Page 91 - Bird R.B. Transport phenomena

76 Chapter 3 The Equations of Change for Isothermal Systems

In §3.2 the equation of motion is developed by making a momentum balance over a
small element of volume and letting the volume element become infinitesimally small.
Here again a partial differential equation is generated. This equation of motion can be
used, along with some help from the equation of continuity, to set up and solve all the
problems given in Chapter 2 and many more complicated ones. It is thus a key equation
in transport phenomena.
In §3.3 and §3.4 we digress briefly to introduce the equations of change for mechani-
cal energy and angular momentum. These equations are obtained from the equation of
motion and hence contain no new physical information. However, they provide a conve-
nient starting point for several applications in this book—particularly the macroscopic
balances in Chapter 7.
In §3.5 we introduce the "substantial derivative." This is the time derivative follow-
ing the motion of the substance (i.e., the fluid). Because it is widely used in books on
fluid dynamics and transport phenomena, we then show how the various equations of
change can be rewritten in terms of the substantial derivatives.
In §3.6 we discuss the solution of flow problems by use of the equations of continu-
ity and motion. Although these are partial differential equations, we can solve many
problems by postulating the form of the solution and then discarding many terms in
these equations. In this way one ends up with a simpler set of equations to solve. In this
chapter we solve only problems in which the general equations reduce to one or more
ordinary differential equations. In Chapter 4 we examine problems of greater complexity
that require some ability to solve partial differential equations. Then in Chapter 5 the
equations of continuity and motion are used as the starting point for discussing turbu-
lent flow. Later, in Chapter 8, these same equations are applied to flows of polymeric liq-
uids, which are non-Newtonian fluids.
Finally, §3.7 is devoted to writing the equations of continuity and motion in di-
mensionless form. This makes clear the origin of the Reynolds number, Re, often men-
tioned in Chapter 2, and why it plays a key role in fluid dynamics. This discussion lays
the groundwork for scale-up and model studies. In Chapter 6 dimensionless numbers
arise again in connection with experimental correlations of the drag force in complex
systems.
At the end of §2.2, we emphasized the importance of experiments in fluid dynamics.
We repeat those words of caution here and point out that photographs and other types
of flow visualization have provided us with a much deeper understanding of flow prob-
1
lems than would be possible by theory alone. Keep in mind that when one derives a
flow field from the equations of change, it is not necessarily the only physically admissi-
ble solution.
Vector and tensor notations are occasionally used in this chapter, primarily for the
purpose of abbreviating otherwise lengthy expressions. The beginning student will find
that only an elementary knowledge of vector and tensor notation is needed for reading
this chapter and for solving flow problems. The advanced student will find Appendix A
helpful in getting a better understanding of vector and tensor manipulations. With re-
gard to the notation, it should be kept in mind that we use lightface italic symbols for
scalars, boldface Roman symbols for vectors, and boldface Greek symbols for tensors.
Also dot-product operations enclosed in ( ) are scalars, and those enclosed in [ ] are
vectors.

1
We recommend particularly M. Van Dyke, An Album of Fluid Motion, Parabolic Press, Stanford
(1982); H. Werle, Ann. Rev. Fluid Mech., 5, 361-382 (1973); D. V. Boger and K. Walters, Rheological
Phenomena in Focus, Elsevier, Amsterdam (1993).