One of the first questions we need to explore is, What is a fluid? Or we might ask, What is
the difference between a solid and a fluid? We have a general, vague idea of the difference.
A solid is “hard”and not easily deformed, whereas a fluid is “soft”and is easily deformed 1we can readily move through air2. Although quite descriptive, these casual observations of
the differences between solids and fluids are not very satisfactory from a scientific or
engineering point of view. A closer look at the molecular structure of materials reveals that
matter that we commonly think of as a solid 1steel, concrete, etc.2has densely spaced molecules
with large intermolecular cohesive forces that allow the solid to maintain its shape, and to
not be easily deformed. However, for matter that we normally think of as a liquid 1water, oil,
etc.2, the molecules are spaced farther apart, the intermolecular forces are smaller than for
solids, and the molecules have more freedom of movement. Thus, liquids can be easily
deformed 1but not easily compressed2 and can be poured into containers or forced through a
tube. Gases 1air, oxygen, etc.2 have even greater molecular spacing and freedom of motion
with negligible cohesive intermolecular forces and as a consequence are easily deformed 1and
compressed2 and will completely fill the volume of any container in which they are placed.
Although the differences between solids and fluids can be explained qualitatively on
the basis of molecular structure, a more specific distinction is based on how they deform
under the action of an external load. Specifically, a fluid is defined as a substance that deforms
continuously when acted on by a shearing stress of any magnitude. A shearing stress 1force
per unit area2 is created whenever a tangential force acts on a surface. When common solids
such as steel or other metals are acted on by a shearing stress, they will initially deform 1usually a very small deformation2, but they will not continuously deform 1flow2. However,
common fluids such as water, oil, and air satisfy the definition of a fluid—that is, they will
flow when acted on by a shearing stress. Some materials, such as slurries, tar, putty, toothpaste,
and so on, are not easily classified since they will behave as a solid if the applied shearing
stress is small, but if the stress exceeds some critical value, the substance will flow. The study
of such materials is called rheology and does not fall within the province of classical fluid
mechanics. Thus, all the fluids we will be concerned with in this text will conform to the
definition of a fluid given previously.Although the molecular structure of fluids is important in distinguishing one fluid from
another, it is not possible to study the behavior of individual molecules when trying to describe
the behavior of fluids at rest or in motion. Rather, we characterize the behavior by considering
the average, or macroscopic, value of the quantity of interest, where the average is evaluated
over a small volume containing a large number of molecules. Thus, when we say that the
velocity at a certain point in a fluid is so much, we are really indicating the average velocity
of the molecules in a small volume surrounding the point. The volume is small compared
with the physical dimensions of the system of interest, but large compared with the average
distance between molecules. Is this a reasonable way to describe the behavior of a fluid? The
answer is generally yes, since the spacing between molecules is typically very small. For
gases at normal pressures and temperatures, the spacing is on the order of and for
liquids it is on the order of The number of molecules per cubic millimeter is on
the order of for gases and for liquids. It is thus clear that the number of molecules
in a very tiny volume is huge and the idea of using average values taken over this volume is
certainly reasonable. We thus assume that all the fluid characteristics we are interested in 1pressure, velocity, etc.2 vary continuously throughout the fluid—that is, we treat the fluid as
a continuum. This concept will certainly be valid for all the circumstances considered in this
text. One area of fluid mechanics for which the continuum concept breaks down is in the
study of rarefied gases such as would be encountered at very high altitudes. In this case the
spacing between air molecules can become large and the continuum concept is no longer
acceptable.
1018 1021
107 mm.
106 mm,
4  Chapter 1 / Introduction
1.1 Some Characteristics of Fluids
A fluid, such as
water or air, deforms
continuously
when acted on by
shearing stresses of
any magnitude.
1.2 Dimensions, Dimensional Homogeneity, and Units
1.2

Since in our study of fluid mechanics we will be dealing with a variety of fluid characteristics,
it is necessary to develop a system for describing these characteristics both qualitatively and
quantitatively. The qualitative aspect serves to identify the nature, or type, of the characteristics
1such as length, time, stress, and velocity2, whereas the quantitative aspect provides a
numerical measure of the characteristics. The quantitative description requires both a number
and a standard by which various quantities can be compared. A standard for length might be
a meter or foot, for time an hour or second, and for mass a slug or kilogram. Such standards
are called units, and several systems of units are in common use as described in the following
section. The qualitative description is conveniently given in terms of certain primary quantities,
such as length, L, time, T, mass, M, and temperature, These primary quantities can
then be used to provide a qualitative description of any other secondary quantity: for example,
and so on, where the symbol is used to
indicate the dimensions of the secondary quantity in terms of the primary quantities. Thus,
to describe qualitatively a velocity, V, we would write
and say that “the dimensions of a velocity equal length divided by time.” The primary
quantities are also referred to as basic dimensions.
For a wide variety of problems involving fluid mechanics, only the three basic dimensions,
L, T, and M are required. Alternatively, L, T, and F could be used, where F is the basic
dimensions of force. Since Newton’s law states that force is equal to mass times acceleration,
it follows that or Thus, secondary quantities expressed in terms
of M can be expressed in terms of F through the relationship above. For example, stress,
is a force per unit area, so that but an equivalent dimensional equation is
Table 1.1 provides a list of dimensions for a number of common physical
quantities.
All theoretically derived equations are dimensionally homogeneous—that is, the dimensions
of the left side of the equation must be the same as those on the right side, and all
additive separate terms must have the same dimensions. We accept as a fundamental premise
that all equations describing physical phenomena must be dimensionally homogeneous.
If this were not true, we would be attempting to equate or add unlike physical quantities,
which would not make sense. For example, the equation for the velocity, V, of a uniformly
accelerated body is
(1.1)
where is the initial velocity, a the acceleration, and t the time interval. In terms of
dimensions the equation is
and thus Eq. 1.1 is dimensionally homogeneous.
Some equations that are known to be valid contain constants having dimensions. The
equation for the distance, d, traveled by a freely falling body can be written as
(1.2)
and a check of the dimensions reveals that the constant must have the dimensions of
if the equation is to be dimensionally homogeneous. Actually, Eq. 1.2 is a special form of
the well-known equation from physics for freely falling bodies,
d  (1.3)
gt
2
2
LT
2
d  16.1t
2
LT
1  LT
1  LT
1
V0
V  V0  at
s  ML1T
2.
s  FL2,
s,
M  FL1 T
F  MLT 2.
2
V  LT
1
velocity  LT density  ML3, 
area  L2, 1,
™.
Fluid characteristics
can be described
qualitatively
in terms of certain
basic quantities
such as length,
time, and mass.
in which g is the acceleration of gravity. Equation 1.3 is dimensionally homogeneous and
valid in any system of units. For the equation reduces to Eq. 1.2 and thus
Eq. 1.2 is valid only for the system of units using feet and seconds. Equations that are restricted
to a particular system of units can be denoted as restricted homogeneous equations, as opposed
to equations valid in any system of units, which are general homogeneous equations. The
preceding discussion indicates one rather elementary, but important, use of the concept of
dimensions:the determination of one aspect of the generality of a given equation simply
based on a consideration of the dimensions of the various terms in the equation. The concept
of dimensions also forms the basis for the powerful tool of dimensional analysis, which is
considered in detail in Chapter 7.