In rigid body mechanics forces act on the center of mass of a
body and Newton's Laws of motion are (somewhat) easily applicable.
Fluid mechanics, on the on the other hand, is considerably more
difficult as the material deforms *continuously* (i.e.,
essentially every "piece" of fluid may respond somewhat differently
to the force).

##### DEFINITION:

The **continuum assumption** requires that a fluid is treated
as a continuous distribution of matter, or a **continuum**,
where properties, velocities, etc. may vary point-by-point.

### Fluid Density at a Point

In an idealized interpretation of the continuum assumption, the
mass ($\Delta$m) per unit volume ($\Delta$V) in a fluid, measured
at a given point, will tend toard a constant value in the limit as
the measuring volume *shrinks* down to zero:

$\rho = \lim_{\Delta V\to 0} \frac{\Delta
m}{\Delta V} = \frac{dm}{dV}$

A problem with this requirement is that, as the volume segment
gets smaller, you reach a size where random molecular (thermal)
motion causes the density to fluctuate, and the continuum
approximation begins to break down:

A more accurate requirement, therefore is that the density
becomes constant as the volume shrinks to some finite value,
$\delta V$:

$\rho(x, y, z) = \frac{dm}{dV}\equiv
\lim_{\Delta V \to \delta V} \frac{\Delta m}{\Delta V}$

The validity of the continuum assumption then requires that
$\delta V$ has a characteristic dimension several orders of
magnitude smaller than the fluid flow domain.

##### OUTCOME:

Explain the continuum hypothesis and the origin of its
breakdown

##### TEST YOURSELF

Give examples of problems where the continuum approximation
breaks down

One obvious answer is the one discussed already: when the system
gets to be close to the size of the molecules. In this sense, some
nanotechnology problems would be poorly approximated as
continua.

A not-so-obvious example is when you have a mixture of gas and
liquid (i.e., a bubbly or frothy system). In some cases, if the
scale of your analysis is too close to the scale of the
heterogeneities (bubbles, droplets) in the flow, you will need to
actually account for **two** continua (the liquid and the gas)
separately!