The Continuum Assumption http://edugen.wiley.com/edugen/courses/crs2436/crowe9771/crowe9771...
1.2 The Continuum Assumption
This section describes how fluids are conceptualized as a continuous medium. This topic is important for
applying the derivative concept to characterize properties of fluids.
While a body of fluid is comprised of molecules, most characteristics of fluids are due to average molecular
behavior. That is, a fluid often behaves as if it were comprised of continuous matter that is infinitely divisible
into smaller and smaller parts. This idea is called the continuum assumption. When the continuum assumption is
valid, engineers can apply limit concepts from differential calculus. Recall that a limit concept, for example,
involves letting a length, an area, or a volume approach zero. Because of the continuum assumption, fluid
parameters such as density and velocity can be considered continuous functions of position with a value at each
point in space.
To gain insight into the validity of the continuum assumption, consider a hypothetical experiment to find
density. Fig. 1.1a shows a container of gas in which a volume has been identified. The idea is to find the
mass of the molecules "M inside the volume and then to calculate density by
The calculated density is plotted in Fig. 1.1b. When the measuring volume is very small (approaching zero),
the number of molecules in the volume will vary with time because of the random nature of molecular motion.
Thus, the density will vary as shown by the wiggles in the blue line. As volume increases, the variations in
calculated density will decrease until the calculated density is independent of the measuring volume. This
condition corresponds to the vertical line at . If the volume is too large, as shown by , then the value of
density may change due to spatial variations.
Figure 1.1 When a measuring volume is large enough for random molecular effects to average
out, the continuum assumption is valid
In most applications, the continuum assumption is valid. For example, consider the volume needed to contain at
least a million (106) molecules. Using Avogadro's number of 6 × 1023 molecules/mole, the limiting volume for
water is 10-13 mm3, which corresponds to a cube less than 10-4 mm on a side. Since this dimension is much
smaller than the flow dimensions of a typical problem, the continuum assumption is justified. For an ideal gas
(1 atm and 20°C) one mole occupies 24.7 liters. The size of a volume with more than 106 molecules would be
10-10 mm3, which corresponds to a cube with sides less than 10-3 mm (or one micrometer). Once again this size
is much smaller than typical flow dimensions. Thus, the continuum assumption is usually valid in gas flows.
The continuum assumption is invalid for some specific applications. When air is in motion at a very low density,
such as when a spacecraft enters the earth's atmosphere, then the spacing between molecules is significant in
comparison to the size of the spacecraft. Similarly, when a fluid flows through the tiny passages in
nanotechnology devices, then the spacing between molecules is significant compared to the size of these
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