Demonstration #3: Motion by the Flow

Another way to look at a
vector fields is to see how objects placed in the flow deform as they
move with the flow. In this demonstration we mark a ring in the fluid,
and watch how it moves along with the fluid.

When watching the demonstrations, it is
interesting to note that the area of the deformed ring always remains
constant. More generally, the area of any closed curve remains constant
as it flows along any of these vector fields. Indeed, let be a smooth closed curve, and let be the resulting curve after it has flowed with the vector field for a time , so that . For each fixed , let

,

be the parametrization of the curve . Then the area of the curve at time is

.

Thus

so that

.

Integration by parts and the fact that the curve is closed imply

.

We recognize this form as

.

Then, using Gauss's Theorem, we see that

where is the region enclosed by the curve . Because for all of the vector fields under consideration, we see that , as claimed.

Flow Around a Cylinder

Point Vortex Flow Around a Cylinder

Flow Around a Cylinder with Circulation

Flow in a Corner

Flow in a Closed Channel

Flow Around a Rankine Half-Body