Effective Boundary Conditions for Heat Equation Arising from Anisotropic and Optimally Aligned Coatings in Three Dimensions

TL;DR

This paper derives effective boundary conditions for the 3D heat equation in domains with thin, anisotropic, and optimally aligned coatings, capturing complex effects like nonlocal operators as the coating thickness approaches zero.

Contribution

It introduces new effective boundary conditions for heat equations with anisotropic coatings, including nonlocal operators, in the thin-layer limit.

Findings

Effective boundary conditions include nonlocal operators like fractional Laplacian.

Anisotropic and aligned coatings significantly influence heat transfer modeling.

The derived conditions generalize classical boundary conditions for layered materials.

Abstract

We discuss the initial boundary value problem for a heat equation in a domain surrounded by a layer. The main features of this problem are twofold: on one hand, the layer is thin compared to the scale of the domain, and on the other hand, the thermal conductivity of the layer is drastically different from that of the bulk; moreover, the bulk is isotropic, but the layer is anisotropic and ``optimally aligned" in the sense that any vector in the layer normal to the interface is an eigenvector of the thermal tensor. We study the effects of the layer by thinking of it as a thickless surface, on which ``effective boundary conditions" (EBCs) are imposed. In the three-dimensional case, we obtain EBCs by investigating the limiting solution of the initial boundary value problem subject to either Dirichlet or Neumann boundary conditions as the thickness of the layer shrinks to zero. These EBCs contain not only the standard boundary conditions but also some nonlocal ones, including the Dirichlet-to-Neumann mapping and the fractional Laplacian. One of the main features of this work is to allow the drastic difference in the thermal conductivity in the normal direction and two tangential directions within the layer.