**3. Viscous terms in dissipative hydrodynamics**

#### **3.1 Account of viscosity relaxation for a fluid**

To take into account fluid viscosity in the equation of motion in the framework of the generalized variational principle it is possible to introduce additional internal parameters to describe the quasi-equilibrium state of the medium, analogous to the Mandelshtam – Leontovich approach (Mandelshtam & Leontovich, 1937). As will be shown, in order to describe both the shear and the volume viscosities simultaneously, this internal parameter needs to possess the properties of a tensor. To simplify the description we consider the case when the temperature variation variable *T* is not essential so that the heat displacement *uT* terms can be omitted. In this case the additional terms associated with the tensor internal parameter *ik* ξ , will appear in the expression for the free energy of an elastic medium (19), and it can be written as:

$$2F(\nabla \vec{u}\_{\prime}, \xi\_{ij}) = 2\mu \varepsilon\_{ik}^2 + \lambda \varepsilon\_{ll}^2 + a\_1 \xi\_{ll}^2 + a\_2 \xi\_{ik}^2 + 2b\_1 \xi\_{kk} \varepsilon\_{ll} + 2b\_2 \xi\_{ik} \varepsilon\_{ki} \tag{19c}$$

where *<sup>i</sup> a* and *<sup>i</sup> b* are some coefficients of a positively determined quadratic form. The kinetic energy is then given by the ordinal expression (4) and the dissipative function in the absence of the temperature term can be written as the following quadratic form:

$$\text{2D}(\xi\_{ij}) = \chi\_1 \xi\_{ll}^2 + \chi\_2 \xi\_{ik}^2 \tag{27}$$

with some coefficients 1 γ , 2 γ.

The system of motion equations, derived on the basis of the generalized variational principle for this case can be rewritten as

$$
\rho\_0 \frac{d}{dt} \ddot{\bar{u}} - \mu \Delta \ddot{u} - (\mathcal{k} + \mu) \nabla (\nabla \bar{u}) - b\_1 \nabla \xi\_{\mathcal{l}l} - b\_2 \frac{\partial \xi\_{\mathcal{l}k}}{\partial x\_k} = 0 \,\,\,\tag{28}
$$

$$\gamma\_1 \mathfrak{J}\_{\dot{k}} \frac{d\mathfrak{E}\_{\text{ll}}}{dt} + \gamma\_2 \frac{d\mathfrak{E}\_{\text{ik}}}{dt} + a\_1 \mathfrak{J}\_{\dot{k}} \mathfrak{E}\_{\text{ll}} + a\_2 \mathfrak{E}\_{\dot{k}} + b\_1 \mathfrak{J}\_{\dot{k}} \nabla \vec{u} + b\_2 \mathfrak{e}\_{\dot{k}} = 0 \ . \tag{29}$$

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* ξ . Convolving the kinetic equation by indexes it is possible to obtain the separate kinetic equation for the spherical part of the internal parameter tensor *ll* ξ:

$$
\tilde{\mathcal{Y}}\frac{d\tilde{\xi}\_{ll}}{dt} + \tilde{a}\tilde{\xi}\_{ll} + \tilde{b}\varepsilon\_{ll} = 0 \tag{30}
$$

where the coefficients with tilde have the following meaning:

42 Hydrodynamics – Advanced Topics

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;

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

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

 γ

To take into account fluid viscosity in the equation of motion in the framework of the generalized variational principle it is possible to introduce additional internal parameters to describe the quasi-equilibrium state of the medium, analogous to the Mandelshtam – Leontovich approach (Mandelshtam & Leontovich, 1937). As will be shown, in order to describe both the shear and the volume viscosities simultaneously, this internal parameter needs to possess the properties of a tensor. To simplify the description we consider the case when the temperature variation variable *T* is not essential so that the heat displacement *uT*

terms can be omitted. In this case the additional terms associated with the tensor internal

22 2 2 12 1 2 2( , ) 2 2 2 *F u ij ik ll ll ik kk ll ik ki* ∇ = ++ + + +

where *<sup>i</sup> a* and *<sup>i</sup> b* are some coefficients of a positively determined quadratic form. The kinetic energy is then given by the ordinal expression (4) and the dissipative function in the

