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[1]. 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* [2],
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.

Tue Jan 21 16:43:41 EST 1997