A Generalized Variational Principle for Dissipative Hydrodynamics and its Application to Biot's Theory for the Description of a Fluid Shear Relaxation

A generalization of Hamilton's and Onsager's variational principles for dissipative hydrodynamical systems is represented in terms of the mechanical and heat displacement fields. A system of equations for these fields is derived from the extreme condition case using a Lagrangian formulation for the difference between the kinetic and the free energies and for the time integral of the dissipation function. The generalized hydrodynamic equation system is then evaluated using the generalized variational principle. At low frequencies this system corresponds to the traditional Navier-Stokes equations and in the high frequency limit it describes propagation of acoustical and heat modes with finite propagation velocities.

Based on the generalized variational principle a system of Biot-type equations is derived that takes into account the fluid shear viscosity relaxation. This leads to the existence of two shear propagation modes in addition to two longitudinal modes as found in the original Biot approach. One of the two shear modes is an acoustical wave, while the other is a low-frequency diffusive wave. The phase velocity and attenuation factor for the second shear mode depend linearly on the frequency. This behavior is different from that of the diffusive longitudinal mode which depends on the square root of the frequency.