Causal implications of viscous damping in compressible fluid flows

Classically, a compressible, isothermal, viscous fluid is regarded as a mat hematical continuum and its motion is governed by the linearized continuity , Navier-Stokes, and state equations. Unfortunately, solutions of this syst em are of a diffusive nature and hence do not satisfy causality. However, i n the case of a half-space of fluid set to motion by a harmonically vibrati ng plate the classical equation of motion can, under suitable conditions, b e approximated by the damped wave equation. Since this equation is hyperbol ic, the resulting solutions satisfy causal requirements. In this work the L aplace transform and other analytical and numerical tools are used to inves tigate this apparent contradiction. To this end the exact solutions, as wel l as their special and limiting cases, are found and compared for the two m odels. The effects of the physical parameters on the solutions and associat ed quantities are also studied. It is shown that propagating wave fronts ar e only possible under the hyperbolic model and that the concept of phase sp eed has different meanings in the two formulations. In addition, discontinu ities and shook waves an noted and a physical system is modeled under both formulations. Overall, it is shown that the hyperbolic form gives a more re alistic description of the physical problem than does The classical theory. Lastly, a simple mechanical analog is given and connections to viscoelasti c fluids an noted. In particular, the research presented hen supports the n otion that linear compressible, isothermal, viscous fluids can, at least in terms of causality, be better characterized as a type of viscoelastic flui d.