Two-dimensional heat conduction in thermodynamics of continua with microtemperature distributions

Abstract

This paper is concerned with the linear theory of heat conduction in continua with microtemperatures. The work is motivated by increasing use of materials which possess thermal variation at a microstructure level. The theory of plane thermal fields in homogeneous and isotropic bodies is investigated. The first part of the paper is devoted to the basic boundary value problems of the stationary theory. The fundamental solutions of the field equations are established and the potentials of single layer and double layer are introduced. The boundary value problems are reduced to the study of singular integral equations for which Fredholm’s theorems hold. Existence and uniqueness results are established. The second part of the paper is devoted to time-dependent problems. First, a solution of Galerkin type of field equations is established. Then a uniqueness theorem and an instability result are presented. The solution of Galerkin type is used to investigate the effects of some concentrated heat sources acting in an infinite medium. The theory is applied to solve the problem of stationary thermal fields in a hollow cylinder.