Energy flux for damped inhomogeneous plane waves in viscoelastic fluids

Abstract

The propagation of inhomogeneous plane waves in the context of the linearized theory of incompressible viscoelastic fluids is considered. The angular frequency and the slowness vector are both assumed to be complex. As in incompressible purely viscous fluids, two kinds of waves may propagate: A “zero pressure wave” for which the increment in pressure due to the wave is zero, and a “universal wave” which is independent of the viscoelastic relaxation modulus. The balance of energy is written using a decomposition of the stress power into a reversible component and a dissipative component proposed namely by P.W. Buchen [J. R. Astr. Soc. 23 (1971) 531–542]. For the inhomogeneous waves, a “weighted mean” energy flux vector, “weighted mean” energy density and “weighted mean” energy dissipation are introduced. It is shown that they satisfy two modulus independent relations. These generalize to the case of viscoelasticity relations previously obtained in other contexts.