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

摘要

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 apriori do not know them. This leads us to the principal apriori assumptions on the model parameters that ensure the H?lder 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.