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
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:
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.