Curl of Nonlinear Flows

• Movie #4: Surface flow in a narrow channel. The flow is modeled by F(x,y,z)=(4-4y^2, 0, 0).
• Movie #5: Surface flow approximating two nonlinear vortices (whirlpools). The flow is modeled by F(x,y,z)=(-y^3, -x+x^3, 0).

Question #3

1. Compute the curl of the velocity field.
2. Plot the third component of the curl as a function of (x,y). (We have provided an example of such a plot.)
3. Does your previous conjecture (Question #2) hold for this flow? In particular, look at trajectories passing through points that have zero curl. What is true about the speed of neighboring trajectories at points of zero curl? At points of positive curl?

Question #4

• Compute the circulation about the circle (centered at the origin) with radius r for the velocity field.
• You now have circulation as a function of radius. Plot this function for r between [0,1].
• For what value of r does the circulation of water most impede your effort. That is, when you swim against the current, you do work. For what radius is the work you do the most negative?