Four Linear Continuum Theories - Continuum Mechanics of Anisotropic Materials - page 157

In the case of pipe flow under a steady pressure gradient the differential equation
corresponding to

2 v 1
@

m @

x 2 ¼ @ p

is, in cylindrical coordinates r and z (see Sect. A.14), is

@x 1

@r

given by r @r r @v z ¼ @ p

@z where the pressure gradient is assumed to be a constant. The
solution to this equation in cylindrical coordinates subject to the “no slip” boundary
condition at the pipe wall is a similar parabolic profile to the one obtained above

r o
4

;

r 2
r o

n z ¼ @

p

1

@ z m

where r o is the radius of the pipe and r and z are two of the three cylindrical
coordinates. The volume flow rate is given by

p

Z r o

r o
8

r o ¼ @

p

Q ¼ 2 p r n z d r ¼ np

m ;

@ z

0

where is the mean velocity.

Historical Note: The solution to the Navier-Stokes equations for steady flow in a pipe is
called Poiseuille flow after Jean-Louis-Marie Poiseuille (1799-1869), a Parisian physician
and physiologist interested in the flow of blood. Poiseuille received his medical degree in
1828 and established his practice in Paris. He developed an improved method for measur-
ing blood pressure. He also is believed to be the first to have used the mercury manometer
to measure blood pressure. In the 1840s Poiseuille experimentally determined the basic
properties of steady laminar pipe flow using water as a substitute for blood. The formula for
Q above is rewritten below with the negative pressure gradient expressed as the change in
pressure

n

D p along the entire length of pipe divided by the pipe length L, thus

r o ¼ D

p p r o

Q ¼ np

L : 8 m

This formula had not been derived when Poiseuille did his very careful experimental
work, which demonstrated its principal features using capillary tubes of glass (models of
the blood capillary vessels). Poiseuille showed the volume flow rate Q was proportional to
the pressure drop along the pipe p, to the fourth power of the radius r o of the pipe and

inversely proportional to the length of the pipe, L. In honor of Poiseuille the unit of viscosity
is call the poise. The poise has the symbol P and it is equal to one (dyne-second)/
(centimeter) 2 or 0.1 Pas .
Problems

D

6.4.1. Verify the calculation of the Navier-Stokes equations ( 6.37 ) by (1)
substituting the rate-of-deformation-velocity (2.32) into the stress-strain
relations (5.11N), then (2) substituting the resulting expression relating the
stress to the first derivatives of the velocity into the three stress equations of
motion ( 6.36 ).
6.4.2. Find the constant c in @ tr D 2

@ tr D ¼ @ D ij D ji @D kk ¼

c tr D

2

6.4.3. Prove @ tr ð dev D Þ

¼

0.
6.4.4. Determine the shear stress acting on the lower plate in example 6.4.2.

@ tr D