We study a system of partial differential equations describing a steady flow of an incompressible generalized Newtonian fluid, wherein the Cauchy stress depends on concentration. Namely, we consider a coupled system of the generalized Navier-Stokes equations (viscosity of power-law type with concentration dependent power index) and convection-diffusion equation with non-linear diffusivity. We focus on the existence analysis 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 (class of Sobolev-Orlicz spaces). Such results is then adapted for a suitable FEM approximation, for which the main tool of proof is a generalization of the Lipschitz approximation method.
