We'll start with the flow of a viscous fluid in a channel. The channel has a
width in the *y*-direction of *a*, a length in the *z*-direction of ,
and a length in the *x*-direction, the direction of flow, of . There is
a pressure drop along the length of the channel, so
that the constant pressure gradient is
(such a pressure gradient could be supplied by gravity, for instance).
Assuming the flow to be steady, . Also,
we'll assume that the flow is of the form ; then . The no-slip boundary condition at
the top and bottom edges of the channel reads . The
Navier-Stokes equation then becomes

Integrating twice, we obtain

where and are integration constants. To determine these, we impose the boundary conditions to obtain

We see that the velocity profile is a parabola, with the fluid in the center
of the channel having the greatest speed. Once we know the velocity profile
we can determine the flow rate *Q*, defined as the volume of fluid which
passes a cross section of the channel per unit time. This is obtained by
integrating the velocity profile over the cross sectional area of the
channel:

The analogous result for flow through a pipe of radius *a* and length *l* in
the presence of a uniform pressure gradient is

The important feature of both of these results is the sensitive dependence
upon either the channel width *a* or the pipe radius *a*. For instance, for
a pipe with a fixed pressure gradient, a 20% reduction in the pipe radius
leads to a 60% reduction of the flow rate! This clearly has important
physiological implications -- small amounts of plaque accumulation in
arteries can lead to very large reductions in the rate of blood flow.

Sun Sep 28 22:13:11 EDT 1997