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Density variations

 

The density of water can, except for some extreme situations, be assumed to be constant. This reduces the equation of continuity to tex2html_wrap_inline4323, and removes the second viscosity term from the Navier-Stokes equation.
  eqnarray180

There are two competing terms in this reduced version of the Navier-Stokes equation, the nonlinear tex2html_wrap_inline4325-term, and the tex2html_wrap_inline4327-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 tex2html_wrap_inline4329 and tex2html_wrap_inline4331. These two values are usually composed to the dimensionless Reynolds number :
 eqnarray200
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 (tex2html_wrap_inline4333), which is quite a bit compared to the typical thickness of the film (tex2html_wrap_inline4335 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.
eqnarray214
where tex2html_wrap_inline4337 and tex2html_wrap_inline4339 are two positive constants.

It is a common assumption, that the Navier-Stokes equation (1.1) is an example of deterministic chaos gif, 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.


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Next: Statistical measures of hydrodynamics Up: Equations of motion for Previous: Equations of motion for

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