Page 131 - Bird R.B. Transport phenomena

116 Chapter 4 Velocity Distributions with More Than One Independent Variable

in which v = /A/p. The initial and boundary conditions are

I.C.: atf<0, v = 0 for all у (4.1-2)
x
B.C.I: aty = 0, v = v for alH > 0 (4.1-3)
x o
B.C. 2: at у = oo, = 0 for all t > 0 (4.1-4)
Vx
Next we introduce a dimensionless velocity ф = v /v , so that Eq. 4.4-1 becomes
x 0
= 1
-
£ 7т (4Л 5)
with ф(у, 0) = 0, ф(0, 0 = 1/ and ф(°°, 0 = 0. Since the initial and boundary conditions con-
tain only pure numbers, the solution to Eq. 4.1-5 has to be of the form ф = ф(у, t; v). However,
since ф is a dimensionless function, the quantities y, f, and v must always appear in a dimen-
sionless combination. The only dimensionless combinations of these three quantities are
y/Wt or powers or multiples thereof. We therefore conclude that

ф = ф(г)), where rj = (4.1-6)

This is the "method of combination of (independent) variables." The "4" is included so that
the final result in Eq. 4.1-14 will look neater; we know to do this only after solving the prob-
lem without it. The form of the solution in Eq. 4.1-6 is possible essentially because there is no
characteristic length or time in the physical system.
We now convert the derivatives in Eq. 4.1-5 into derivatives with respect to the "com-
bined variable" rj as follows:

=
=
~at Jn^i ~2~t~dr] ( 4 Л " 7 )
ay dr) dy dr) y/4pt
Substitution of these expressions into Eq. 4.1-5 then gives

(^ф dф
-—: + 2г)-у- = 0 (4.1-9)
drf dr)
This is an ordinary differential equation of the type given in Eq. C.I-8, and the accompanying
boundary conditions are
B.C1: at 77 = 0, ф = \ (4.1-10)
B.C. 2: at 77 = °o, ф = 0 (4.1-11)
The first of these boundary conditions is the same as Eq. 4.1-3, and the second includes
Eqs. 4.1-2 and 4. If now we let dф/dr) = ф, we get a first-order separable equation for ф,
and it may be solved to give

2
^ рг] ) (4.1-12)
A second integration then gives

ф = d Гехр(-^ ) d^ + C 2 (4.1-13)
2
Jo
The choice of 0 for the lower limit of the integral is arbitrary; another choice would lead
to a different value of C , which is still undetermined. Note that we have been careful
2
to use an overbar for the variable of integration (rj) to distinguish it from the r\ in the upper
limit.