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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.

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  • $\begingroup$ You should try asking this on the Physics SE, tbh, your question is mostly about theoretical fluid mechanics. And in general, whenever you need any of the rarer named numbers (Knudsen, Froude, etc), double check if you're still doing engineering at all :) $\endgroup$
    – AEhere supports Monica
    May 6, 2019 at 12:51
  • $\begingroup$ @AEhere Oh, I see. I understand. $\endgroup$
    – Karan
    May 19, 2019 at 14:15

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The key is what's called "scale separation". For continuum hydrodynamics to work, there has to exist a length scale that is much smaller than the smallest length scale on which any interesting fluid-mechanical phenomena happen, so that it's safe to treat that length scale as differentially small; yet that same length scale must also be much larger than the length scale on which matter is discrete, so that a "local" variable that is constructed by spatial averaging over regions whose size (in each spatial dimension) is given by that length scale is an average over a very large number of atoms/molecules. In very violent turbulence, or in shocks with a very high upstream fluid velocity, or in very rarefied gases, it may be that no such length scale exists, in which case continuum hydrodynamics will fail; but in pretty much all engineering applications, we're safe.

I have vague memories of "Callister" as the name of an author of a textbook in which I've seen a nifty explanation of this.

ETA: No, it wasn't Callister, that's a book about materials science.

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    $\begingroup$ And the separation metric is the Knudsen Number $\endgroup$
    – Phil Sweet
    Feb 21, 2021 at 22:26
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    $\begingroup$ @PhilSweet Or at least a Knudsen number: the one where the length scale you choose to put in the denominator is the smallest length scale of any fluid-mechanical phenomenon that's taking place. $\endgroup$
    – Daniel Hatton
    Feb 21, 2021 at 23:05
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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?

Undeniably, yes. If there are not enough particles in the control volume to achieve a sufficient Hamiltonian to represent the system (continuum check), then the measurement may not be ergodic even in a system at dynamic equilibrium. Without invoking further fluid dynamic parameters, i.e. [insert name's] number, this is the minimum scale possible for measurement.

Please tell me if this interpretation is correct or not and flaw in reasoning.

I would say this interpretation is flawed. Infinitesimal elements are the basic element and do not have a finite size regardless of the system dimensions and this does not limit itself to the Eulerian perspective. I would suggest decoupling measurement from discrete system representation in your mind.

Further, fluid systems do not necessarily need "hard" instruments to measure dynamics. Generally microfluidics uses optical methods for flow dynamics and so has only the wavelength of light as a limitation. Further reduction in measurement scale is possibly with MRI (among others) but is extremely technical and has temporal and spatial limitations.

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