A fluid is categorized as a substance that cannot sustain a shear force when at rest. However, it can sustain and transmit a shear force when in motion. The proportionality between the shear force per unit area (or stress ) and an appropriate velocity gradient in a shearing flow defines the viscosity of the fluid as . Called constitutive equation of the fluid, this relation holds true both for Newtonian and non-Newtonian fluids, except that the viscosity is constant for Newtonian and rather a function of the velocity gradient for non-Newtonian fluids. Thus, it is highly non-linear for the latter category.
Although governing equations describe the fluid motion, often simplifications of these equations are used to define the fluid's category, e.g. incompressible flow is associated with very small values of the Mach number. Similarly, in Newtonian mechanics, the Reynolds number is important to determine if the flow is laminar or turbulent. For viscoelastic fluids (non-Newtonian), the key dimensionless number is the Deborah number (De) which is interpreted as the ratio of the magnitude of the elastic forces to that of the viscous forces. The understanding of the constitutive properties of the non-Newtonian fluids has been a subject of both experimental and numerical studies. Often test problems, such as contraction flow, are used to test the numerical scheme's ability to deal with the singularities in the geometries and initial velocity profiles. A common problem is that numerical schemes break down when De number reaches a certain limit. One approach proposed by Malkus et all , is the inclusion of spurt phenomena in the model: velocity gradient can contain jumps. In order to analyze the spurt phenomena, though, fully dynamic equations need to be considered. Solving fully dynamic equations is a computationally intensive problem, and indeed that has been the reason behind the motivation for developing a parallel scheme.