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 , | 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.
Highlights
1 Introduction Motivated by lubrication or injection/extrusion industrial processes, we consider in this paper an unsteady incompressible non-isothermal flow problem with non-linear boundary conditions of friction type for a general class of non-Newtonian fluids
We assume that the stress tensor is given by σ = 2μ θ, υ, D(υ) D(υ) p–2D(υ) – π IdR3, (1.1)
The case of stationary non-Newtonian fluids satisfying the general power law (1.1) is considered in [9], and thermal effects lead to a coupled fluid flow/heat transfer problem
Summary
Motivated by lubrication or injection/extrusion industrial processes, we consider in this paper an unsteady incompressible non-isothermal flow problem with non-linear boundary conditions of friction type for a general class of non-Newtonian fluids. Where μ is a given mapping, θ is the temperature, π is the pressure, υ is the velocity, D(υ) is the strain rate tensor, and p ∈ (1, +∞) is a real parameter
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