Page 70 - Bird R.B. Transport phenomena

§2.4 Flow Through an Annulus 55

Integration of this first-order separable differential equation then gives

v y = —

We now evaluate the two constants of integration, Л and C by using the no-slip condi-
2/
tion on each solid boundary:
B.C. 1: at r = KR, V Z = 0 (2.4-7)
B.C. 2: at r = R, v z = 0 (2.4-8)

Substitution of these boundary conditions into Eq. 2.4-6 then gives two simultaneous
equations:
2
0 = к 2 - 2Л In к + C ; 0 = 1 + C 2 (2.4-9,10)
2
From these the two integration constants A and C are found to be
2
C 2 = - 1 ; 2A 2 = ! ~ " 2 (2.4-11,12)

These expressions can be inserted into Eqs. 2.4-4 and 2.4-6 to give the momentum-flux
distribution and the velocity distribution 1 as follows:
1— / ч 2
к
-
I
т — I I I 2 In (l /к)Щ (2.4-13)
2L [\Rj
(Ob — Ob \T?2 Г / > 2 2 )l
\ I - к (2.4-14)
/ ln(l/K)

Note that when the annulus becomes very thin (i.e., к only slightly less than unity), these
results simplify to those for a plane slit (see Problem 2B.5). It is always a good idea to
check "limiting cases" such as these whenever the opportunity presents itself.
The lower limit of к —> 0 is not so simple, because the ratio 1пСК/г)/1п(1/к) will al-
ways be important in a region close to the inner boundary. Hence Eq. 2.4-14 does not
simplify to the parabolic distribution. However, Eq. 2.4-17 for the mass rate of flow does
simplify to the Hagen-Poiseuille equation.
Once we have the momentum-flux and velocity distributions, it is straightforward
to get other results of interest:
(i) The maximum velocity is

2
2
[I - A (l - In A )] (2.4-15)
4/iX
2
where A is given in Eq. 2.4-12.
(ii) The average velocity is given by
• rdrd0
Jo J KR 2
П CR rdrdO
2ir

JO J
(iii) The mass rate of flow is w = irR (l - /c)p(z;>, or
2
2
2
4
h)R P Г
w = 8/xL (2.4-17)
1
H. Lamb, Hydrodynamics, Cambridge University Press, 2nd edition (1895), p. 522.