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EXAMPLE 4 Horizontal channel flow Consider the velocity field v = (0,1 – x², 0), for |x| < 1 and |z| < 1, which represents a horizontal flow in the y-direction (Figure 17.65a). a. Suppose you place a paddle wheel at the point P(, 0, 0). Using physical arguments, in which of the coordinate directions should the axis of the wheel point in order for the wheel to spin? In which direction does it spin? What happens if you place the wheel at e(-4, 0, 0)? b. Compute and graph the curl of v and provide an interpretation. Horizontal channel flow v = (0, 1 – x², 0) P. Paddle wheel with vertical axis spins clockwise, forx >0, and counterclockwise, for x<0.

EXAMPLE 4 Horizontal channel flow Consider the velocity field v = (0,1 – x², 0), for |x| < 1 and |z| < 1, which represents a horizontal flow in the y-direction (Figure 17.65a). a. Suppose you place a paddle wheel at the point P(, 0, 0). Using physical arguments, in which of the coordinate directions should the axis of the wheel point in order for the wheel to spin? In which direction does it spin? What happens if you place the wheel at e(-4, 0, 0)? b. Compute and graph the curl of v and provide an interpretation. Horizontal channel flow v = (0, 1 – x², 0) P. Paddle wheel with vertical axis spins clockwise, forx >0, and counterclockwise, for x<0.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter8: Applications Of Trigonometry
Section8.3: Vectors
Problem 60E
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In the given problem, explain why a paddle wheel with its axis
aligned with the z-axis does not spin when placed on the y-axis.

EXAMPLE 4 Horizontal channel flow Consider the velocity field v = (0,1 – x², 0), for |x| < 1 and |z| < 1, which represents a horizontal flow in the y-direction (Figure 17.65a). a. Suppose you place a paddle wheel at the point P(, 0, 0). Using physical arguments, in which of the coordinate directions should the axis of the wheel point in order for the wheel to spin? In which direction does it spin? What happens if you place the wheel at e(-4, 0, 0)? b. Compute and graph the curl of v and provide an interpretation.
Transcribed Image Text:EXAMPLE 4 Horizontal channel flow Consider the velocity field v = (0,1 – x², 0), for |x| < 1 and |z| < 1, which represents a horizontal flow in the y-direction (Figure 17.65a). a. Suppose you place a paddle wheel at the point P(, 0, 0). Using physical arguments, in which of the coordinate directions should the axis of the wheel point in order for the wheel to spin? In which direction does it spin? What happens if you place the wheel at e(-4, 0, 0)? b. Compute and graph the curl of v and provide an interpretation.
Horizontal channel flow v = (0, 1 – x², 0) P. Paddle wheel with vertical axis spins clockwise, forx >0, and counterclockwise, for x<0.
Transcribed Image Text:Horizontal channel flow v = (0, 1 – x², 0) P. Paddle wheel with vertical axis spins clockwise, forx >0, and counterclockwise, for x<0.
Expert Solution
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The given vector field is v=0,1-x2,0 which represents flow in the y-direction.

(a) Suppose the paddle wheel is placed on P12,0,0, then in what should the axis of the wheel be placed so that wheel can spin. Explain same when it is placed on Q-12,0,0.

(b) To find: Curl of the vector field and explain its interpretation.

 

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