fluid_mechanics

392 Chapter 8 ■ Viscous Flow in Pipes
Newtonian fluid, the shear stress is simply proportional to the velocity gradient,
1see Section 1.62. In the notation associated with our pipe flow, this becomes
“t m dudy”
t m du
dr
(8.6)
The negative sign is included to give t 7 0 with dudr 6 0 1the velocity decreases from the pipe
centerline to the pipe wall2.
Equations 8.3 and 8.6 represent the two governing laws for fully developed laminar flow of
a Newtonian fluid within a horizontal pipe. The one is Newton’s second law of motion and the
other is the definition of a Newtonian fluid. By combining these two equations we obtain
which can be integrated to give the velocity profile as follows:
or
du ¢p
a
dr 2m/ b r
du ¢p
2m/ r dr
u a ¢p
4m/ b r 2 C 1
where C 1 is a constant. Because the fluid is viscous it sticks to the pipe wall so that u 0 at
r D2. Thus, C 1 1¢p16m/2D 2 . Hence, the velocity profile can be written as
u1r2 a ¢pD2
2
bc1 a2r
16m/ D b d V c c 1 a 2r 2
D b d
(8.7)
Under certain restrictions
the velocity
profile in a pipe
is parabolic.
R
dr
r
dA = 2 πr
dr
where V c ¢pD 2 116m/2 is the centerline velocity. An alternative expression can be written by using
the relationship between the wall shear stress and the pressure gradient 1Eqs. 8.5 and 8.72 to give
where R D2 is the pipe radius.
This velocity profile, plotted in Fig. 8.9, is parabolic in the radial coordinate, r, has a maximum
velocity, V c , at the pipe centerline, and a minimum velocity 1zero2 at the pipe wall. The volume
flowrate through the pipe can be obtained by integrating the velocity profile across the pipe.
Since the flow is axisymmetric about the centerline, the velocity is constant on small area elements
consisting of rings of radius r and thickness dr as shown in the figure in the margin. Thus,
or
Q u dA
rR
u1r2 t wD
4m c 1 a r R b 2
d
r0
u1r22pr dr 2p V c R
Q pR2 V c
2
By definition, the average velocity is the flowrate divided by the cross-sectional area,
V QA QpR 2 , so that for this flow
0
c 1 a r R b 2
d r dr
and
V pR2 V c
2pR 2
V c
2 ¢pD2
32m/
Q pD4 ¢p
128m/
(8.8)
(8.9)