Go with the Flow

Mike O'Leary

Towson University

Objective

To provide a collection of examples of physically significant vector fields that can be used to illustrate important topics in the study of vector calculus.

Level

Multivariable calculus or vector calculus.

Prerequisites

Basic notions of vector fields, including the notion of a potential function. The material on pressure and resultant forces requires a knowledge of line integrals, as does the proof that a shape flowing with the vector field conserves area.

Platform

There are two versions- a non-interactive version that requires only a web browser capable of rendering animated .gif files, and a fully interactive version that requires Mathematica.

Instructor's Notes

Vector fields find use in many
applications, especially in mathematical physics. Despite this, many
examples in current textbooks present vector fields exclusively as
algebraic, mathematical objects, divorced from their physical roots.
With this in mind, we will present a sequence of vector fields with
real significance in fluid dynamics, and we will use the physical
properties of the fluid flow to illustrate the mathematical concepts
that we present.

Our examples are two-dimensional, inviscid,
incompressible flows. Inviscid means that we ignore the effects of
friction, which is represented in fluid dynamics by the fluid's
viscosity. A vector field is incompressible if it satisfies .
This is the case for most physical fluids, including water and air far
from the speed of sound. Thus, each of our examples represents the two
dimensional motion of a physical fluid like water, provided we ignore
all effects arising from friction.

The examples are

- 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, and

- Flow around a Rankine half-body.

For each example, we begin by describing the flow, graphing the vector
field, and investigating how the different physical parameters change
the flow. We then demonstrate how to find the equipotential curves and
the streamlines for each flow. Next, we demonstrate how the objects
placed in the fluid will be moved and distorted by the flow, using a
sequence of animations. Finally we determine the corresponding pressure
field for each flow, and use it to determine the force the fluid flow
exerts on any obstacles in its path.

We use a number of fairly sophisticated Mathematica commands to generate the demonstrations; these are extensively commented.

The Examples

Demonstration #1: Potentials

Demonstration #2: Streamlines

Demonstration #3: Motion by the Flow

Demonstration #4: Pressure and Force

References