On Existence analysis of steady flows of generalized Newtonian fluids with concentration dependent power-law index

We study a system of partial differential equations describing a steady flow of an incompressible generalized Newtonian fluid, wherein the Cauchy stress is concentration dependent. Namely, we consider a coupled system of the generalized Navier–Stokes equations and convection–diffusion equation with non-linear diffusivity. We prove the existence of a weak solution for certain class of models by using a generalization of the monotone operator theory which fits into the framework of generalized Sobolev spaces with variable exponent. Such a framework is involved since the function spaces, where we look for the weak solution, are “dependent” of the solution itself, and thus, we a priori do not know them. This leads us to the principal a priori assumptions on the model parameters that ensure the Hölder continuity of the variable exponent. We present here a constructive proof based on the Galerkin method that allows us to obtain the result for very general class of models.