The Navier-Stokes equation is a rather complicated-looking nonlinear partial
differential equation. In order to make some progress in understanding the
possible solutions, it is useful to do some dimensional analysis and to make
some estimates. For this purpose, we want to compare the inertial terms on
the left hand side to the viscous terms on the right hand side. Let's assume
that the flow is steady, so that . Let *U* be
a characteristic speed of the flow (usually taken to be the free-stream
speed far from the object), and let *L* be some characteristic dimension of
the flow (the typical size of an object in the fluid, say). Then for the
inertial term,

For the viscous term,

Both of these quantities have the same dimension, so we can define a
dimensionless number, the *Reynolds number* Re, by taking the ratio:

with n = h/r the kinematic viscosity.

To get an idea of the magnitude of the Reynolds number, consider a car
moving at a speed of 55 mi/hr (= 24 m/s). An appropriate value of *L*
would be the typical width of the car, which I would estimate to be about
1.5 m. The Reynolds number for the flow of air about the car is then

This is fairly typical--large Reynolds number flow tends to be the rule in most situations.

Sun Sep 28 22:13:11 EDT 1997