Analysis and approximation of incompressible chemically reacting non-Newtonian fluids - Ko, S (2018)
In this thesis, we consider a system of nonlinear partial differential equations modelling the motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a convection-diffusion equation for the concentration and the generalized Navier-Stokes equations, where the viscosity coeffiicient is a power-law-type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. The only established mathematical result for this model is the existence of weak solutions for the stationary model. As a subsequent mathematical study, we first construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in two and three space dimensions. Key technical tools include discrete counterparts of the Bogovski operator, De Giorgi-type regularity theorem in two dimensions, and the Acerbi-Fusco Lipschitz truncation of Sobolev functions. Then we move to the unsteady problem, and 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 theory exploits the framework of generalized Lebesgue and Sobolev spaces with variable-integrability exponent.