Exact solutions of non-linear differential-difference equations of a viscous fluid with finite relaxation time

We have found a number exact of solutions to non-linear differential-difference equations of a viscous incompressible fluid with finite relaxation time. The results are used to solve a few hydrodynamic problems; in particular, we describe a one-dimensional flow arising from longitudinal periodic oscillations of a rigid plane and a two-dimensional periodic flow with pressure gradient near a fixed porous plate. We discuss issues of hydrodynamic instability of solutions as well as ways to enhance differential-difference fluid and related heat and diffusion models. The modified relaxation fluid model presented may theoretically explain one of the causes of the onset of turbulence. We also give a few exact solutions to the Racke-Saal non-linear hyperbolic fluid equations.