Page 137 - Bird R.B. Transport phenomena

122 Chapter 4 Velocity Distributions with More Than One Independent Variable

pressure by taking the curl of the equation of motion, after making use of the vector
identity [v • Vv] = ^V(v • v) - [v X [V X v]], which is given in Eq. A.4-23. For fluids of
constant viscosity and density, this operation gives

4 [V X v] - [V X [v X [V X v]]] = X v] (4.2-1)
ot
This is the equation of change for the vorticity [V X v]; two other ways of writing it are
given in Problem 3D.2.
For viscous flow problems one can then solve the vorticity equation (a third-order
vector equation) together with the equation of continuity and the relevant initial and
boundary conditions to get the velocity distribution. Once that is known, the pressure
distribution can be obtained from the Navier-Stokes equation in Eq. 3.5-6. This method
of solving flow problems is sometimes convenient even for the one-dimensional flows
previously discussed (see, for example, Problem 4B.4).
For planar or axisymmetric flows the vorticity equation can be reformulated by in-
troducing the stream function ф. То do this, we express the two nonvanishing compo-
nents of the velocity as derivatives of ф in such a way that the equation of continuity is
automatically satisfied (see Table 4.2-1). The component of the vorticity equation corre-
sponding to the direction in which there is no flow then becomes a fourth-order scalar
equation for ф. The two nonvanishing velocity components can then be obtained after
the equation for the scalar ф has been found. The most important problems that can be
treated in this way are given in Table 4.1-I. 1
The stream function itself is not without interest. Surfaces of constant ф contain the
2
streamlines, which in steady-state flow are the paths of fluid elements. The volumetric
rate of flow between the surfaces ф = ф and ф = ф is proportional to ф — ф\.
х
2
2
In this section we consider, as an example, the steady, creeping flow past a station-
ary sphere, which is described by the Stokes equation of Eq. 3.5-8, valid for Re < < 1 (see
the discussion right after Eq. 3.7-9). For creeping flow the second term on the left side of
Eq. 4.2-1 is set equal to zero. The equation is then linear, and therefore there are many
methods available for solving the problem. 3 We use the stream function method based
on Eq. 4.2-1.

EXAMPLE 4.2-1 Use Table 4.2-1 to set up the differential equation for the stream function for the flow of a
Newtonian fluid around a stationary sphere of radius R at Re << 1. Obtain the velocity and
Creeping Flow around pressure distributions when the fluid approaches the sphere in the positive z direction, as in
a Sphere Fig. 2.6-1.

1 For a technique applicable to more general flows, see J. M. Robertson, Hydrodynamics in Theory and
Application, Prentice-Hall, Englewood Cliffs, N.J. (1965), p. 77; for examples of three-dimensional flows
using two stream functions, see Problem 4D.5 and also J. P. Sorensen and W. E. Stewart, Chem. Eng. Sci.,
29, 819-825 (1974). A. Lahbabi and H.-C. Chang, Chem. Eng. Sci., 40, 434^47 (1985) dealt with high-Re
flow through cubic arrays of spheres, including steady-state solutions and transition to turbulence.
W. E. Stewart and M. A. McClelland, AIChE Journal, 29, 947-956 (1983) gave matched asymptotic
solutions for forced convection in three-dimensional flows with viscous heating.
See, for example, G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press
2
(1967), §2.2. Chapter 2 of this book is an extensive discussion of the kinematics of fluid motion.
The solution given here follows that given by L. M. Milne-Thomson, Theoretical Hydrodynamics,
3
Macmillan, New York, 3rd edition (1955), pp. 555-557. For other approaches, see H. Lamb, Hydrodynamics,
Dover, New York (1945), §§337, 338. For a discussion of unsteady flow around a sphere, see R. Berker, in
Handbuch der Physik, Volume VIII-2, Springer, Berlin (1963), §69; or H. Villat and J. Kravtchenko,
Leqons sur les Fluides Visqueux, Gauthier-Villars, Paris (1943), Chapter VII. The problem of finding the
forces and torques on objects of arbitrary shapes is discussed thoroughly by S. Kim and S. J. Karrila,
Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinemann, Boston (1991), Chapter II.