Page 66 - Bird R.B. Transport phenomena

§2.3 Flow Through a Circular Tube 51

This first-order separable differential equation may be integrated to give

The constant C is evaluated from the boundary condition
2
B.C. 2: at r = R, v z = 0 (2.3-17)
2
(
From this C is found to be (SPQ —3> )R /4IJLL. Hence the velocity distribution is
2
L
(2.3-18)

We see that the velocity distribution for laminar, incompressible flow of a Newtonian
fluid in a long tube is parabolic (see Fig. 2.3-2).
Once the velocity profile has been established, various derived quantities can be
obtained:
(i) The maximum velocity v occurs at r = 0 and is
zmax
(2.3-19)

(ii) The average velocity (v ) is obtained by dividing the total volumetric flow rate by
z
the cross-sectional area
Г2тг fR
v
J J rdrd6 (9 _ ^ )R2
n rdrdd ~ Sal ~ 2 2 max
z
'
J Q J Q
2
(iii) The mass rate of flow w is the product of the cross-sectional area TTR , the density
p, and the average velocity (v )
z
„ = «»'-»№ (2 .3-21)
SfJLL
2
This rather famous result is called the Hagen-Poiseuille equation. It is used, along
with experimental data for the rate of flow and the modified pressure difference,
to determine the viscosity of fluids (see Example 2.3-1) in a "capillary viscometer."
(iv) The z-component of the force, F,, of the fluid on the wetted surface of the pipe is
just the shear stress r integrated over the wetted area
rz
2
F 2 = f -/x, ^ 4 = TTR (% - <3> )
L
2
= TTR ( 2 PO - p ) + irR Lpg (2.3-22)
L
This result states that the viscous force F z is counterbalanced by the net pres-
sure force and the gravitational force. This is exactly what one would obtain
from making a force balance over the fluid in the tube.

2 G. Hagen, Ann. Phys. Chem., 46,423-442 (1839); J. L. Poiseuille, Comptes Rendus, 11, 961 and 1041
(1841). Jean Louis Poiseuille (1799-1869) (pronounced "Pwa-ztf'-yuh," with б is roughly the "oo" in
book) was a physician interested in the flow of blood. Although Hagen and Poiseuille established the
dependence of the flow rate on the fourth power of the tube radius, Eq. 2.3-21 was first derived by E.
Hagenbach, Pogg. Annalen der Physik u. Chemie, 108,385-426 (1860).