Page 102 - Bird R.B. Transport phenomena

§3.6 Use of the Equations of Change to Solve Flow Problems 87

In Chapter 1 we gave the components of the stress tensor in Cartesian coordinates,
and in this chapter we have derived the equations of continuity and motion in Cartesian
coordinates. In Tables B.I, 4, 5, 6, and 7 we summarize these key equations in three
much-used coordinate systems: Cartesian {x, y, z), cylindrical (г, 0, z), and spherical (r, 0,
ф). Beginning students should not concern themselves with the derivation of these equa-
tions, but they should be very familiar with the tables in Appendix В and be able to use
them for setting up fluid dynamics problems. Advanced students will want to go
through the details of Appendix A and learn how to develop the expressions for the var-
ious V-operations, as is done in §§A.6 and A.7.
In this section we illustrate how to set up and solve some problems involving the
steady, isothermal, laminar flow of Newtonian fluids. The relatively simple analytical
solutions given here are not to be regarded as ends in themselves, but rather as a prepa-
ration for moving on to the analytical or numerical solution of more complex problems,
the use of various approximate methods, or the use of dimensional analysis.
The complete solution of viscous flow problems, including proofs of uniqueness and
criteria for stability, is a formidable task. Indeed, the attention of some of the world's best
applied mathematicians has been devoted to the challenge of solving the equations of con-
tinuity and motion. The beginner may well feel inadequate when faced with these equa-
tions for the first time. All we attempt to do in the illustrative examples in this section is to
solve a few problems for stable flows that are known to exist. In each case we begin by
making some postulates about the form for the pressure and velocity distributions: that is,
we guess how p and v should depend on position in the problem being studied. Then we
discard all the terms in the equations of continuity and motion that are unnecessary ac-
cording to the postulates made. For example, if one postulates that v x is a function of у
2
alone, terms like dvjdx and d v /dz 2 can be discarded. When all the unnecessary terms
x
have been eliminated, one is frequently left with a small number of relatively simple equa-
tions; and if the problem is sufficiently simple, an analytical solution can be obtained.
It must be emphasized that in listing the postulates, one makes use of intuition. The
latter is based on our daily experience with flow phenomena. Our intuition often tells us
that a flow will be symmetrical about an axis, or that some component of the velocity is
zero. Having used our intuition to make such postulates, we must remember that the
final solution is correspondingly restricted. However, by starting with the equations of
change, when we have finished the "discarding process" we do at least have a complete
listing of all the assumptions used in the solution. In some instances it is possible to go
back and remove some of the assumptions and get a better solution.
In several examples to be discussed, we will find one solution to the fluid dynamical
equations. However, because the full equations are nonlinear, there may be other solutions
to the problem. Thus a complete solution to a fluid dynamics problem requires the specifi-
cation of the limits on the stable flow regimes as well as any ranges of unstable behavior.
That is, we have to develop a "map" showing the various flow regimes that are possible.
Usually analytical solutions can be obtained for only the simplest flow regimes; the re-
mainder of the information is generally obtained by experiment or by very detailed nu-
merical solutions. In other words, although we know the differential equations that govern
the fluid motion, much is yet unknown about how to solve them. This is a challenging area
of applied mathematics, well above the level of an introductory textbook.
When difficult problems are encountered, a search should be made through some of
the advanced treatises on fluid dynamics. 1

1
R. Berker, Handbuch der Physik, Volume VIII-2, Springer, Berlin (1963), pp. 1-384; G. K. Batchelor,
An Introduction to Fluid Mechanics, Cambridge University Press (1967); L. Landau and E. M. Lifshitz, Fluid
Mechanics, Pergamon Press, Oxford, 2nd edition (1987); J. A. Schetz and A. E. Fuhs (eds.), Handbook of Fluid
Dynamics and Fluid Machinery, Wiley-Interscience, New York (1996); R. W. Johnson (ed.), The Handbook of
Fluid Dynamics, CRC Press, Boca Raton, Fla. (1998); С Y. Wang, Ann. Revs. Fluid Mech., 23,159-177 (1991).