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Unformatted text preview: 4 C1 ln r C2 The constants are found from the two noslip conditions
u(r a) 0 1 2K
a
4 C1 ln a C2 u(r b) 0 1 2K
b
4 C1 ln b C2 The final solution for the velocity profile is
u 1
4 d
(p
dx gz) a2 a2 b2
a
ln
ln (b/a)
r r2 r=a
u(r) r
r=b
x u(r)  v v Fig. 6.15 Fully developed flow
through a concentric annulus.  eText Main Menu  Textbook Table of Contents  Study Guide (6.91) 6.6 Flow in Noncircular Ducts 363 The volume flow is given by
a Q u2 r dr d
(p
dx 8 b gz) a4 b4 (a2 b2)2
ln (a/b) (6.92) The velocity profile u(r) resembles a parabola wrapped around in a circle to form a
split doughnut, as in Fig. 6.15. The maximum velocity occurs at the radius
a2 b2
2 ln (a/b) r 1/2 u umax (6.93) This maximum is closer to the inner radius but approaches the midpoint between cylinders as the clearance a b becomes small. Some numerical values are as follows:
b
a
r
a 0.01
b
b 0.1 0.2 0.5 0.8 0.9 0.99 0.323 0.404 0.433 0.471 0.491 0.496 0.499 Also, as the clearance becomes small, the profile approaches a parabolic distribution,
as if the flow were between two parallel plates [Eq. (4.143)].
It is confusing to base the friction factor...
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 Spring '08
 Sakar

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