Existence Analysis for a Model Describing Flow of an Incompressible Chemically Reacting Non-Newtonian Fluid

Abstract

We consider a system of PDEs describing steady motions of an incompressible chemically reacting non-Newtonian fluid. The system of governing equations is composed of the convection-diffusion equation for concentration and generalized Navier--Stokes equations where the generalized viscosity depends polynomially on the shear rate (the modulus of the symmetric part of the velocity gradient) and the coupling is due to the dependence of the power-law index on the concentration. Namely, we assume that the viscosity is of the form $\nu(c,|\boldsymbol{D}\boldsymbol{v}|) \sim \nu_0 (\kappa_1+\kappa_2 |\boldsymbol{D}\boldsymbol{v}|^2)^{\frac{p(c(x))-2}{2}}.$ This dependence of the power-law index on the solution itself causes the main difficulties in the analysis of the relevant boundary value problem. We generalize the Lipschitz approximation method and show the existence of a weak solution provided that the minimal value of the power-law exponent is bigger than $d/2$.