Non-isothermal non-Newtonian flow problem with heat convection and Tresca's friction law

Abstract

We consider an incompressible non-isothermal fluid flow with non-linear slip boundary conditions governed by Tresca's friction law. We assume that the stress tensor is given as where θ is the temperature, π is the pressure, u is the velocity and is the strain rate tensor of the fluid while p is a real parameter. The problem is thus given by the p-Laplacian Stokes system with subdifferential type boundary conditions coupled to a elliptic equation describing the heat conduction in the fluid. We establish first an existence result for a family of approximate coupled problems where the coupling term in the heat equation is replaced by a bounded one depending on a parameter , by using a fixed point technique. Then we pass to the limit as δ tends to zero and we prove the existence of a solution to our original coupled problem in Banach spaces depending on p for any .