On linear non-local thermo-viscoelastic waves in fluids

Abstract

The following is an elaboration on the linear non-local model of viscoelastic fluids proposed in a previous work (Goddard, 2010, Int. J. Eng. Sci. 48, 1279). As a recapitulation of that work, the basic theory is presented in terms of the temporal frequency and spatial wave number in the Laplace-Fourier domain. Taylor-series expansions in these variables provides a weakly non-local theory in spatio-temporal gradients that is more comprehensive than the “bi-velocity” model of Brenner. The linearized Chapman-Enskog kinetic theory is shown to provide a confirmation of the more general theory, from which one can reconstruct a fully non-local integral model. Following the work of Davis and Brenner (2012, J. Acoust. Soc. Am. 132, 2963). the general theory is employed to derive dispersion relations for acoustic, thermal and shear-wave propagation in compressible viscoelastic fluids. At Burnett order the Chapman-Enskog theory gives a cubic polynomial in wave number squared which reduces in the dissipative quasi-static limit to a quadratic like that given by the classical Navier-Stokes-Fourier model and the bi-velocity modification of that model. With minor modification, the present analysis applies to viscoelastic shear and dilatational wave propagation in solids with higher-gradient and Cosserat effects, where it may, for example, find application to the field of rotational seismology.