Math 294


Lecture 3

Given by Prof. Alan Zehnder on 9/6/95 at 10:10 in Kaufmann Auditorium


Circulation and Flux


In Lecture 2 we talked about one type of line integral, the work integral. In this lecture we cover two more line integrals: the Circulation integral and the flux integral.


Flow and Circulation Integrals

If given a Vector Field, F, representing the flow of a fluid, we can define an integral of flow along a path C by:

where F (dot) T is the component of fluid velocity parallel to the path C.

This integral is exactly the same as the Work integral from

We can get a physical feel for what the circulation integral represents if we look at two situations: one where there is uniform flow, and one where the flow is rotating.

Here it is clear that the circulation is 0, as there is no flow around the path.

Here the circulation is greater than 0, since the flow clearly has a component along the path. By convention, the circulation is calculated around the path in counterclockwise direction. Flow in a clockwise direction would yield a negative circulation.


The Flux Integral

The flux integral measures flow across a path C.

The velocity of a fluid perpendicular to a path is F(dot)n, where n is the unit normal to the path C.

The flux integral is

Recall from before that T = ( dx/dt i + dy/dt j ) / ||v|| = v / ||v||

n = T x k = ( ( dx/dt i + dy/dt j ) x k ) / ||v||
= ( dy/dt i - dx/dt j ) / ||v||

Alternatively, n can be found by inspection. For example, Given a semicircle as below, we can easily pick out the normal vectors indicated.

An alternative method of computing the flux integral is as follows:

Pictorially the flux can be looked at in the following way:
With fluid leaving the area enclosed by the path, the flux is > 0.
With fluid entering the area enclosed, the flux < 0
If fluid flow is uniform, as much fluid enters the area as exits, and the flux is 0.


Example:

Find the Circulation and Flux for F = xi + yj and C is a semicircle:

We can do the integral in two parts - over C1, the circular portion of the path, and over C2, the flat part of the curve along the x-axis.

Paramatrizing the Curves:
C1: r1(t) = a cos t i + a sin t j; 0<t<Pi

C2: r2(t) = ati; -1<t<1

On C1: T1 = v(t) / ||v(t)|| = - a sin ti + a cos t j / ||v||
n1 = a cos t i + a sin t j / ||v||

On C2: T2 = i
n2 = -j


Note that the circulation was zero, as expected for a radial field - there is no flow around the path, and the flux was positive, as expected for a field expanding outward from the area enclosed by the path.


Lecture written by Lawrence C. Weintraub on Wednesday, September 13, 1995
Edited by Aric Shafran on Thursday, September 14, 1995
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