The Fluid as a Continuum

The Fluid as a Continuum

We have already used technical terms such as fluid pressure and density without a rig-

orous discussion of their definition. As far as we know, fluids are aggregations of mol-

ecules, widely spaced for a gas, closely spaced for a liquid. The distance between mol-

ecules is very large compared with the molecular diameter. The molecules are not fixed

in a lattice but move about freely relative to each other. Thus fluid density, or mass per

unit volume, has no precise meaning because the number of molecules occupying a given

volume continually changes. This effect becomes unimportant if the unit volume is large

compared with, say, the cube of the molecular spacing, when the number of molecules

within the volume will remain nearly constant in spite of the enormous interchange of

particles across the boundaries. If, however, the chosen unit volume is too large, there

could be a noticeable variation in the bulk aggregation of the particles. This situation is

illustrated in Fig. 1.4, where the “density” as calculated from molecular mass m within

a given volume -

 is plotted versus the size of the unit volume. There is a limiting vol-

ume -

* below which molecular variations may be important and above which aggre-

gate variations may be important. The density of a fluid is best defined as

One atmosphere equals 2116 lbf/ft2 -

 101,300 Pa.

Fig. 1.4 The limit definition of

continuum fluid density: (a) an

elemental volume in a fluid region

of variable continuum density;

(b) calculated density versus size

of the elemental volume.

The limiting volume -

* is about 10 9 mm3 for all liquids and for gases at atmo-

spheric pressure. For example, 10 9 mm3 of air at standard conditions contains approx-

imately 3 107 molecules, which is sufficient to define a nearly constant density

according to Eq. (1.1). Most engineering problems are concerned with physical dimen-

sions much larger than this limiting volume, so that density is essentially a point func-

tion and fluid properties can be thought of as varying continually in space, as sketched

in Fig. 1.4a. Such a fluid is called a continuum, which simply means that its varia-

tion in properties is so smooth that differential calculus can be used to analyze the

substance. We shall assume that continuum calculus is valid for all the analyses in

this book. Again there are borderline cases for gases at such low pressures that molec-

ular spacing and mean free path3 are comparable to, or larger than, the physical size

of the system. This requires that the continuum approximation be dropped in favor of

a molecular theory of rarefied gas flow [18]. In principle, all fluid mechanics problems

can be attacked from the molecular viewpoint, but no such attempt will be made here.

Note that the use of continuum calculus does not preclude the possibility of discon-

tinuous jumps in fluid properties across a free surface or fluid interface or across a

shock wave in a compressible fluid (Chap. 9). Our calculus in analyzing fluid flow

must be flexible enough to handle discontinuous boundary conditions.