Page 73 - Bird R.B. Transport phenomena

58 Chapter 2 Shell Momentum Balances and Velocity Distributions in Laminar Flow

These distributions are shown in Fig. 2.5-1. If both viscosities are the same, then the ve-
locity distribution is parabolic, as one would expect for a pure fluid flowing between
parallel plates (see Eq. 2B.3-2).
The average velocity in each layer can be obtained and the results are

(Po - (2.5-20)
+ д"

(2.5-21)
1 2 H
M
From the velocity and momentum-flux distributions given above, one can also calculate
the maximum velocity, the velocity at the interface, the plane of zero shear stress, and
the drag on the walls of the slit.

1 2 3 4
§2.6 CREEPING FLOW AROUND A SPHERE ' ' '
In the preceding sections several elementary viscous flow problems have been solved.
These have all dealt with rectilinear flows with only one nonvanishing velocity compo-
nent. Since the flow around a sphere involves two nonvanishing velocity components, v r
and v , it cannot be conveniently understood by the techniques explained at the begin-
e
ning of this chapter. Nonetheless, a brief discussion of flow around a sphere is warranted
here because of the importance of flow around submerged objects. In Chapter 4 we show
how to obtain the velocity and pressure distributions. Here we only cite the results and
show how they can be used to derive some important relations that we need in later dis-
cussions. The problem treated here, and also in Chapter 4, is concerned with "creeping
flow"—that is, very slow flow. This type of flow is also referred to as "Stokes flow."
We consider here the flow of an incompressible fluid about a solid sphere of radius
R and diameter D as shown in Fig. 2.6-1. The fluid, with density p and viscosity /x, ap-

Radius of sphere = R
At every point there are ^~~-^ -?*ъъ^ Point in space
pressure and friction / ^ • Щ ^ x>(x,y,z)or
forces acting on the PC
sphere surface / л

Щ^^^^Я- - _L Projection
j^^^^^V of point on
^ ^ ^ ^ ^ r ry-plane Fig. 2.6-1 Sphere of radius R
around which a fluid is flow-
ing. The coordinates г, в, and ф
Fluid approaches are shown. For more informa-
from below with tion on spherical coordinates,
velocity v x see Fig. A.8-2.

G. G. Stokes, Trans. Cambridge Phil. Soc, 9, 8-106 (1851). For creeping flow around an object of
1
arbitrary shape, see H. Brenner, Chem. Engr. Sci., 19, 703-727 (1964).
2
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd edition, Pergamon, London (1987), §20.
G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press (1967), §4.9.
3
S. Kim and S. J. Karrila, Microhydrodynamics: Principles and Selected Applications, Butterworth-
4
Heinemann, Boston (1991), §4.2.3; this book contains a thorough discussion of "creeping flow" problems.