1 2 2( ) *D ij ll ik*

The system of motion equations, derived on the basis of the generalized variational

, will appear in the expression for the free energy of an elastic medium (19),

 *aa b b* ξξ

2 2

 γξ

 ξ ε

, (19c)

ξ ε

= + (27)

γ

conductivity coefficient. It is remarkable that the coefficient in the dissipation function

= − *c* 1 , <sup>2</sup>

= *C C P V* / , and

λ+ = 2

0 0

χκρ

 *c* , ( ) 2 2 0 0 κρ

 γ= − *c* 1 , (26)

= / <sup>0</sup>*CV* is the temperature

βis

μργ

Zhdanov & Roldugin 1998).

2 0 0 <sup>2</sup> 1

 γ

ρ*c*

χ= − ,

β

parameter *ik*

ξ

and it can be written as:

with some coefficients 1

where γ

of coefficients for these equations.

parameters by the following expressions

( )

θ

is the specific temperature ratio,

**3. Viscous terms in dissipative hydrodynamics** 

**3.1 Account of viscosity relaxation for a fluid** 

ξ

γ , 2 γ.

principle for this case can be rewritten as

 με  λε

absence of the temperature term can be written as the following quadratic form:

ξ γξ

0 1 *T* γ

inversely proportional to the temperature conductivity coefficient.

<sup>−</sup> = − , ( ) <sup>2</sup> 0 0 αρ

α

$$
\tilde{\gamma} = \Im \gamma\_1 + \mathcal{\gamma}\_2, \ \tilde{a} = \Im a\_1 + a\_2, \ \bar{b} = \Im b\_1 + b\_2 \tag{31}
$$

Kinetic equation (29) is an inhomogeneous ordinary differential equation of the first order. Its solution can be written as:

$$\xi\_{ll} = -\frac{\tilde{b}}{\tilde{\mathcal{P}}} \int\_{-\infty}^{t} e^{-\frac{\tilde{a}}{\tilde{\mathcal{P}}}(t-t')} \varepsilon\_{ll}(t') \, dt' \tag{32}$$

For the other components of the internal tensor parameter *ik* ξ we can also obtain a kinetic equation of similar form to equation (29), but with added inhomogeneous terms, i.e.

$$
\gamma\_2 \frac{d\tilde{\xi}\_{ik}}{dt} + a\_2 \tilde{\xi}\_{ik} + b\_2 \varepsilon\_{ik} + \tilde{a}\_1 \delta\_{ik} \tilde{\xi}\_{ll} + \tilde{b}\_1 \delta\_{ik} \varepsilon\_{ll} = 0 \tag{33}
$$

where the following notations are introduced

$$
\tilde{a}\_1 = \left(a\_1 - \tilde{a}\frac{\mathcal{Y}\_1}{\mathcal{Y}}\right),
$$

$$
\tilde{b}\_1 = \left(b\_1 - \tilde{b}\frac{\mathcal{Y}\_1}{\mathcal{Y}}\right) \tag{34}
$$

Again, the solution of equation (33) has a form analogous to expression (32) with additional contributions from the terms with multipliers 1*a* and 1 *b* . Specifically,

$$\xi\_{\rm ik} = -\frac{b\_2}{\gamma\_2} \int\_{-\infty}^{t} dt' e^{-\frac{a\_2}{\gamma\_2}(t-t')} \left( \varepsilon\_{\rm ik} - \delta\_{\rm ik} \varepsilon\_{\rm ll} \left( 1 - \frac{\tilde{b}}{b\_2} \frac{(a\_1 \gamma\_2 - a\_2 \gamma\_1)}{(\tilde{a} \gamma\_2 - a\_2 \tilde{\gamma})} \right) \right) - \delta\_{\rm ik} \frac{\tilde{b}}{\tilde{\mathcal{I}}} \frac{(a\_1 \tilde{\mathcal{I}} - \tilde{a} \gamma\_1)}{(\tilde{a} \gamma\_2 - a\_2 \tilde{\gamma})} \int\_{-\infty}^{t} dt' e^{-\frac{\tilde{a}}{\tilde{\mathcal{I}}} (t-t')} \varepsilon\_{\rm ll} \tag{35}$$

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

Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity

mentioned internal parameters, responsible for relaxation.

