Existence of global weak solutions for unsteady motions of incompressible chemically reacting generalized Newtonian fluids

TL;DR

This paper proves the existence of global weak solutions for a complex system modeling unsteady, chemically reacting non-Newtonian fluids with variable rheological properties, relevant to biological and industrial applications.

Contribution

It introduces a novel existence proof for solutions to a nonlinear PDE system with variable power-law index using advanced mathematical techniques.

Findings

Existence of global weak solutions established.

Applicable to rheology of synovial fluids.

Utilizes generalized Sobolev spaces with variable exponents.

Abstract

We study a system of nonlinear partial differential equations describing the unsteady motions of incompressible chemically reacting non-Newtonian fluids. The system under consideration consists of the generalized Navier-Stokes equations with a power-law type stress-strain relation, where the power-law index depends on the concentration of a chemical, coupled to a convection-diffusion equation for the concentration. This system of equations arises in the rheology of the synovial fluid found in the cavities of synovial joints. We prove the existence of global weak solutions of the non-stationary model by using the Galerkin method combined with generalized monotone operator theory and parabolic De Giorgi-Nash-Moser theory. As the governing equations involve a nonlinearity with a variable power-law index, our proofs exploit the framework of generalized Sobolev spaces with variable integrability exponent.