Steady Flow of a Non-Newtonian Fluid through a Contraction

The equations describing the steady two-dimensional flow of a dilute suspension of macromolecules, a non-Newtonian fluid, are numerically modeled using a finite-difference technique. The flow domain is composed of a parallel-walled inflow region, a contraction region in which the walls are rectangular hyperbolas and a parallel-walled outflow region. The problem is formulated in terms of the vorticity and stream function along with explicit dependence on the deviatoric stresses. For a Newtonian fluid the relationship between stress and deformation rate is a linear one; however, for a non-Newtonian fluid the relationship is more complex. Here the constitutive equation used is a three constant Oldroyd equation which is valid for dilute polymer solutions. Due to a singularity in the transformation to a natural set of coordinates (with respect to the contraction boundary) a Cartesian grid system is used throughout the domain. The irregular grid structure at the curved boundaries necessitates developing a method for determining the boundary values of the vorticity, stream function and stresses. An explicit differencing scheme is used to model the governing equations, with the advection terms in the equations modeled using upstream differencing. The effect of numerical viscosity on the flow structure is examined with respect to the boundary layer thickness along the curved boundary and, in addition, contour plots of the flow variables are presented for both the non-Newtonian fluid and the Newtonian solvent fluid.