A non-steady system with friction boundary conditions for flow of heat-conducting incompressible viscous fluids

Abstract

In this paper we are concerned with a non-steady system for motion of incompressible viscous Newtonian heat-conducting fluids under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak and one-sided leak conditions based on total stress, velocity, total pressure, rotation, total stress together and the conditions for temperature may include Dirichlet, Neumann and Robin conditions together. Relying on the relations among strain, rotation, normal derivative of velocity and shape of boundary surface, we get variational formulation. The formulation consists of a variational inequality for velocity due to the boundary conditions of friction type and a variational equation for temperature. We first study existence, estimation and relative compactness of solutions to an approximate equation. By passing to limits as the parameter for approximation goes to zero, we get existence of a solution with “defect measure” for temperature.