Missing mathematical basis of continuum hypothesis (fluid mechanics)

Batchelor (1967) discusses that numerous experimental evidences have pointed to the assumption which every fluid dynamicist make in day-to-day life: assuming fluid flow as smooth and continuous. The length scale for such assumption should be large enough to accomodate molecular fluctuations of density and other properties and small enough to not any effect of spatial distribution. Theoretically, it is difficult to deduce properties of such a hypothetical continuous medium from the real fluid – it requires extensive use of kinetic theory of gased/liquids. While for gases it studied, for liquids the mathematical basis of continuum treatment is incomplete, as remarked by Batchelor:

It seems important to connect MD simulations (if not kinetic theory) to the NS theory. It has been done by many references in this work. A critical length scale of 10 nm for liquids is suggested below which NS theory doesn’t yield the results corresponding to MD simulations. However, to provide an experimental proof of the continuum hypothesis, it seems important to find a mathematical underpinning of NS theory which gets disconnected below 10 nm because of influence of fluctuations of properties.

Especially, Nanofluidics book (2009) talks in good details about how MD simulations or random process theory (stochastic) hint about applicability of NS theory to nanochannels – one needs to have atleast 10^4 molecules in a control volume to have less than 1% of significant statistical deviations in the measurement of transport properties. The minimum width of nanochannel should be 10 times the mean free path of liquid/gas to enable the assumption of continuum. It turns out to be 4 nm for water and 100 microns for gas. Knudsen number is also a good indicator of continuum assumption. In nutshell, the bottom-up approach towards continuum hypothesis is fairly established, as summarised below:

Now, it really seems important to understand why continuum assumption across different length scales seems true.