calculus. Whether in solid mechanics or in fluid mechanics, engineers encounter the concepts of gradients, divergence, and curl. Our personal experience has been that engineering students are often exceptionally talented at manipulating and computing with these vector operations, but are much less comfortable with the relationship between the symbols and the geometry of fluid flow. As a result when students encounter that advanced theorems of Green, Gauss, and Stokes, the theorems have no meaning, other than a set of symbols to memorize.

A module on the geometry of fluid flow consists of five labs in which the students investigate the geometric meaning of gradients, divergence, curl, flux, and path integrals. For example, in the lab on divergence, the students are lead to the realization that the divergence at a sink is negative and the divergence at a source is positive. They explore the geometry of fluid flux across a boundary and discover for themselves that the flux across the boundary of a region is equal to the divergence over the region (this is known as the divergence theorem.) They also collect data that leads them to conjecture that the divergence at a point is a limit of the flux across the boundary of a ball centered at the point of interest as the radius of that ball shrinks to zero.

A subsequent lab introduces students to the geometry of curl, and shows them movies of model fluid flows (see Figure 2). By computing the curl at various locations and comparing their answers to the behavior of the fluid near those points in the movies, students develop an intuition for the geometry of curl. Essentially, the main idea is to imagine yourself on a raft floating down a river. You drop a tennis ball to the left of the raft and another to your right. If the tennis balls move at different speeds than your raft, then the river has curl. The analogous situation for three dimensional fluid flow in more complicated, but also accessible.

Figure 2: One frame in an on-line movie depicting an ideal fluid flow in a narrow channel. Students use movies such as this to help them understand the geometry of curl.

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Author: Frederick J. Wicklin <fjw@geom.umn.edu>
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Created: Fri Aug 18 1995 --- Last modified: Jul 21 1996
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