In the given problem, explain why a paddle wheel with its axisaligned with the z-axis does not spin when placed on the y-axis. EXAMPLE 4 Horizontal channel flow Consider the velocity fieldv = (0,1 – x², 0), for |x| < 1 and |z| < 1, which represents a horizontal flow in they-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 thewheel to spin? In which direction does it spin? What happens if you place the wheel ate(-4, 0, 0)?b. Compute and graph the curl of v and provide an interpretation. Horizontal channel flowv = (0, 1 – x², 0)P.Paddle wheel with vertical axisspins clockwise, forx >0, andcounterclockwise, for x<0.

Answered: EXAMPLE 4 Horizontal channel flow…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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.

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

Step 1

The given vector field is $v = <0, 1 - x^{2}, 0>$ which represents flow in the y-direction.

(a) Suppose the paddle wheel is placed on $P (\frac{1}{2}, 0, 0)$ , then in what should the axis of the wheel be placed so that wheel can spin. Explain same when it is placed on $Q (- \frac{1}{2}, 0, 0)$ .

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

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