Fluids will invariably swirl around, and it is often useful to characterize
the amount of swirling. This is done by introducing the *circulation* , defined as

with the line integral taken is around a closed circuit *C* in a
counter-clockwise sense. By using Stokes' theorem we may write this as a
surface integral,

where the *vorticity* of the fluid
is defined as

If the vorticity of the fluid is zero, we say that the fluid is *irrotational* :

We've already mentioned that there are some useful analogies between
magnetostatics and incompressible fluid flow. I've collected these together
in Table 2.1 below. To transform the magnetostatics expressions
into those for fluid flow, we make the substitutions , , and (where here is the mass density of the fluid). If a fluid is
both incompressible *and* irrotational, then
and , which are the same as the equations of
electrostatics and magnetostatics in free space (i.e., in the absence of
charges and currents).

Magnetostatics | Incompressible fluid flow | |

(Ampére's Law) | (circulation) | |

(energy density of the magnetic field) | (kinetic energy density of a fluid) |

As an example of fluid flow with vorticity, we can consider a single *vortex* , which would be produced when draining a bathtub,
and which is illustrated in Fig. 2.5.

**Figure 2.5:** Velocity field of a vortex.
Flow lines are drawn for equal increments of the speed, as shown by arrows. Click here to see animations.

In cylindrical coordinates there is only one component of the velocity field, . In calculating the circulation, the line element , so that . If the circulation is independent of the integration path, then we must have , with

so that . Therefore, the velocity field of a vortex is

By making the substitutions suggested above, we can use this result to find
the magnetic field produced by a wire carrying a current *I*:

Wed Sep 10 01:02:02 EDT 1997