Page 135 - Bird R.B. Transport phenomena

120 Chapter 4 Velocity Distributions with More Than One Independent Variable

Those readers who are encountering the method of separation of variables for the
first time will have found the above sequence of steps rather long and complicated.
However, no single step in the development is particularly difficult. The final solution in
Eq. 4.1-40 looks rather involved because of the infinite sum. Actually, except for very
small values of the time, only the first few terms in the series contribute appreciably.
Although we do not prove it here, the solution to this problem and that of the pre-
ceding problem are closely related. 2 In the limit of vanishingly small time, Eq. 4.1-40 be-
comes equivalent to Eq. 4.1-15. This is reasonable, since, at very small time, in this
problem the fluid is in motion only very near the wall at у = 0, and the fluid cannot
"feel" the presence of the wall at у = b. Since the solution and result in Example 4.1-1 are
far simpler than those of this one, they are often used to represent the system if only
small times are involved. This is, of course, an approximation, but a very useful one. It is
often used in heat- and mass-transport problems as well.

EXAMPLE 4.1-3 A semi-infinite body of liquid is bounded on one side by a plane surface (the xz-plane). Ini-
tially the fluid and solid are at rest. At time t = 0 the solid surface is made to oscillate sinu-
Unsteady Laminar soidally in the x direction with amplitude X and (circular) frequency w. That is, the
o
Flow near an displacement X of the plane from its rest position is
Oscillating Plate
X(f) = X sin wt (4.1-41)
o
and the velocity of the fluid at у = 0 is then

V x (0, t) = —r- = X o O) COS O)t (4.1-42)
at
We designate the amplitude of the velocity oscillation by v 0 = X io and rewrite Eq. 4.1-42 as
0
iu)t
^ ( 0 , t) = v cos o>t = v dl{e } (4.1-43)
o
0
where Ш{г] means "the real part of z."
For oscillating systems we are generally not interested in the complete solution, but only
the "periodic steady state" that exists after the initial "transients" have disappeared. In this
state all the fluid particles in the system will be executing sinusoidal oscillations with fre-
quency IO, but with phase and amplitude that are functions only of position. This "periodic
steady state" solution may be obtained by an elementary technique that is widely used. Math-
ematically it is an asymptotic solution for t —> °o.

SOLUTION Once again the equation of motion is given by

(4.1-44)
dy 1
and the initial and boundary conditions are given n by
by
I.C.: at t < 0, v x = 0 for all у (4.1-45)
B.C.I: aty = 0, v = v for alH > 0 (4.1-46)
x z
B.C. 2: at у = °°, v = 0 for all f > 0 (4.1-47)
x
The initial condition will not be needed, since we are concerned only with the fluid response
after the plate has been oscillating for a long time.
We postulate an oscillatory solution of the form
v (y, t) = (4.1-48)
x

See H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, 2nd edition
2
(1959), pp. 308-310, for a series solution that is particularly good for short times.