Page 130 - Bird R.B. Transport phenomena

§4.1 Time-Dependent Flow of Newtonian Fluids 115
The first example illustrates the method of combination of variables (or the method of
similarity solutions). This method is useful only for semi-infinite regions, such that the ini-
tial condition and the boundary condition at infinity may be combined into a single new
boundary condition.
The second example illustrates the method of separation of variables, in which the partial
differential equation is split up into two or more ordinary differential equations. The so-
lution is then an infinite sum of products of the solutions of the ordinary differential
equations. These ordinary differential equations are usually discussed under the heading
of "Sturm-Liouville" problems in intermediate-level mathematics textbooks. 1
The third example demonstrates the method of sinusoidal response, which is useful in
describing the way a system responds to external periodic disturbances.
The illustrative examples are chosen for their physical simplicity, so that the major
focus can be on the mathematical methods. Since all the problems discussed here are lin-
ear in the velocity, Laplace transforms can also be used, and readers familiar with this
subject are invited to solve the three examples in this section by that technique.

EXAMPLE 4.1-1 A semi-infinite body of liquid with constant density and viscosity is bounded below by a hor-
izontal surface (the xz-plane). Initially the fluid and the solid are at rest. Then at time t = 0,
Flow near a Wall the solid surface is set in motion in the positive x direction with velocity v as shown in Fig.
0
Suddenly Set in Motion 4.4-1. Find the velocity v x as a function of у and t. There is no pressure gradient or gravity
force in the x direction, and the flow is presumed to be laminar.

SOLUTION For this system v x = v (y, t), v y = 0, and v z = 0. Then from Table B.4 we find that the equation
x
of continuity is satisfied directly, and from Table B.5 we get
2
dV x d V x
(4.1-1)

УК
f < 0
Fluid at
rest

Wall set in
motion
v o

t>0
Fluid in
unsteady
flow Fig. 4.1-1. Viscous flow of a fluid near a wall
suddenly set in motion.

1
See, for example, M. D. Greenberg, Foundations of Applied Mathematics, Prentice-Hall, Englewood
Cliffs, NJ. (1978), §20.3.