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** = **j**_{e},
where **j**_{e} 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.

- Ideal fluids and Euler's equation
- Bernoulli's equation
- Applications of Bernoulli's equation
- The Pitot tube
- The Venturi effect
- Atomizers
- Hurricanes
- Flapping flags and the Kelvin-Helmholtz instability

- Drag and D'Alembert's ``paradox''

Wed Sep 10 01:02:02 EDT 1997