Numerical simulations and global existence of solutions of two-dimensional flows of fluids with pressure- and shear-dependent viscosities

There is a considerable amount of experimental evidence that unequivocally shows that there are fluids whose viscosity depends on both the mean normal stress (pressure) and the shear rate. Recently, global existence of solutions for the flow of such fluids for the three-dimensional case was established by Málek, Nečas and Rajagopal. Here, we present a proof for the global existence of solutions for such fluids for the two-dimensional case. After establishing the global-in-time existence, we discretize the equations via the finite element method, outline the Newton type iterative method to solve the non-linear algebraic equations and provide numerical computations of the steady flow of such fluids in geometries that have technological significance.