The density of water can, except for some extreme situations, be assumed to be
constant. This reduces the equation of continuity to ,
and removes the second viscosity term from the Navier-Stokes equation.
There are two competing terms in this reduced version of the
Navier-Stokes equation, the nonlinear -term, and the
-term. The latter of the two even out variations in the velocity
field, whereas the former has a tendency to enhance them. If you want to
estimate the relative
importance of the two terms, and thus see if velocity variations have a
tendency to grow or disappear, you should find the typical velocity and length
for the flow system, and then compare the magnitudes of
and . These two values are
usually composed to the dimensionless Reynolds
For low values of the Reynolds number you will see a laminar flow , whereas for for high values of the Reynolds number you will see a chaotic flow. The actual definition of what is a high Reynolds number depends on the shape of the system.
The bulk of a soap film consists mostly of water, so if you are looking at it
as a three-dimensional fluid, you can use the equations
1.3 and 1.4, but
if you want to study soap films as an example of (almost) two-dimensional
fluids it is not that simple. The thickness of a
soap film can vary by at least
one wavelength of the yellow sodium line (), which is quite
a bit compared to the typical thickness of the film (
depending on the circumstances). In two dimensions the density in the
equations of motion is not mass per volume but mass per area, so the
density of the film is proportional to the thickness of the film, and it is
not constant. I.e. we have to use the complete Navier-Stokes equation,
although in two dimensions. It is also necessary to add an equation that
describes the density variations. It should consist of a surface
tension term proportional to the curvature of the
surface and of a pressure term proportional to the Laplacian of the pressure.
where and are two positive constants.
It is a common assumption, that the Navier-Stokes equation (1.1) is an example of deterministic chaos , and that this is the cause for the difficulties we have with predicting the behaviour of fluid flows (and the weather). It must be noted that no actual proof of this assumption is available.
Thus small differences in the initial conditions will grow exponentially in time, and make it unrealistic to expect to be able to recreate the same flow pattern in two succeeding experiments. This leads to the study of hydrodynamic systems through statistical measures of the velocity field, where the exact flow pattern is of less importance.