Continuum hypothesis confusion (fluid dynamics)

Question

For the quantity $\rho(x,t)$ with the continuum hypothesis am I taking the average value of the density at each point in the small volume surrounding the point $x$ or am I taking the average density of each molecule in the small volume. If it's the latter what does this intuitively mean?

Answer 1

It's the average of a small enough volume. In order to quantify the validity of continuum hypothesis you can use Knudsen number. $K_n=\lambda/L$ where $\lambda$ is the mean free path of a material particle and $L$ is the characteristic length scale of the problem. if Knudsen is larger than one, you can't use continuum hypothesis.