ITP: The Continuum Approximation

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).


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.


Explain the continuum hypothesis and the origin of its breakdown


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!