Two Stationary Radiative-Conductive Heat Transfer Problems for a System of Two-Dimensional Plates

Abstract

UDC 517.95 We consider two nonlinear stationary radiative-conductive heat transfer problems in a system of two-dimensional heat-conducting plates of width e separated by vacuum in- terlayers. We establish comparison theorems and obtain estimates for the weak solu- tion, in particular, the two-sided estimate umin u umax and estimates of the form � DxuL 2 (G e ) = O( √ e) andDxuL 2 (G e ) = O ( √ e/λ). Bibliography :1 0titles. In applications, it is important to study the heat transfer process in media containing vacuum interlayers through which the heat transfer is realized by radiation. A numerical solution of such problems with a large number of heat transferring elements is, in fact, impossible, especially for two- and three-dimensional structures. Therefore, it is important to construct efficient approxi- mation methods which, in particular, could be based on constructing special homogenizations of the problem under consideration. Such approximations were constructed and justified in (1)-(6). We note that for studying the properties of approximation problems, it is necessary to perform a preliminary analysis of properties of the original problem. Such an analysis involves the proof of special estimates for the solutions and their derivatives in terms of the data. In this paper, we consider two boundary value problems describing the stationary radiative- conductive heat transfer in a system of n two-dimensional heat-conductive grey plates of width e = X/n and height Y separated by infinitely thin vacuum interlayers. We prove comparison theorems and estimate the solution. In particular, we obtain the two-sided estimate umin u umax and estimates of the formDxuL2(Ge) = O( √ e )a ndDxuL2(Ge) = O( √ e/λ). We emphasize that such estimates play an important role in the justification of semidiscrete and asymptotic approximations (cf., for example, (4, 6)).