**hydrodynamical description of continuum mechanics** 

introduce some tensor internal parameter *ik*

conservation law of angular momentum *M*

medium is required for application of such approach.

is represented.

and angular momentum.

*P* 

from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics 45

fluids, including the simplest of them, is described by the Navier-Stokes equation, then the only available value, which could relax in all cases, and hence could be considered as common scalar internal parameter, is the mean distance between molecules in gas or liquid. In the condensed and especially in the solid media the mutual space placement of atoms becomes to be essential, hence a space variation of their mutual positions, holding rotational invariance of a body as whole, has to be described by symmetrical tensor of the second order. Hence the corresponding internal parameter could be the same tensor. Thus, the discrete structure of medium on the kinetic level predetermines existence, at least, of

**3.2 Shear viscosity as a consequence of the angular momentum relaxation for the** 

ξ

As shown in the previous section, it is possible to derive the system of hydrodynamical equations on the GVP basis for viscous, compressible fluid in the form of Navier-Stokes equations. However for the account of terms responsible for viscosity it is required to

approach (Mandelshtam & Leontovich, 1937). Relaxation of this internal parameter provides appearance of viscous terms in the Navier-Stokes equation. It is worth mentioning that the developed approach allowed to generalize the Navier-Stokes equation with constant viscosity coefficient to more general case accounting for viscosity relaxation in analogy to the Maxwell's model (Landau & Lifshitz, 1972). However the physical interpretation of the tensor internal parameter, which should be enough universal due to general character of the Navier-Stokes equation, requires more clear understanding. On the intuition level it is clear that corresponding internal parameter should be related with neighbor order in atoms and molecules placement and their relaxation. In the present section such physical interpretation

As was mentioned in Introduction the system of hydrodynamical equations in the form of Navier-Stokes is usually derived on the basis of conservation laws of mass *M* , momentum

laying in the basis of traditional hydrodynamics. In this connection it is interesting to

description. It is worth mentioning that equation for angular momentum appeared in hydrodynamics early (Sorokin, 1943; Shliomis, 1966) and arises and develops in the momentum elasticity theory. The Cosserat continuum is an example of such description (Kunin, 1975; Novatsky, 1975; Erofeev, 1998). However some internal microstructure of

In the hydrodynamical description as a partial case of continuum mechanics the definition of material point is introduced as sifficiently large ensemble of structural elements of medium (atoms and molecules) that on one hand one has to describe properties of this ensemble in statistical way and on the other one has to consider the size of material point as small in comparison with specific scales of the problem. A material point itself as closed ensemble of particles possesses the following integrals of motion: mass, momentum, energy

The basic independent variables, in terms of which the hydrodynamical description should be constructed, are the values which can be determined for separate material point in

understand the role of conservation law of angular momentum *M*

 and energy *E* . The correctness of equations of the traditional hydrodynamics is confirmed by the large number of experiments where it is adequate. However the

in agreement with Mandelshtam-Leontovich

is absent among the mentioned balance laws

in hydrodynamical

$$\begin{split} \frac{\partial \xi\_k}{\partial \mathbf{x}\_k} &= -\frac{b\_2}{\mathcal{Y}\_2} \int\_{-\infty}^t dt' e^{-\frac{\tilde{\theta}\_2}{\tilde{\mathcal{Y}}\_2}(t-t')} \Big( \frac{1}{2} (\Delta \vec{u} + \nabla(\nabla \vec{u})) - \nabla(\nabla \vec{u}) \Big( 1 - \frac{\tilde{b}}{b\_2} \frac{(a\_1 \mathcal{Y}\_2 - a\_2 \mathcal{Y}\_1)}{(\tilde{a} \mathcal{Y}\_2 - a\_2 \mathcal{Y}\_1)} \Big) \Big) - \\\\ &- \frac{\tilde{b}}{\tilde{\mathcal{Y}}\_1} \frac{(a\_1 \tilde{\mathcal{Y}} - \tilde{a} \mathcal{Y}\_1)}{(\tilde{a} \mathcal{Y}\_2 - a\_2 \tilde{\mathcal{Y}})} \int\_{-\infty}^t dt' e^{-\frac{\tilde{\mathcal{Y}}}{\tilde{\mathcal{Y}}}(t-t')} \nabla(\nabla \vec{u}) \end{split} \tag{36}$$

