Channel flow The flow in a long shallow channel is modeled by the velocity field F = (0, 1 – x 2 ), where R = {( x , y ): | x | ≤ 1 and |y| = 5}. a. Sketch R and several streamlines of F . b. Evaluate the curl of F on the lines x = 0, x = 1 4 , x = 1 2 , and x = 1. c. Compute the circulation on the boundary of R. d. How do you explain the fact that the curl of F is nonzero at points of R, but the circulation is zero?
Channel flow The flow in a long shallow channel is modeled by the velocity field F = (0, 1 – x 2 ), where R = {( x , y ): | x | ≤ 1 and |y| = 5}. a. Sketch R and several streamlines of F . b. Evaluate the curl of F on the lines x = 0, x = 1 4 , x = 1 2 , and x = 1. c. Compute the circulation on the boundary of R. d. How do you explain the fact that the curl of F is nonzero at points of R, but the circulation is zero?
Solution Summary: The author illustrates the region R and several streamlines of the velocity field F=langle 0,1-x2rangle .
calculate div(F) and curl(F).
F = (xy, yz, y² – x³)
Evaluate the circulation of G = xyi + zj + 5yk around a square of side 9, centered at the origin, lying in the yz-plane, and
oriented counterclockwise when viewed from the positive x-axis..
Circulation =
- 1.² F.dr = 328
Analyze the flow given by Φ(z) = z3
Chapter 17 Solutions
Calculus: Early Transcendentals, Books a la Carte, and MyLab Math with Pearson eText -- Title-Specific Access Card Package (3rd Edition)
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