**1. Introduction**

34 Hydrodynamics – Advanced Topics

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A system of hydrodynamic equations for a viscous, heat conducting fluid is usually derived on the basis of the mass, the momentum and the energy conservation laws (Landau & Lifshitz, 1986). Certain assumptions about the form of the viscous stress tensor and the energy density flow vector are made to derive such a system of equations for the dissipative viscous, heat conductive fluid. The system of equations based on the mass, the momentum and the energy conservation laws describes adequately a large set of hydrodynamical phenomena. However, there are some aspects which suggest that this system is only an approximation.

For example, if we consider propagation of small perturbations described by this system, then it is possible to separate formally the longitudinal, shear and heat or entropy waves. The coupling of the longitudinal and heat waves results in their splitting into independent acoustic-thermal and thermo-acoustic modes. For these modes the limits of phase velocities tends to infinity at high frequencies so that the system is in formal contradiction with the requirements for a finite propagation velocity of any perturbation which the medium can undergo. Thus it is possible to suggest that such a hydrodynamic equation system is a mere low frequency approximation. Introducing the effects of viscosity relaxation (Landau & Lifshitz, 1972), guarantees a limit for the propagation velocity of the shear mode, and the introduction of the heat relaxation term (Deresiewicz, 1957; Nettleton, 1960; Lykov, 1967) in turn ensures finite propagation velocities of the acoustic-thermal and thermo-acoustic modes. However, the introduction of such relaxation processes requires serious effort with motivation.

Classical mechanics provides us with the Lagrange's variational principle which allows us to derive rigorously the equations of motion for a mechanical system knowing the forms of kinetic and potential energies. The difference between these energies determines the form of the Lagrange function. This approach translates directly into continuum mechanics by introduction of the Lagrangian density for non-dissipative media. In this approach the dissipation forces can be accounted for by the introduction of the dissipation function derivatives into the corresponding equations of motion in accordance with Onsager's

Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity

, 2 2 2 2 *<sup>U</sup>* = +

*dt* ρ

 μ

α

δα

*dt* α

α

relaxation of a thermodynamic system to its equilibrium state, i.e.:

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

λε

2 2 <sup>0</sup> 2( ) *Ku u* = ρ

0 is the density of the medium, and

**2.2 Onsager's variational principle** 

thermodynamic relaxation process

α, i.e.

rate of change of

(Landau & Lifshitz, 1986).

Lifshitz, 1964) we have

quadratic forms

derived:

where ρ

from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics 37

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

<sup>0</sup> *dL L dt u u* ∂ ∂ +∇ =

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

 *ll ik* με

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

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

 λμ

λ and μ

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

> α

The kinetic equation can then be derived from variational principle (6) to describe the

() 2() *<sup>d</sup> s D*

The above equation satisfies strictly the symmetry principle for the kinetic coefficients

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 &

> *dL L D dt u u u* ∂∂∂ +∇ =−

 α

, <sup>1</sup>

ε

*ik*

2

<sup>∂</sup> ∂∇ . (3)

*i k*

(4)

 ∂ ∂ = + ∂ ∂

− Δ − + ∇∇ = , (5)

, and the dissipation function *D* as a function of the

[*s D* () () 0 − =] . (6)

∂ ∂ ∂∇ . (8)

= . (7)

are the Lamé's constants.

*k i u u x x*

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.
