**4. Conclusion**

The first part of the chapter presents an original formulation of the generalized variational principle (GVP) for dissipative hydrodynamics (continuum mechanics) as a direct combination of Hamilton's and Onsager's variational principles. The GVP for dissipative continuum mechanics is formulated as Hamilton's variational principle in terms of two independent field variables i.e. the mean mass and the heat displacement fields. It is important to mention that existence of two independent fields gives us opportunity to consider a closed mechanical system and hence to formulate variational principle. Dissipation plays only a role of energy transfer between the mean mass and the heat displacement fields. A system of equations for these fields is derived from the extreme condition for action with a Lagrangian density in the form of the difference between the kinetic and the free energies minus the time integral of the dissipation function. All mentioned potential functions are considered as a general positively determined quadratic forms of time or space derivatives of the mean mass and the heat displacement fields. The generalized system of hydrodynamical equations is then evaluated on the basis of the GVP. At low frequencies this system corresponds to the traditional Navier – Stokes equation system. It allowed us to determine all coefficients of quadratic forms by direct comparison with the Navier – Stokes equation system.

The second part of the chapter is devoted to consistent introduction of viscous terms into the equation of fluid motion on the basis of the GVP. A tensor internal parameter is used for description of relaxation processes in vicinity of quasi-equilibrium state by analogy with the Mandelshtam – Leontovich approach. The derived equation of motion describes the viscosity relaxation phenomenon and generalizes the well known Navier – Stokes equation. At low frequencies the equation of fluid motion reduces exactly to the form of Navier – Stokes equation. Nevertheless there is still a question about physical interpretation of the used internal parameter. The answer on this question is presented in the last section of the chapter.

It is shown that the internal parameter responsible for shear viscosity can be interpreted as a consequence of relaxation of angular momentum of material points constituting a mechanical continuum. Due to angular momentum balance law 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. When dissipation prevails over momentum of inertion this approach describes local relaxation of angular momentum and corresponds to the sense of the internal parameter. It is important that such principal parameter of Cosserat continuum as the inertia moment of intrinsic microstructure can completely vanish from the description for dissipative continuum. The independent equation of motion for angular momentum in this case reduces to local relaxation and after its substitution into the momentum balance equation leads to the viscous terms in Navier – Stokes equation. Thus, it is shown that the nature of viscosity phenomenon can be interpreted as relaxation of angular momentum of material points on the kinetic level.
