Page 79 - Bird R.B. Transport phenomena

64 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow
2B.4 Laminar slit flow with a moving wall ("plane Couette flow"). Extend Problem 2B.3 by al-
lowing the wall at x = В to move in the positive z direction at a steady speed % Obtain (a) the
shear-stress distribution and (b) the velocity distribution. Draw carefully labeled sketches of
these functions.

- <3> L )B :
Answers: T = ]Х = НУН 1
XZ ~Ж'^ 2/xL
2B.5 Interrelation of slit and annulus formulas. When an annulus is very thin, it may, to a good
approximation, be considered as a thin slit. Then the results of Problem 2B.3 can be taken over
with suitable modifications. For example, the mass rate of flow in an annulus with outer wall
of radius R and inner wall of radius (1 - e)R, where e is small, may be obtained from Problem
2B.3 by replacing 2B by sR, and Wby 2тг(1 - \e)R. In this way we get for the mass rate of flow:
Vp
IV = (2B.5-1)

Show that this same result may be obtained from Eq. 2.4-17 by setting к equal to 1 - e every-
where in the formula and then expanding the expression for w in powers of s. This requires
using the Taylor series (see §C2)
In (\-e)= -e- \e 2 - \е ъ - \e A (2B.5-2)

and then performing a long division. The first term in the resulting series will be Eq. 2B.5-1. Cau-
tion: In the derivation it is necessary to use the first four terms of the Taylor series in Eq. 2B.5-2.
2B.6 Flow of a film on the outside of a circular tube (see Fig. 2B.6). In a gas absorption experi-
ment a viscous fluid flows upward through a small circular tube and then downward in lami-
nar flow on the outside. Set up a momentum balance over a shell of thickness Ar in the film,

Velocity
distribution
inside tube Velocity
distribution
outside
in film

Ar

z-Momentum z-Momentum
into shell out of shell
of thickness of thickness
Ar Ar

T Gravity force
acting on
the volume
2тгтАгЬ

Fig. 2B.6 Velocity distribution and z-momentum
balance for the flow of a falling film on the outside
of a circular tube.