If we substitute (36) and (32) in the motion equation (28), we can write:

$$\begin{split} \rho\_0 \frac{d}{dt} \dot{\overline{u}} - \mu \Delta \overline{u} - (\lambda + \mu) \nabla(\nabla \overline{u}) &= -\frac{\tilde{b}}{\tilde{\mathcal{I}}} \Big( b\_1 - b\_2 \frac{(a\_1 \tilde{\mathcal{I}} - \tilde{a} \mathcal{I} \chi\_1)}{(\tilde{a} \mathcal{Y}\_2 - a\_2 \tilde{\mathcal{I}})} \Big) \Big\Big\|\_{\to \alpha}^{t} dt' e^{-\frac{\tilde{b}}{\tilde{\mathcal{I}}} (t - t')} \nabla(\nabla \overline{u}) - \\ -\frac{b\_2^2}{\mathcal{Y}\_2} \int\_{-\infty}^{t} dt' e^{-\frac{a\_2}{\mathcal{Y}\_2} (t - t')} \Big( \frac{1}{2} (\Delta \overline{u} + \nabla(\nabla \overline{u})) - \nabla(\nabla \overline{u}) \Big( 1 - \frac{\tilde{b}}{b\_2} \frac{(a\_1 \chi\_2 - a\_2 \chi\_1)}{(\tilde{a} \mathcal{Y}\_2 - a\_2 \tilde{\mathcal{Y}})} \Big) \Big) \end{split} \tag{37}$$

In the low frequency limit, at times greater than the relaxation times /γ *a* and γ 2 2 /*a* , it is possible to derive an equation analogous to the Navier – Stokes motion equation with shear and volume viscosities:

$$
\rho\_0 \frac{d}{dt} \dot{\overline{\mu}} - \not{\mu} \Delta \overline{\iota} - (\not{\mathcal{L}} + \not{\mu}) \nabla(\nabla \overline{\iota}) = \not{\eta} \Delta \dot{\overline{\iota}} + (\not{\zeta} + \frac{\not{\eta}}{3}) \nabla(\nabla \dot{\overline{\iota}}) \tag{38}
$$

where the effective elastic moduli λ and μ and coefficients of shear and volume viscosities are expressed as

$$\begin{aligned} \tilde{\mu} &= \mu - \frac{b\_2^2}{2a\_2}, \quad \tilde{\lambda} = \lambda + \frac{b\_2^2}{2a\_2} - \frac{\tilde{b}}{\tilde{a}} \left( b\_1 - b\_2 \frac{\{a\_1 \tilde{\gamma} - \tilde{a}\gamma\_1\}}{\left(\tilde{a}\gamma\_2 - a\_2 \tilde{\gamma}\right)} \right), \quad \tilde{\eta} = \frac{1}{2}\gamma\_2 \frac{b\_2^2}{a\_2^2}, \\\\ \tilde{\xi} + \frac{\tilde{\eta}}{3} &= \tilde{\mathcal{Y}} \frac{\tilde{b}}{\tilde{a}^2} \left( b\_1 - b\_2 \frac{\{a\_1 \tilde{\gamma} - \tilde{a}\gamma\_1\}}{\left(\tilde{a}\gamma\_2 - a\_2 \tilde{\gamma}\right)} \right) - \gamma\_2 \frac{b\_2}{a\_2^2} \left( \frac{b\_2}{2} - \tilde{b} \frac{\{a\_1 \chi\_2 - a\_2 \chi\_1\}}{\left(\tilde{a}\gamma\_2 - a\_2 \tilde{\gamma}\right)} \right) \end{aligned} \tag{39}$$

It is important to note that the structure of the effective shear modulus μ in (39) is determined by a difference, which can be equal to zero, in which case equation (38) becomes completely equivalent to the Navier – Stokes equation for a viscous fluid. Thus the condition

$$
\mu = \frac{b\_2^2}{2a\_2} \tag{40}
$$

should be satisfied to consider a solid with shear relaxation like a viscous fluid. If 0 μ > , then we have the case of elastic medium with a shear viscosity (the Voight's model) or with relaxation in the more general case (37). Thus, in the framework of the uniform approach it is possible to describe viscous fluids and solids with visco-elastic properties.

