MIXED BOUNDARY VALUE PROBLEMS OF THE SYSTEM FOR STEADY FLOW OF HEAT-CONDUCTING INCOMPRESSIBLE VISCOUS FLUIDS WITH DISSIPATIVE HEATING

In this paper we are concerned with the equation for steady flow of heat-conducting incompressible viscous Newtonian fluids with dissipative heating under mixed boundary conditions. The boundary conditions for fluid may include Tresca slip, leak condition, one-sided leak conditions, velocity, pressure, rotation, 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 formulations consisted of a variational inequality for velocity and a variational equation for temperature, which are equivalent to the original PDE problems for smooth solutions. Then, we study the existence of solutions to the variational problems. To this end, first we study the existence of solutions to auxiliary problems including a parameter for approximation and two or three parameters concerned with the norms of velocity and temperature. Then we determine the parameters concerned with the norms of velocity and temperature in accordance with the data of problems, and we get the existence of solutions by passing to limits as the parameter for approximation goes to zero.