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(a) Expression for circulation about a closed contour of height h and length L
(b) Evaluate part (a) for h = b/2 and h = b
(c) Show that the same result is obtained from area integral of Stokes Theorem (Eq. 5.14)
Problem 5.85
[Difficulty: 3]
Given:
Velocity field for pressuredriven flow between stationary parallel plates
Find:
Solution:
We will apply the definition of circulation to the given velocity field.
s
d
V
Governing
Equations:
(Definition of circulation)
s
d
V
dA
V
A
(Stokes Theorem)
Assumptions:
(1) Steady flow
4
3
2
1
s
d
V
s
d
V
s
d
V
s
d
V
From the definition of circulation we break up the integral:
dx
b
y
b
y
U
j
dy
i
dx
i
b
y
b
y
U
s
d
V
1
ˆ
ˆ
ˆ
1
The integrand is equal to:
Therefore, the circulation is equal to:
Γ
0
L
x
U
0
b
1
0
b
d
L
0
x
U
h
b
1
h
b
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This note was uploaded on 10/19/2011 for the course EGN 3353C taught by Professor Lear during the Fall '07 term at University of Florida.
 Fall '07
 Lear
 Fluid Mechanics

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