Unique Solvability of a Two-Dimensional Nonstationary Problem for the Convection Equations with Temperature-Dependent Viscosity

Abstract

Numerous papers deal with the mathematical justication of models of viscous uid dynamics. Various aspects of the theory of the viscous incompressible uid equations, including the construction of generalized solutions, are presented in the monographs [1, 2] and also in the monograph [3], devoted to inhomogeneous uids (see also the bibliography therein). Important results in the mathematical theory of viscous incompressible ow with constant viscosity coecient were obtained in most of these papers. In problems of hydrodynamics and heat and mass transfer arising in models of many technological processes, it is important to take account of the dependence of the transport coecients on temperature. The importance of the analysis of mathematical models of these phenomena is justied by numerous applications, experimental data, and numerical investigations [4{6]. Problems in which the transport coecients depend on temperature were discussed in [7], where a result on the existence of at least one generalized solution in a model of an inhomogeneous uid was apparently announced for the rst time. It is very dicult to take account of the dependence of the transport coecients on temperature in theoretical analysis of mathematical models, since this results in an additional nonlinearity in the equations. The papers [8, 9] deal with the existence and uniqueness issues for stationary boundary value problems for the free convection equations with temperature-dependent viscosity coecients. It was proved in [10] that the three-dimensional nonstationary problem for these equations is solvable. The solution obtained there is a Hopf type solution. In the present paper, we continue the analysis of well-posedness of the nonstationary boundary value problem for equations generalizing the Oberbeck{Boussinesq equations to the case in which the viscosity coecient depends on temperature: ~t + ~