Fluid Mechanics-Differential Analysis of Fluid Flow

In the previous chapter attention is focused on the use of finite control volumes for the solution
of a variety of fluid mechanics problems. This approach is very practical and useful,
since it does not generally require a detailed knowledge of the pressure and velocity variations
within the control volume. Typically, we found that only conditions on the surface of
the control volume entered the problem, and thus problems could be solved without a detailed
knowledge of the flow field. Unfortunately, there are many situations that arise in which
the details of the flow are important and the finite control volume approach will not yield
the desired information. For example, we may need to know how the velocity varies over the
cross section of a pipe, or how the pressure and shear stress vary along the surface of an airplane
wing. In these circumstances we need to develop relationships that apply at a point, or
at least in a very small region 1infinitesimal volume2 within a given flow field. This approach,
which involves an infinitesimal control volume, as distinguished from a finite control volume,
is commonly referred to as differential analysis, since 1as we will soon discover2 the
governing equations are differential equations.
In this chapter we will provide an introduction to the differential equations that describe
1in detail2 the motion of fluids. Unfortunately, we will also find that these equations
are rather complicated, partial differential equations that cannot be solved exactly except in
a few cases, at least without making some simplifying assumptions. Thus, although differential
analysis has the potential for supplying very detailed information about flow fields,
this information is not easily extracted. Nevertheless, this approach provides a fundamental
basis for the study of fluid mechanics. We do not want to be too discouraging at this point,
since there are some exact solutions for laminar flow that can be obtained, and these have
proved to be very useful. A few of these are included in this chapter. In addition, by making
some simplifying assumptions many other analytical solutions can be obtained. For example,
in some circumstances it may be reasonable to assume that the effect of viscosity is small
and can be neglected. This rather drastic assumption greatly simplifies the analysis and provides
the opportunity to obtain detailed solutions to a variety of complex flow problems.
Some examples of these so-called inviscid flow solutions are also described in this chapter.
It is known that for certain types of flows the flow field can be conceptually divided
into two regions—a very thin region near the boundaries of the system in which viscous
effects are important, and a region away from the boundaries in which the flow is essentially
inviscid. By making certain assumptions about the behavior of the fluid in the thin layer near
the boundaries, and using the assumption of inviscid flow outside this layer, a large class of
problems can be solved using differential analysis. These boundary layer problems are discussed
in Chapter 9. Finally, it is to be noted that with the availability of powerful digital
computers it is feasible to attempt to solve the differential equations using the techniques of
numerical analysis. Although it is beyond the scope of this book to delve into this approach,
which is generally referred to as computational fluid dynamics 1CFD2, the reader should be
aware of this approach to complex flow problems. A few additional comments about CFD
and other aspects of differential analysis are given in the last section of this chapter.
We begin our introduction to differential analysis by reviewing and extending some of
the ideas associated with fluid kinematics that were introduced in Chapter 4. With this background
the remainder of the chapter will be devoted to the derivation of the basic differential
equations 1which will be based on the principle of conservation of mass and Newton’s
second law of motion2 and to some applications.