Solved: Stream function and vorticity The rotation of a

Chapter 14, Problem 62

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Stream function and vorticity The rotation of a threedimensional velocity field V = 8u, v, w9 is measured by the vorticity V = _ * V. If V = 0 at all points in the domain, the flow is irrotational. a. Which of the following velocity fields is irrotational: V = 82, -3y, 5z9 or V = 8y, x - z, -y9? b. Recall that for a two-dimensional source-free flow V = 1u, v, 02, a stream function c1x, y2 may be defined such that u = cy and v = -cx. For such a two-dimensional flow, let z = k # _ * V be the k-component of the vorticity. Show that _2c = _ # _c = -z. c. Consider the stream function c1x, y2 = sin x sin y on the square region R = 51x, y2: 0 x p, 0 y p6. Find the velocity components u and v; then sketch the velocity field. d. For the stream function in part (c), find the vorticity function z as defined in part (b). Plot several level curves of the vorticity function. Where on R is it a maximum? A minimum?