Page 110 - Bird R.B. Transport phenomena

§3.6 Use of the Equations of Change to Solve Flow Problems 95

This equation gives the pressure at all points within the liquid. Right at the liquid-air inter-
face, p = p alm/ and with this substitution Eq. 3.6-43 gives the shape of the liquid-air interface:

2 - Z n = (3.6-44)

This is the equation for a parabola. The reader can verify that the free surface of a liquid in a
rotating annular container obeys a similar relation.

EXAMPLE 3.6-5 A solid sphere of radius R is rotating slowly at a constant angular velocity П in a large body
of quiescent fluid (see Fig. 3.6-7). Develop expressions for the pressure and velocity distribu-
Floiu near a Slowly tions in the fluid and for the torque T required to maintain the motion. It is assumed that the
z
Rotating Sphere sphere rotates sufficiently slowly that it is appropriate to use the creeping flow version of the
equation of motion in Eq. 3.5-8. This problem illustrates setting up and solving a problem in
spherical coordinates.

SOLUTION The equations of continuity and motion in spherical coordinates are given in Tables B.4 and
B.6, respectively. We postulate that, for steady creeping flow, the velocity distribution will
have the general form v = Ъ у (г, 0), and that the modified pressure will be of the form
ф ф
ty = ^(r, 0). Since the solution is expected to be symmetric about the z-axis, there is no depen-
dence on the angle ф.
With these postulates, the equation of continuity is exactly satisfied, and the components
of the creeping flow equation of motion become

r-component (3.6-45)

^-component (3.6-46)

ф-component + 1 1 1 (3.6-47)
dr I у^двХ sin 0
*дв
The boundary conditions may be summarized as

B.C. 1: at r = R, v r = O,y o = 0, У Ф = RCL sin 0 (3.6-48)
B.C. 2: a s r ^ ° o , vr _> 0, y e -» 0, у ф -» 0 (3.6-49)
B.C. 3: a s r ^ o o , 9> -»p (3.6-50)
n
where $P = p + pgz, and p is the fluid pressure far from the sphere at z = 0.
0
Equation 3.6-47 is a partial differential equation for у (г, в). То solve this, we try a solu-
ф
tion of the form v = f(r) sin в. This is just a guess, but it is consistent with the boundary con-
(f)
dition in Eq. 3.6-48. When this trial form for the velocity distribution is inserted into Eq. 3.6-47
m-*-* (3.6-51)
we get the following ordinary differential equation for/(r):

Torque T is required
z
to make the sphere
rotate

Fig. 3.6-7. A slowly rotating sphere in an infinite expanse
of fluid. The primary flow is у ф = £lR(R/r) 2 sin 6.