Chapter 6
SOLUTION OF VISCOUS-FLOW PROBLEMS
6.1 Introduction
T
HE previous chapter contained derivations of the relationships for the con-
servation of mass and momentum—the
equations of motion
—in rectangular,
cylindrical, and spherical coordinates. All the experimental evidence indicates
that these are indeed the most fundamental equations of ﬂuid mechanics, and that
in principle they govern any situation involving the ﬂow of a Newtonian ﬂuid.
Unfortunately, because of their all-embracing quality, their solution in analytical
terms is diﬃcult or impossible except for relatively simple situations. However,
it is important to be aware of these “Navier-Stokes equations,” for the following
reasons:
1. They lead to the analytical and exact solution of some simple, yet important
problems, as will be demonstrated by examples in this chapter.
2. They form the basis for further work in other areas of chemical engineering.
3. If a few realistic
simplifying assumptions
are made, they can often lead to
approximate solutions that are eminently acceptable for many engineering
purposes. Representative examples occur in the study of boundary layers,
waves, lubrication, coating of substrates with ﬁlms, and inviscid (irrotational)
ﬂow.
4. With the aid of more sophisticated techniques, such as those involving power
series and asymptotic expansions, and
particularly
computer-implemented nu-
merical methods, they can lead to the solution of moderately or highly ad-
vanced problems, such as those involving injection-molding of polymers and
even the incredibly diﬃcult problem of weather prediction.
The following sections present exact solutions of the equations of motion for
several relatively simple problems in rectangular, cylindrical, and spherical coor-
dinates. Throughout, unless otherwise stated, the ﬂow is assumed to be
steady
,
laminar
and
Newtonian,
with
constant
density and viscosity. Although these as-
sumptions are necessary in order to obtain solutions, they are nevertheless realistic
in many cases.
All of the examples in this chapter are characterized by low Reynolds numbers.
That is, the viscous forces are much more important than the inertial forces, and
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