Chapter 14.5, Problem 62E

Stream function and vorticity The rotation of a three-dimensional velocity field V = 〈u, v, w〉 is measured by the vorticity ω = ▿ × V. If ω = 0 at all points in the domain, the flow is irrotational.

a.    Which of the following velocity fields is irrotational:

 V = 〈2, –3y, 5z〉 or V = 〈y, x – z, – y〉?

b.    Recall that for a two-dimensional source-free flow V = (u, v, 0), a stream function ѱ(x, y) may be defined such that u = ѱ_y and v = – ѱ_r. For such a two-dimensional flow, let ζ = k ·▿ × V be the k-component of the vorticity. Show that ▿² ѱ = ▿ · ▿ ѱ = – ζ.

c.    Consider the stream function ѱ (x, y) = sin x sin y on the square region R = {(x, y): 0 ≤ x ≤ π, 0 ≤ y ≤ π}. Find the velocity components u and v; then sketch the velocity field.

d.    For the stream function in part (c), find the vorticity function ζ as defined in part (b). Plot several level curves of the vorticity function. Where on R is it a maximum? A minimum?