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Conservation of Momentum

We now consider the motion of the substance. Since we can expect the
conservation of momentum to apply we model motion through momentum.

Again looking at a finite volume we see that rate of change in momentum in
the volume must somehow equal the ``momentum flux'' crossing the boundaries,
i.e. the rate of momentum entering the volume minus the rate of momentum
exiting.

This handles advective mass transport, where mass pushes mass ahead.
Unfortunately this does not account for an important phenomena: Imagine a fast
moving flow streaming through a slower moving volume of flow. Naturally, even
though both flows move in the same direction, some interaction will take place
along the boundary between the flows, resulting in a diffusion of momentum as
friction affects the two flows by decreasing the momentum on the boundary layer
of the faster moving flow, and increasing it on the boundary of the slower
moving flow, see figure 2.1

Figure 2.1: Example of shear in a two dimensional
laminar flow. The viscosity of the media causes shear between the fast moving
flow on the top and slow moving flow at the bottom. The arrows illustrate
velocities, the dotted line the velocity profile.

. This effect, which stems from
intermolecular collision, is known as shear stress, and may be modeled using
Newton's law of viscosity2.7, which when considering a simple laminar flow
in two dimensions state that the shear stress is proportional to the negative velocity gradient and
viscosity :

In a general three dimensional case things get a little more complicated.
Shear stress now occur in two dimensions on the plane tangent to the point
considered. The traditional derivation is non-trivial and quite lengthy, a
derivation of a similar expression can be found in [9],
p. 38-43. In numerical simulations, see [2]
and [20],
a slightly different, but analogous, form is used:

Here Sij represents the
deformation. Note for later reference that the viscosity can be removed from the shear stress tensor through the kinematic viscosity and density , since . Also note that later we introduce an approximation
to the true dynamic viscosity, which then becomes dependent of dimensions. For
this reason we already now introduce the notation , where the indices may be ignored in the case of uniform
viscosity.

As one could expect gravity and pressure also influence the momentum of
matter. We will not indulge in explaining the correlations between the three
phenomena of stress, gravity, and pressure induced movement, but only note that
they can be coupled into the equation in a straightforward manner, if the drains
and sinks are interpreted as the forces acting upon the volume. The manner in
which they enter are a direct consequence of this physical formulation.
Collectively we find2.8 for mass density and mass velocity vector u, shear stress tensor , pressure p, gravity vector g:

It is practical to rewrite this equation by collecting the
tensors under the divergence operator:

The velocity may be determined by dividing the momentum with the density, all
though one must be aware that this does become ill-defined if in a vacuum of
e.g. a post-shock region.

Peter Søndergaard & Jan Vittrup Hansen