Conduction of heat in inhomogeneous solids

Abstract

In this letter we present a method for calculation of linear heat flow in inhomogeneous solids. The method is based on the evaluation of transfer matrices for each layer in a multilayered structure from the Laplace transformation of the partial differential equation of heat conduction. The multilayered structure is then described by a matrix obtained as a chain of products of individual layer transfer matrices and corresponding boundary thermal resistivity matrices. The analytic expression for the nth power of the multilayered transfer matrix is found, describing a periodic multilayered structure composed of n equal multilayered structures. The application of the presented method for calculation of photothermal signals is also shown. Dispersion relation for the thermal waves in inhomogeneous solids is obtained from the matrix elements of the transfer matrix. Finally, from the dispersion relation explicit expressions for the effective values of thermal diffusivity and conductivity of both the discontinuously and continuously inhomogeneous solids are evaluated.