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 Overscript[v, ⇀] is incompressible if it satisfies div Overscript[v, ⇀] = 0. 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