Page 146 - Bird R.B. Transport phenomena

130 Chapter 4 Velocity Distributions with More Than One Independent Variable

Note that the modified pressure distribution is symmetric about the x-axis; that is, for poten-
7
tial flow there is no form drag on the cylinder (d'Alembert's paradox). Of course, we know
now that this is not really a paradox, but simply the result of the fact that the inviscid fluid
does not permit applying the no-slip boundary condition at the interface.

EXAMPLE 4.3-2 Show that the inverse function
Flow Into a z(w) = — + — (4.3-26)
Rectangular Channel
represents the potential flow into a rectangular channel of half-width b. Here i> is the magni-
x
tude of the velocity far downstream from the entrance to the channel.

SOLUTION First we introduce dimensionless distance variables
iry
• II Л. -w z = x + /y = 7TZ (4.3-27)
~Т ~'ь
and the dimensionless quantities
ттф
Ф = (4.3-28)
bv x bv x
The inverse function of Eq. 4.3-26 may now be expressed in terms of dimensionless quantities
and split up into real and imaginary parts
Z = W + e w = (Ф + е ф cos 40 + /(¥ + е ф sin 40 (4.3-29)
Therefore

X = Ф + е ф cos 4> Y = 4' + е ф sin (4.3-30,31)
We can now set 4^ equal to a constant, and the streamline У = У(Х) is expressed parametrically
in Ф. For example, the streamline 4^ = 0 is given by

X = Ф У = 0 (4.3-32, 33)
As Ф goes from -oo to +°°, X also goes from -oo to +o°; hence the X-axis is a streamline.
Next, the streamline ^ = IT is given by
X = Ф - е ф Y=7T (4.3-34, 35)

As Ф goes from — oo to +°o, X goes from — oo to —1 and then back to — oo; that is, the stream-
line doubles back on itself. We select this streamline to be one of the solid walls of the rectan-
gular channel. Similarly, the streamline 4^ = -тт is the other wall. The streamlines 4> = C,
where -тт < С < тт, then give the flow pattern for the flow into the rectangular channel as
shown in Fig. 4.3-2.
Next, from Eq. 4.3-29 the derivative -dz/dzv can be found:

w
~ =-!-%,= - J T (1 + e ) = ~ (1 + е ф cos 4' + /У" sin 40 (4.3-36)
dw v * dW v r- v *
Comparison of this expression with Eq. 4.3-15 gives for the velocity components
v v ~ = -(1 4- е cos 40 -^ = ~(е ф sin 40 (4.3-37)
ф
r
These equations have to be used in conjunction with Eqs. 4.3-30 and 31 to eliminate Ф and
in order to get the velocity components as functions of position.

7 Hydrodynamic paradoxes are discussed in G. Birkhoff, Hydrodynamics, Dover, New York (1955).