Analytical aspects of heat flow in random two-component laminates
Analytical aspects are examined for the coupled reaction-diffusion equations arising for the expected conditional temperatures for heat flow in a random two-component laminate with heat flowing in a direction perpendicular to the laminae. The basic equations derived by Clarke [1] involve new terms not previously encountered within the contexts of either diffusion or heat flows. Various results and solution procedures are developed for both the coupled system and the underlying fourth-order equation. Basic source solutions for the coupled system are derived which are extended to general solutions of the coupled system with the new terms. These general solutions involve two arbitrary heat functions and display explicitly the dependence of solutions on the various parameters of the model. However, these general solutions appear not to be as useful in the context of the solution of boundary value problems as the corresponding results for the coupled system without the new terms. A simple illustrative example is provided for the utilization of such general solutions for a specific problem.