Synovial fluid is a polymeric liquid which generally behaves as a viscoelastic fluid due to the presence of hyaluronan molecules. We restrict ourselves to the regime in which the fluid responds as a viscous fluid. A novel generalized power-law fluid model is developed wherein the power-law exponent depends on the concentration of the hyaluronan. Such a model will be adequate to describe the flows of such fluids as long as they are not subjected to instantaneous stimuli. Assuming two different structures for the form of the power law exponent, both in keeping with physical expectations, we numerically solve for the flow of the synovial fluid (described by the constraint of incompressibility, the balance of linear momentum, and a convection-diffusion equation for the concentration of hyaluronan) in a rectangular cavity. The solutions obtained with our models are compared with the predictions of those based on a model that has been used in the past to describe synovial fluids. While all the three models seem to agree well with available experimental results, one of the models proposed by us seems to fit the data the best; it would, however, be hasty to pass judgment based on this one particular experimental correlation.

We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier-Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentrationdependence 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. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method in the case of two space dimensions. Key technical tools include discrete counterparts of the Bogovskiȋ operator, De Giorgi's regularity theorem in two dimensions, and the Acerbi-Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

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We consider a system of PDEs describing steady motions of an incompressible chemically reacting non-Newtonian fluid. The system of governing equations is composed of the convectiondiffusion equation for concentration and generalized Navier-Stokes equations where the generalized viscosity depends polynomially on the shear rate (the modulus of the symmetric part of the velocity gradient) and the coupling is due to the dependence of the power-law index on the concentration.Namely, we assume that the viscosity is of the form ν(c, |Dv|) ∼ ν 0 (κ 1 + κ 2 |Dv| 2 ). This dependence of the power-law index on the solution itself causes the main difficulties in the analysis of the relevant boundary value problem. We generalize the Lipschitz approximation method and show the existence of a weak solution provided that the minimal value of the power-law exponent is bigger than d/2.