A combined finite element and finite difference approach has been developed to include the capability to model fluid and thermal transport for the filling of a die cavity by a fluid that has differing sensitivities to extensional and shear deformation rates. This is referred to here as a dual viscosity fluid. For the case of mildly convergent or divergent quasi two-dimensional flows, a viscosity model is described that has such a dual-viscosity character and in which shear and extension rate sensitivities are nearly separated. Filling simulation results can be generated rapidly in a modest computational environment. The range of cavities and molding materials that may be modeled realistically is widened by the inclusion of a dual viscosity model. The effect of wall quenching (freezing) increased with decreasing filling rate, while the effect of dual viscosity increased with increasing filling rate.