The Heat Equation Solution Near the Interface Between Two Media

Abstract

Physical phenomena that arise near the boundaries of media with different characteristics, for example, changes in temperature at the water-air interface, require the creation of models for their adequate description. Therefore, when setting model problems one should take into account the fact that the environment parameters undergo changes at the interface. In particular, experimentally obtained temperature curves at the water-air interface have a kink, that is, the derivative of the temperature distribution function suffers a discontinuity at the interface. A function with this feature can be a solution to the problem for the heat equation with a discontinuous thermal diffusivity and discontinuous function describing heat sources. The coefficient of thermal diffusivity in the water-air transition layer is small, so a small parameter appears in the equation prior to the spatial derivative, which makes the equation singularly perturbed. The solution of the boundary value problem for such an equation can have the form of a contrast structure, that is, a function whose domain contains a subdomain, where the function has a large gradient. This region is called an internal transition layer. The existence of a solution with the internal transition layer of such a problem requires justification that can be carried out with the use of an asymptotic analysis. In the present paper, such an analytic investigation was carried out, and this made it possible to prove the existence of a solution and also to construct its asymptotic approximation.