  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.
