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Unformatted text preview: (R2 r2) (6.40) The laminarflow profile is thus a paraboloid falling to zero at the wall and reaching
a maximum at the axis
umax R2
4 d
(p
dx gz) (6.41) It resembles the sketch of u(r) given in Fig. 6.10.
The laminar distribution (6.40) is called HagenPoiseuille flow to commemorate the
experimental work of G. Hagen in 1839 and J. L. Poiseuille in 1940, both of whom
established the pressuredrop law, Eq. (6.1). The first theoretical derivation of Eq. (6.40)
was given independently by E. Hagenbach and by F. Neumann around 1859.
Other pipeflow results follow immediately from Eq. (6.40). The volume flow is
R Q R u dA
0 1
umax R2
2 0 umax 1
R4
8 r2
2 r dr
R2
d
(p
dx gz)  v v Thus the average velocity in laminar flow is onehalf the maximum velocity  eText Main Menu  Textbook Table of Contents  Study Guide (6.42) Chapter 6 Viscous Flow in Ducts Q
A V
For a horizontal tube ( z
iment, Eq. (6.1): Q
R2 1
umax
2 (6.43) 0), Eq. (6.42) is of the form predicted by Hagen’s exper8 LQ
R4 p (6.44) The wall shear is computed from the wall velocity gradient
du
dr w rR 2 umax
R 1d
R
(p
2 dx gz) (6.45) This gives an exact theory for laminar Darcy friction factor
f 8w
V2 or 8(8 V/d)
V2
flam 64
Vd 64
Red (6.46) This is plotted late...
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This note was uploaded on 10/27/2009 for the course MAE 101a taught by Professor Sakar during the Spring '08 term at UCSD.
 Spring '08
 Sakar

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