fluid_mechanics

6.9 Some Simple Solutions for Laminar, Viscous, Incompressible Fluids 315
V6.14 Complex
pipe flow
and therefore
Equation 6.148 for
yield
(6.149)
can be substituted into Eq. 6.149, and the resulting equation integrated to
(6.150)
This relationship can be expressed in terms of the pressure drop, ¢p, which occurs over a length,
/, along the tube, since
and therefore
v z
R
Q 2p v z r dr
Q pR 4
8m a 0p
0z b
0
¢p
/ 0p 0z
Q pR 4 ¢p
8m/
(6.151)
R
Poiseuille’s law relates
pressure drop
and flowrate for
steady, laminar flow
in circular tubes.
r
v z
v max
= 1 – (
r_
R)
2
v max
For a given pressure drop per unit length, the volume rate of flow is inversely proportional to the
viscosity and proportional to the tube radius to the fourth power. A doubling of the tube radius
produces a 16-fold increase in flow! Equation 6.151 is commonly called Poiseuille’s law.
In terms of the mean velocity, V, where V QpR 2 , Eq. 6.151 becomes
V R2 ¢p
8m/
The maximum velocity v max occurs at the center of the tube, where from Eq. 6.148
so that
v max R2
4m a 0p
0z b R2 ¢p
4m/
v max 2V
The velocity distribution, as shown by the figure in the margin, can be written in terms of
v z
v max
1 a r R b 2
(6.152)
(6.153)
as
(6.154)
As was true for the similar case of flow between parallel plates 1sometimes referred to as
plane Poiseuille flow2, a very detailed description of the pressure and velocity distribution in
tube flow results from this solution to the Navier–Stokes equations. Numerous experiments
performed to substantiate the theoretical results show that the theory and experiment are in
agreement for the laminar flow of Newtonian fluids in circular tubes or pipes. In general, the
flow remains laminar for Reynolds numbers, Re rV12R2m, below 2100. Turbulent flow in
tubes is considered in Chapter 8.
v max
F l u i d s i n t h e N e w s
Poiseuille’s law revisited Poiseuille’s law governing laminar
flow of fluids in tubes has an unusual history. It was developed in
1842 by a French physician, J. L. M. Poiseuille, who was interested
in the flow of blood in capillaries. Poiseuille, through a
series of carefully conducted experiments using water flowing
through very small tubes, arrived at the formula, Q K¢p D 4 / .
In this formula Q is the flowrate, K an empirical constant, ¢p the
pressure drop over the length /, and D the tube diameter. Another
formula was given for the value of K as a function of the water
temperature. It was not until the concept of viscosity was introduced
at a later date that Poiseuille’s law was derived mathematically
and the constant K found to be equal to p8m,
where
m is the fluid viscosity. The experiments by Poiseuille have long
been admired for their accuracy and completeness considering
the laboratory instrumentation available in the mid nineteenth
century.