Page 160 - Bird R.B. Transport phenomena

144 Chapter 4 Velocity Distributions with More Than One Independent Variable

(d) Evaluate the total force of the fluid on the sphere to obtain
F z = \irR pg + 4TT/JLRV X (4B.3-5)
2>
This result may be obtained by the method of §2.6 or by integrating the z-component of - [n • IT]
over the sphere surface (n being the outwardly directed unit normal on the surface of the
sphere).
4B.4 Use of the vorticity equation.
(a) Work Problem 2B.3 using the y-component of the vorticity equation (Eq. 3D.2-1) and the
following boundary conditions: at x = ±B, v z = 0 and at x = 0, v z = v Zimax . Show that this
leads to
v z = i> , [l ~ (x/B) ] (4B.4-1)
2
2 max
Then obtain the pressure distribution from the z-component of the equation of motion.
(b) Work Problem 3B.6(b) using the vorticity equation, with the following boundary con-
ditions: at r = R, v z = 0 and at r = KR, V = v . 0 In addition an integral condition is needed
Z
to state that there is no net flow in the z direction. Find the pressure distribution in the
system.
(c) Work the following problems using the vorticity equation: 2B.6, 2B.7, 3B.1, 3B.10, ЗВ.16.

4B.5 Steady potential flow around a stationary sphere. 2 In Example 4.2-1 we worked through the
creeping flow around a sphere. We now wish to consider the flow of an incompressible, invis-
cid fluid in irrotational flow around a sphere. For such a problem, we know that the velocity
potential must satisfy Laplace's equation (see text after Eq. 4.3-11).
(a) State the boundary conditions for the problem.
(b) Give reasons why the velocity potential ф can be postulated to be of the form ф(г, в) =
f(r) cos 0.
(c) Substitute the trial expression for the velocity potential in (b) into Laplace's equation for
the velocity potential.
(d) Integrate the equation obtained in (c) and obtain the function f(r) containing two con-
stants of integration; determine these constants from the boundary conditions and find
3S0 (4В.5-1)
L W ^ \ ' / J
(e) Next show that
3
Г /j?\ l
- [ y l cos0 (4В.5-2)
v = -vj 1 + 1 f £ j sin 0 (4B.5-3)
0

(f) Find the pressure distribution, and then show that at the sphere surface
2
9> - 0> x = kpvlil ~ I sin 0) (4B.5-4)
4B.6 Potential flow near a stagnation point (Fig. 4B.6).
2
(a) Show that the complex potential w = -VQZ describes the flow near a plane stagnation point.
(b) Find the velocity components v x(x, y) and v y{x, y).
(c) Explain the physical significance of %

2
L. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon, Boston, 2nd edition (1987), pp. 21-26,
contains a good collection of potential-flow problems.