Solution of the spherically symmetric linear thermoviscoelastic problem in the inertia-free limit

TL;DR

This paper presents an analytical solution for the spherically symmetric thermoviscoelastic problem in the inertia-free limit, simplifying the analysis of frequency-dependent thermal and mechanical properties in viscous liquids.

Contribution

It introduces a transfer matrix approach to solve the problem analytically, enabling easier derivation of boundary condition effects on thermoviscoelastic measurements.

Findings

Analytical solution for spherical thermoviscoelastic problem

Transfer matrix formulation links boundary conditions to measurable properties

Simplifies analysis of frequency-dependent specific heats

Abstract

The coupling between mechanical and thermal properties due to thermal expansion complicates the problem of measuring frequency-dependent thermoviscoelastic properties, in particular for highly viscous liquids. A simplification arises if there is spherical symmetry where - as detailed in the present paper - the thermoviscoelastic problem may be solved analytically in the inertia-free limit, i.e., the limit where the sample is much smaller than the wavelength of sound waves at the frequencies of interest. As for the one-dimensional thermoviscoelastic problem [Christensen et al., Phys. Rev. E 75, 041502 (2007)], the solution is conveniently formulated in terms of the so-called transfer matrix, which directly links to the boundary conditions that can be experimentally controlled. Once the transfer matrix has been calculated, it is fairly easy to deduce the equations describing various experimentally relevant special cases (boundary conditions that are adiabatic, isothermal, isochoric, etc.). In most situations the relevant frequency-dependent specific heat is the longitudinal specific heat, a quantity that is in between the isochoric and isobaric frequency-dependent specific heats.