitz, 1972):

$$2\operatorname{F}(\nabla \vec{u}, T) = 2\mu \varepsilon\_{\text{ik}}^2 + \lambda \varepsilon\_{\text{ll}}^2 + \mathfrak{k} \left(\frac{T - T\_0}{\theta T\_0}\right)^2 + 2\mathcal{dc}\_{\text{ll}} \left(\frac{T - T\_0}{\theta T\_0}\right) \tag{19a}$$

or with substitution of expression (13) instead of the temperature terms:

40 Hydrodynamics – Advanced Topics

for simplification of the expression for the dissipation function. Thus, the divergence of the

The above example (12), (13) of the derivation of the equation of motion for dissipative systems on the basis of Hamilton's variational principle with the Lagrange's function (9) suggests the possibility of formulating a generalized variational principle for dissipative hydrodynamical systems. This formulation can be obtained by a simple combination of Hamilton's variational principle (eqs. (1) and (2)) and Onsager's variational principle (eq. (6)), if the latter is integrated over time and multiplied by a temperature term (Maximov , 2008, 2010, originally formulated by Maximov , 2006). The Lagrangian density in this case

> *L K E T s Ddt K F T Ddt* = −+ − = −− ′ ′

where *E* and *F* are the internal energy (potential for the dissipationless case) and the free energy respectively. For the non-dissipative case, the Lagrangian depends on the time and

variable in this formulation. For the dissipative case, the temperature should be considered as an additional independent variable for a complete description. Hence, a free energy and dissipation function should also depend on the temperature variations. But temperature by itself is not a convenient variable here. Instead it is more convenient to consider the heat

(, , ) () ( , ) (, )

*L u u u K u F u u T D u u dt T TT* ∇∇ = − ∇∇ − ′

It is important to note here that the opportunity to formulate the variational principle for a dissipative system arises due to the energy conservation for two interacting fields: the mean

In this way the motion equations derived by variation of action with the Lagrangian (17),

*dK F D <sup>T</sup> dt u u u* ∂∂ ∂ −∇ =− ∂ ∂ ∂∇ ,

<sup>0</sup> 0 *T T D F <sup>T</sup> u u* ∂ ∂ −∇ = <sup>∂</sup> ∂∇ .

0

In this case the generalized Lagrangian can be written in the following form:

and the heat displacement *uT*

role in the transformation rate between these fields.

can be expressed in the following forms

0 0

*T T*

*T*

**2.5 Generalized variational principle (GVP) for dissipative hydrodynamics** 

is some dimensionless constant which is specially introduced in definition (15a)

*T*

*u*

0 0

, introduced in previous section, of which the divergence will give us

0 0

*t*

. (17)

*t t*

θ

determines temperature deviation from its equilibrium level

<sup>−</sup> =− ∇ . (15b)

, (16)

, which is a basic independent

. The dissipation function only plays a

(18)

where

θ

displacements *uT*

mass displacement *u*

temperature.

heat displacement field *uT*

can be written in the following form:

spatial derivatives of the mean mass displacement field *u*

$$2F(\nabla \vec{u}, \nabla \vec{u}\_T) = 2\mu \varepsilon\_{ik}^2 + \lambda \varepsilon\_{ll}^2 + \tilde{\kappa} \left(\nabla \vec{u}\_T\right)^2 + 2\tilde{\alpha} \varepsilon\_{ll} \left(\nabla \vec{u}\_T\right)\_{,i} \tag{19b}$$

The dissipation function is the square of the difference between the mean mass and the heat displacements

$$2D(\dot{\vec{\mu}}, \dot{\vec{\mu}}\_T) = \beta(\dot{\vec{\mu}} - \dot{\vec{\mu}}\_T)^2 \tag{20}$$

The meanings of the coefficients κ , α and β in quadratic forms (19), (20) will be defined in the next section by comparison with the classical Navier-Stokes hydrodynamical system of equations.

In this case the motion equations for the mean displacement field and for the temperature field derived on the basis of the generalized variational principle are just equivalent (at μ= 0 ) to the linearized traditional system of hydrodynamics equations:

$$
\rho\_0 \frac{d}{dt} \dot{\bar{u}} - \mu \Delta \bar{u} - (\lambda + \mu + \hat{\alpha}) \nabla(\nabla \bar{u}) = (\tilde{\alpha} + \tilde{\kappa}) / (\theta T\_0) \nabla T \tag{21}
$$

$$
\mathcal{A}\mathcal{B}(\dot{\Gamma} - T\_0 \theta \nabla \dot{\overline{u}}) - \tilde{\kappa}\Delta T = \tilde{\alpha}^\* T\_0 \theta \Delta \nabla \ddot{u} \,. \tag{22}
$$

