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 $\sigma = 2 \mu ( \theta , \upsilon , \| D(\upsilon ) \|) \|D( \upsilon ) \|^{p-2} D(\upsilon ) - \pi \mathrm{Id}$ where θ is the temperature, π is the pressure, υ 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 $0 < \delta \ll 1$ , 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 δ tends to zero.