3.1 Basic Physical Laws
of Fluid Mechanics
Motivation.
In analyzing fluid motion, we might take one of two paths: (1) seeking to
describe the detailed flow pattern at every point (
x
,
y
,
z
) in the field or (2) working
with a finite region, making a balance of flow in versus flow out, and determining gross
flow effects such as the force or torque on a body or the total energy exchange. The
second is the “control-volume” method and is the subject of this chapter. The first is
the “differential” approach and is developed in Chap. 4.
We first develop the concept of the control volume, in nearly the same manner as
one does in a thermodynamics course, and we find the rate of change of an arbitrary
gross fluid property, a result called the
Reynolds transport theorem
. We then apply this
theorem, in sequence, to mass, linear momentum, angular momentum, and energy, thus
deriving the four basic control-volume relations of fluid mechanics. There are many
applications, of course. The chapter then ends with a special case of frictionless, shaft-
work-free momentum and energy: the
Bernoulli equation
. The Bernoulli equation is a
wonderful, historic relation, but it is extremely restrictive and should always be viewed
with skepticism and care in applying it to a real (viscous) fluid motion.
It is time now to really get serious about flow problems. The fluid-statics applications
of Chap. 2 were more like fun than work, at least in my opinion. Statics problems ba-
sically require only the density of the fluid and knowledge of the position of the free
surface, but most flow problems require the analysis of an arbitrary state of variable
fluid motion defined by the geometry, the boundary conditions, and the laws of me-
chanics. This chapter and the next two outline the three basic approaches to the analy-
sis of arbitrary flow problems:
1.
Control-volume, or large-scale, analysis (Chap. 3)
2.
Differential, or small-scale, analysis (Chap. 4)
3.
Experimental, or dimensional, analysis (Chap. 5)
The three approaches are roughly equal in importance, but control-volume analysis is
“more equal,” being the single most valuable tool to the engineer for flow analysis. It
gives “engineering” answers, sometimes gross and crude but always useful. In princi-
129
Chapter 3
Integral Relations
for a Control Volume