In Eulerian description of fluid flow, we observe how the properties of the fluid are changing at a point. Since a point has zero dimension hence a zero area, how can we make sense of the 'finite pressure at the point' phrase?
Since fluids are discrete in nature, a collection of molecules, the way I think about it is that Imagine that we are measuring the pressure with a measurement device [assume] which is cylindrical in shape having diameter enough so that Knudsen number is quite low [continuum hypothesis is valid]. At the molecular scale, Pressure is collisions of the fluid molecules with the measurement device. We position the device in the fluid with direction of fluid flow in such a way that diameter is the characteristic length. Let's say we keep on decreasing the diameter of the device hence the Knudsen number will keep on increasing and it will reach a point when the continuum hypothesis is not a good approximation anymore.
Shouldn't it be that the diameter should approach a finite threshold value [no matter how small it is] rather than zero for continuum hypothesis to hold true?
[On the same note, when a pitot tube measures local velocity, it does not measure velocity at a point 'strictly' because it has a finite area of opening for the fluid to enter and I guess surface tension effects will also come in to play if we decrease the area of the opening. There is only so much area we can decrease theoretically for the continuum hypothesis to hold true.]
Hence the way I visualize a point in Eulerian description of fluid flow is that it is a collection of the minimum number of molecules for which the continuum hypothesis must hold true. Since molecules occupy space hence the point in this context does not have a zero area but a finite positive area [no matter how small it is].
Please tell me if this interpretation is correct or not and flaw in reasoning.