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

Abstract

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 $\sigma =2\mu (\theta ,\upsilon ,\parallel D(\upsilon )\parallel ){\parallel D(\upsilon )\parallel }^{p-2}D(\upsilon )-\pi \mathrm{Id}$ 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.