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 θ is the temperature, π 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 δ 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 $p>3/2$.

COMMUNICATED BY:

• M. Shillor

Keywords:

• Incompressible non-Newtonian fluids
• heat transfer
• Tresca's friction law
• existence and regularity result

AMS CLASSIFICATIONS:

• 76A05
• 35Q35
• 35Q79
• 35J87
• 49J40

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by Université Jean Monnet de Saint-Etienne [UJM].