Page 65 - Essentials of physical chemistry

Viscosity of Laminar Flow 27

P dr P

r R
P P

d

FIGURE 2.2 Calculus diagram of ‘‘sliding sleeves’’ of fluid flowing in a cylinder.

Note this maintains the units described above for the sliding layers since derivative (dv=dr) has units
of velocity=length and the total expression is a drag force. However, we need a sign reversal to
account for the fact that the velocity decreases as the radius increases. (Let us use d ¼ l to avoid
conflict with d=dr.):

dv dv
f ¼ hA ¼ h(2prl) :
dr dr
What is the force driving the fluid? We know from above that it is a difference in pressure but for the
time being let us just call P ¼ (P 1 P 2 ) and note that a pressure is a (force=area), so we need to
multiply the pressure by the cross-sectional area of the tube:

dv
2
f ¼ h(2prl) ¼ P(pr ):
dr
By canceling pr and splitting the derivative into separate differentials, we obtain the variation of
velocity in terms of the radius as

P
rdr,
2hl
dv ¼
v r P P r
ð ð ð
which can be integrated as dv ¼ rdr ¼ rdr.
0 R 2hl 2hl R
Here, we use v ¼ 0at r ¼ R because the velocity is zero at the outer wall of the pipe. With these
limits, the integrals are easy to do and we get
2 2
P r R P 2 2
(R r ):
v(r) ¼ ¼
2hl 2 4hl