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 $\sigma =2\mu (\theta ,u,|D(u)|)|D(u){|}^{p-2}D(u)-\pi \mathrm{Id}$ where $\theta$ is the temperature, $\pi$ is the pressure, $u$ is the velocity and $D(u)$ 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 $L^1$ elliptic equation describing the heat conduction in the fluid. We establish first an existence result for a family of approximate coupled problems where the $L^1$ coupling term in the heat equation is replaced by a bounded one depending on a parameter $0<\delta <<1$, by using a fixed point technique. Then we pass to the limit as $\delta$ tends to zero and we prove the existence of a solution $(u,\pi ,\theta )$ to our original coupled problem in Banach spaces depending on $p$ for any $p>3/2$.