Go with the Flow Demos with Positive Impact www.mathdemos.org Flow around a cylinder with circulation.

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

• 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. Access  to the labs is provided at the following links.
Vector Fields:
http://www.towson.edu/math/courses_textbooks/Calculus3Web/Lab%2012/index.html
Vector Fields and Line Integrals:
http://www.towson.edu/math/courses_textbooks/Calculus3Web/Lab%2013/index.html

• 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.

Credits:  This demo was submitted by Michael O'Leary, Department of Mathematics, Towson University and is included in Demos with Positive Impact with his permission.

DRH 8/16/2005     Last updated 1/4/2006

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