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:
Appropriate for 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 are used 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 present a sequence of vector fields with real significance in fluid
dynamics, and we use the physical properties of the fluid flow to
illustrate the mathematical concepts that are presented.
Our examples are two-dimensional, inviscid,
incompressible flows. Inviscid means that we ignore the effects of
friction (see the Comments below), 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
- 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.
A number of fairly sophisticated Mathematica commands
were used to generate the demonstrations; these are extensively commented in
the Mathematica notebook included with this collection of demos. (To
download the Mathematica notebook, click here.
Carefully follow the directions that accompany this file in order to use it
with Mathematica or a compatible application.)
The Examples
Demonstration #1: Potentials
Demonstration #2: Streamlines
Demonstration #3: Motion by the Flow
Demonstration #4: Pressure and Force
Comments
-
A slimmer version of this demo is
used in the Calculus III course at Towson University. Computer laboratory
exercises are used in each calculus course in weekly labs. A portion of the
demo materials included here are used for a pair of labs that provide an
active learning experience. The students are primarily mathematics majors,
with some other science majors.
-
Extensions or further investigations related to the
material in this demo could include examples of viscous effects. Such
developments would be quite instructive and provide a visual comparison of a
viscous and non-viscous flow side-by-side.
-
This demo also can be used in a course on fluid
mechanics to aid in discussions of streamlines and velocity potentials for
basic plane potential flows. In addition, the demo is appropriate for
corresponding topics in heat transfer.
References
• Chorin, A.J. and Marsden, J.E.,
A Mathematical Introduction to Fluid
Mechanics, Springer-Verlag, 1990.
• Feistauer, M., Mathematical Methods in
Fluid Mechanics, Longman Scientific and Technical, 1993.
• Granger, R.M., Fluid Mechanics,
Dover Publications, 1995.