As a final remark of this section it is possible to say several words about physical sense of the introduced internal parameter. Since in the low frequency limit the majority of gases and

( ) <sup>2</sup> 12 21 2 2 22

*b b <sup>a</sup> <sup>a</sup> dt e uu u x b a a*

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

( ) 1 1

−∞ <sup>−</sup> <sup>−</sup> ′ ∇ ∇ <sup>−</sup>

( ) ( ) ( ) *<sup>a</sup> <sup>t</sup> t t ba a dt e u*

γ

− − ′

1 ( ) ( ) ( )1 2 ( )

2 2 ( ) ( )( ) ( ) ( )

γ γ

1 ( ) ( ) ( )1 2 ( )

 γγ

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

 

<sup>2</sup> ( ) <sup>2</sup> 12 21 2 2 22

<sup>−</sup> <sup>−</sup> ′ Δ +∇ ∇ −∇ ∇ − <sup>−</sup>

possible to derive an equation analogous to the Navier – Stokes motion equation with shear

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

*uu u u u*

2 1 1 1 2 2 2 2

<sup>−</sup> =+ − −

3 ( ) 2( ) *b a a bb a a b b <sup>b</sup> a a a a a a*

determined by a difference, which can be equal to zero, in which case equation (38) becomes completely equivalent to the Navier – Stokes equation for a viscous fluid. Thus

− − += − − −

γ

 γ

μ

should be satisfied to consider a solid with shear relaxation like a viscous fluid. If 0

then we have the case of elastic medium with a shear viscosity (the Voight's model) or with relaxation in the more general case (37). Thus, in the framework of the uniform approach it

As a final remark of this section it is possible to say several words about physical sense of the introduced internal parameter. Since in the low frequency limit the majority of gases and

2 () *bb aa b b aa a a*

γ

− −

 η

*<sup>a</sup> <sup>t</sup> t t b b <sup>a</sup> <sup>a</sup> dt e uu u*

γ

 γ

( ) 1 1

−∞

*ba a*

and coefficients of shear and volume viscosities

η

 γ

γ

= (40)

1 2

 γ<sup>=</sup> ,

> γ

> γ

μ

in (39) is

μ> ,

*b a*

 γ

η

 ζ− Δ − + ∇∇ = Δ + + ∇∇ (38)

( )

 γ

γ γ

1 1 2 2 12 21

2 2 2 2 2 () ( )

γ

−

,

γ

γ

− − ′

 γ

γ

γ

2 2 /*a* , it is

γ*a* and

 γ

γ

(36)

(37)

(39)

( )

2 2

 γ

*<sup>a</sup> <sup>t</sup> t t d b <sup>a</sup> <sup>a</sup> u u u bb dt e u*

γ

( )

*a a*

γ γ

γγ

0 1 2

 λμ

> 2 2

− − ′

γ

*d*

*dt*

 μ

λ λ

ρ

If we substitute (36) and (32) in the motion equation (28), we can write:

*dt a a*

In the low frequency limit, at times greater than the relaxation times /

 λμ

 and μ

2

2 2 1 2 2

 γ

γ

It is important to note that the structure of the effective shear modulus

is possible to describe viscous fluids and solids with visco-elastic properties.

λ

2 2

− − ′

γ

*<sup>a</sup> <sup>t</sup> t t ik*

−∞

γ

 μ

γ

where the effective elastic moduli

μ μ= − ,

ζ

η

 γ

−∞

*k*

ξ

ρ

and volume viscosities:

are expressed as

the condition

fluids, including the simplest of them, is described by the Navier-Stokes equation, then the only available value, which could relax in all cases, and hence could be considered as common scalar internal parameter, is the mean distance between molecules in gas or liquid. In the condensed and especially in the solid media the mutual space placement of atoms becomes to be essential, hence a space variation of their mutual positions, holding rotational invariance of a body as whole, has to be described by symmetrical tensor of the second order. Hence the corresponding internal parameter could be the same tensor. Thus, the discrete structure of medium on the kinetic level predetermines existence, at least, of mentioned internal parameters, responsible for relaxation.
