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Steady flows of incompressible, nonviscous fluids

We want to first understand the behavior of some simple fluid flows. To begin with, we'll assume that the fluid is incompressible, which is not a particularly restrictive condition, and has zero viscosity (i.e., we consider nonviscous  fluids), which is a restrictive condition.

We recall that in an incompressible fluid, where the density r is just a constant, the equation of continuity reduces simply to Ñ·v = 0. If we know the vorticity w, then v can be determined from the equations

Ñ·v = 0

Ñ×v = w

much in the same way as B can be determined from Ñ·B = 0, Ñ×B = je, where je is the electric current. Historically, fluid dynamics was worked out first, by Euler and others, and then it turned out, amazingly, that very similar equations apply for electromagnetism.
There is still the question: what determines the vorticity w? We cannot answer this yet, because vorticity is generated by viscous forces. Even after we introduce viscosity (in the next chapter), we will have a hard time predicting w, because most vortices are generated in complicated, turbulent flow patterns. However, there are two classes of simple flows that we can already treat: irrotational flows, where w = 0, and steady flows, where v does not change with time and w, which is also time-independent, can be taken as given (of course the simplest flows are both steady and irrotational). An example of steady flow is the single vortex line we have already discussed.

We will show in the next section that irrotational flow is stable in an incompressible, nonviscous fluid; i.e., w remains 0 at all times if it is 0 initially. However, if vorticity is present it will in general move about and only some distributions of vorticity can stay fixed (w/t = 0) and give rise to a steady flow. Here are examples:

• a straight, fixed vortex line gives a steady flow, as we have seen; a straight vortex line moving with uniform speed gives a non-steady flow, which however looks steady when viewed by an observer moving along with the vortex line;
• a vortex ring of fixed radius, on the other hand, cannot sit still (in an infinite fluid), but must move with a well-defined speed, and the flow looks steady when viewed by an observer moving along with this speed.

We will also address the important question: what is the pressure inside the moving fluid? It would seem that we need to know the pressure on each fluid element before we can decide how it moves, but actually for an incompressible fluid the motion is strongly constrained by the equation of continuity, and the pressure adjusts itself to the fact that Ñ·v = 0. This has surprising consequences that we discuss.

V. Celli, Univ. of Virginia
Wed Sep 10 01:02:02 EDT 1997