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Unformatted text preview: p
2 dx gz) (const)(r) (6.35) 2  v v Ask your instructor to explain this to you if necessary.  eText Main Menu  Textbook Table of Contents  Study Guide 6.4 Flow in a Circular Pipe 341 Thus the shear varies linearly from the centerline to the wall, for either laminar or turbulent flow. This is also shown in Fig. 6.10. At r R, we have the wall shear
1
R
2 w p gz
L (6.36) which is identical with our momentum relation (6.27). We can now complete our study
of pipe flow by applying either laminar or turbulent assumptions to fill out Eq. (6.35). LaminarFlow Solution Note in Eq. (6.35) that the HGL slope d( p
gz)/dx is negative because both pressure and height drop with x. For laminar flow,
du/dr, which we substitute in
Eq. (6.35)
du
dr 1
rK
2 K d
(p
dx gz) (6.37) Integrate once
1 2K
r
4 u C1 (6.38) The constant C1 is evaluated from the noslip condition at the wall: u
1 2K
R
4 0 0 at r C1 R
(6.39) 12
or C1
4 R K/ . Introduce into Eq. (6.38) to obtain the exact solution for laminar
fully developed pipe flow 1
4 u d
(p
dx gz)...
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 Spring '08
 Sakar

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