Unsteady non-Newtonian fluid flow with heat transfer and Tresca's friction boundary conditions

We consider an unsteady non-isothermal flow problem for a general class of non-Newtonian fluids. More precisely the stress tensor follows a power law of parameter p, namely sigma = 2 mu(theta,upsilon, ||D(upsilon)||)||D(upsilon)||Dp-2(upsilon) - pi Id where theta is the temperature, pi is the pressure, upsilon is the velocity, and D(upsilon) is the strain rate tensor of the fluid. The problem is then described by a non-stationary p-Laplacian Stokes system coupled to an L-1-parabolic equation describing thermal effects in the fluid. We also assume that the velocity field satisfies non-standard threshold slip-adhesion boundary conditions reminiscent of Tresca's friction law for solids. First, we consider an approximate problem (P-delta), where the L-1 coupling term in the heat equation is replaced by a bounded one depending on a small parameter 0 < delta << 1, and we establish the existence of a solution to (P-delta) by using a fixed point technique. Then we prove the convergence of the approximate solutions to a solution to our original fluid flow/heat transfer problem as delta tends to zero.