Page 133 - Bird R.B. Transport phenomena

118 Chapter 4 Velocity Distributions with More Than One Independent Variable
It is convenient to introduce the following dimensionless variables:

^o' b' b 2
The choices for dimensionless velocity and position ensure that these variables will go from 0
to 1. The choice of the dimensionless time is made so that there will be no parameters occur-
ring in the transformed partial differential equation:

^ = Л (4.1-21)
д д
дт)
дт
The initial condition is ф = 0 at r = 0, and the boundary conditions are ф = 1 at 17 = 0 and
ф = 0atT7 = 1.
We know that at infinite time the system attains a steady-state velocity profile #00(17) so
that at r = 00 Eq. 4.1-21 becomes
0 = — % (4.1-22)

with фес = 1 at 77 = 0, and ф = 0 at 77 = 1. We then get
х
Ф» = 1 - v (4.1-23)
for the steady-state limiting profile.
We then can write
) (4Л-24)
where ф is the transient part of the solution, which fades out as time goes to infinity. Substi-
(
tution of this expression into the original differential equation and boundary conditions then
gives for ф {
% = Щ (4.1-25)

with ф[ = ф at т = 0, and ф = 0 at 17 = 0 and 1.
{
ж
То solve Eq. 4.1-25 we use the "method of separation of (dependent) variables," in which
we assume a solution of the form
(4.1-26)
Substitution of this trial solution into Eq. 4.1-25 and then division by the product fg gives
1
dg
d f
2
1
=
"
U m (4Л 27)
The left side is a function of т alone, and the right side is a function of 17 alone. This means that
both sides must equal a constant. We choose to designate the constant as -c 2 (we could equally
2
well use с or +c , but experience tells us that these choices make the subsequent mathematics
somewhat more complicated). Equation 4.1-27 can then be separated into two equations
j- = ~c g (4.1-28)
2
2
^- + c f=0 (4.1-29)
drf
These equations has the following solutions (see Eqs. C.l-1 and 3):
g = Ае~ с2т (4.1-30)
/ = В sin ст7 + С cos C77 (4.1-31)
in which A, B, and С are constants of integration.