Finite difference calculation of a planar 4:1 contraction flow of the Leonov-like fluid (i.e., the 1-mode Giesekus model with mobility parameter \(\alpha =)\) has been performed on a non-uniform grid system. In discretizing the constitutive equations, an upwind corrected scheme involving an artificial viscosity term is introduced to attain second- order accuracy and unconditional stability, so that the whole system of the governing equations and the boundary conditions can be made second- order accurate for creeping flow, and nearly so for small Reynolds number flows.
In the case of creeping flow, the numerical convergence and stability can be obtained up to higher values of the Weissenberg number, compared with those obtained by discretizing the constitutive equations with a simple upwind difference scheme on a uniform grid system. As the Weissenberg number increases, the corner vortex growth and the percent axial velocity overshoot along the center plane increase, which agrees qualitatively well with extant flow visualization and velocity measurement experiments using shear-thinning fluids. Besides, the distributions of the first normal stress difference for large Weissenberg number flows in the narrow channel exhibit qualitatively similar patterns as observed in existing flow birefringence experiments. For non-vanishing Reynolds number flows, it has been predicted that, as the flow rate, i.e., the viscoelastic Mach number, increases the corner vortex growth will decrease for small elasticity number fluid and increase for large elasticity number fluid.