#### **2.6 Comparison with the system of hydrodynamics equations**

Coefficients of the quadratic forms in equations (19) and (20) can be determined by comparison between the system of equation (21) and (22) and the linearized system of hydrodynamics equations (Landau & Lifshitz, 1986) considering the variables *u T*, :

$$
\rho = \rho\_0 (1 - \nabla \vec{u}) \,, \tag{23}
$$

$$
\rho\_0 \frac{d^2 \vec{u}}{dt^2} - \rho\_0 c\_0^2 \Delta \vec{u} = -\rho\_0 a \nabla T + \eta \Delta \dot{\vec{u}} + \left(\check{\zeta} + \frac{\eta}{3}\right) \nabla \left(\nabla \dot{\vec{u}}\right), \tag{24}
$$

$$
\rho\_0 \mathbf{C}\_V \frac{dT}{dt} + \rho\_0 T\_0 \alpha \nabla \dot{\vec{u}} - \kappa \Delta T' = 0 \,\,\,\,\tag{25}
$$

where 0*c* is the isothermal sound velocity, *CV* - the heat capacity at constant volume, κ the heat conductivity coefficient, and α the thermal expansion coefficient. In the absence of viscosity η = 0 and ς = 0 , which was not taken into account in the dissipation function (20), the structure of equations (21), (22) nearly coincides with the second (24) and the third (25) equations in the system of hydrodynamics equations (Landau & Lifshitz, 1986). The only

Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity

 λμ

 μ

*dt dt*

 γ

 ξ

equation for the spherical part of the internal parameter tensor *ll*

where the coefficients with tilde have the following meaning:

γ γγ

Its solution can be written as:

ξ

ρ

γδ

from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics 43

0 1 <sup>2</sup> ( )( ) 0 *ik*

1 2 1 21 2 0 *ll ik ik ik ll ik ik ik d d a a b ub*

Here in the first equation (28) we safe for shortness the tensor notation for vector obtained as divergence of internal parameter tensor. Equation (28) is the motion equation for an elastic medium. Equation (29) is the kinetic equation for the internal parameter tensor *ik*

Convolving the kinetic equation by indexes it is possible to obtain the separate kinetic

0 *ll ll ll*

Kinetic equation (29) is an inhomogeneous ordinary differential equation of the first order.

*<sup>a</sup> <sup>t</sup> t t ll ll*

γ

−∞

equation of similar form to equation (29), but with added inhomogeneous terms, i.e.

*<sup>d</sup> aba b*

 ε

1 1 *a aa*

1 1 *b bb*

 = − ,

 = − 

Again, the solution of equation (33) has a form analogous to expression (32) with additional

( ) ( ) <sup>2</sup> 12 21 1 1

γ

 γ

−∞ −∞ − − = − ′ ′ − − <sup>−</sup> − −

*<sup>a</sup> <sup>a</sup> <sup>t</sup> <sup>t</sup> t t t t ik ik ik ll ik ll b b <sup>a</sup> <sup>a</sup> <sup>b</sup> <sup>a</sup> <sup>a</sup> dt e dt e*

 γ

− − ′ − − ′

γ

2 2 22 2 2

Taking the divergence of tensor (35), we obtain the following vector

( )

− − ′

= − ′ ′ 

2 221 1 0 *ik ik ik ik ll ik ll*

 δξ

1

γ

1

γ

( ) () <sup>1</sup> () ()

δ

γ γ

*ba a a a*

γ

γ

*<sup>b</sup> e t dt* γ

 ε ( )

*<sup>d</sup> a b*

 ξε

*dt* ξ γ

1 2

ξ

 ξ

For the other components of the internal tensor parameter *ik*

*dt* ξ γ

contributions from the terms with multipliers 1*a* and 1 *b*

ε δε

2 2

γ

ξ

γ

where the following notations are introduced

<sup>∂</sup> − Δ − + ∇∇ − ∇ − =

 ξ

*<sup>d</sup> u u ub b dt x*

> δ ξ

*ll*

 ξ

> δ

*k*

 ε

ξ:

++= , (30)

(32)

we can also obtain a kinetic

(34)

γ

(35)

γ

ε

= + <sup>3</sup> , 1 2 *a aa* = + <sup>3</sup> , 1 2 *b bb* = + <sup>3</sup> (31)

ξ

. Specifically,

 γγ

 δε+++ + = (33)

ξ.

ξ

∂ , (28)

+ + + + ∇+ = . (29)

difference is the additional term in the right part of equation (22) in comparison with (25). We note here briefly that the reason for the introduction of this term is related to a generalized form of the Fourier law for heat energy flow. Besides the term of the temperature gradient in the Fourier law, an additional density or pressure gradient term should appear in spite of the contradicting argument presented in (Landau & Lifshitz, 1986). The independent support of this result can be found in refs. (Martynov, 2001; Zhdanov & Roldugin 1998).

The coefficients of equations (21), (22) and (24), (25) for the fluid case (*rot u*() 0 <sup>=</sup> ) can be found by comparison. One needs to take into account the different dimensions of equation (22) and (25), and, hence, the presence of common dimension multiplier in the comparison of coefficients for these equations.

The parameters of the quadratic forms are expressed explicitly in terms of the physical parameters by the following expressions

$$\rho = \frac{\rho\_0 c\_0^2}{\varkappa} \left(\chi^2 - 1\right), \ \theta = -\frac{\mathcal{V} - 1}{\alpha \Gamma\_0}, \ \tilde{\alpha} = \rho\_0 c\_0^2 \left(\mathcal{V} - 1\right), \ \lambda + 2\mu = \rho\_0 c\_0^2 \mathcal{V}, \ \tilde{\kappa} = \rho\_0 c\_0^2 \left(\mathcal{V}^2 - 1\right), \tag{26}$$

where γ is the specific temperature ratio, γ = *C C P V* / , and χκρ = / <sup>0</sup>*CV* is the temperature conductivity coefficient. It is remarkable that the coefficient in the dissipation function β is inversely proportional to the temperature conductivity coefficient.
