Nonlinear iterative approximation of steady incompressible chemically reacting flows

Abstract

We consider a system of nonlinear partial differential equations modelling steady flow of an incompressible chemically reacting non-Newtonian fluid, whose viscosity depends on both the shear-rate and the concentration; in particular, the viscosity is of power-law type, with a power-law index that depends on the concentration. We prove that the weak solution, whose existence was already established in the literature, is unique, given some strengthened assumptions on the diffusive flux and the stress tensor, for small enough data. We then show that the uniqueness result can be applied to a model describing the synovial fluid. Furthermore, in the latter context, we prove the convergence of a nonlinear iteration scheme; the proposed scheme is remarkably simple and it amounts to solving a linear Stokes–Laplace system at each step. Numerical experiments are performed, which confirm the theoretical findings.

Keywords

• Fixed point iteration
• Incompressible flow
• Non-Newtonian fluids
• Chemically reacting flow
• Synovial fluid