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

Abstract

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 σ = 2 μ ( θ , υ , ∥ D ( υ ) ∥ ) ∥ D ( υ ) ∥ p − 2 D ( υ ) − π Id $\sigma = 2 \mu ( \theta , \upsilon , \| D(\upsilon ) \|) \|D( \upsilon ) \|^{p-2} D(\upsilon ) - \pi \mathrm{Id}$ where θ is the temperature, π is the pressure, υ is the velocity, and D ( υ ) $D(\upsilon )$ 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 $L^{1}$ -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 δ ) $(P_{\delta })$ , where the L 1 $L^{1}$ coupling term in the heat equation is replaced by a bounded one depending on a small parameter 0 < δ ≪ 1 $0 < \delta \ll 1$ , and we establish the existence of a solution to ( P δ ) $(P_{\delta })$ 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.

Keywords

• Unsteady non-Newtonian fluid flow
• Shear thickening and shear thinning fluids
• Heat transfer
• Tresca’s friction boundary conditions
• Non-linear parabolic variational inequality
• Fixed point technique