Generalized Variational Principle for Dissipative Hydrodynamics: Shear Viscosity from Angular Momentum Relaxation in the Hydrodynamical Description of Continuum Mechanics

Abstract

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