Math and Engineering is available with “continuum” assumption, How to do the opposite?

[Rao]

The “Continuum” assumption in propulsion and fluid dynamics is that their atomic structure will be ignored and they will be considered as capable of being subdivided into infinitesimal pieces of identical structure. In this way it is legitimate to speak of the properties of a continuum – e.g., density, pressure, velocity – as point properties. Now, this assumption needs to be modified, for example if the “Non-Continuum” assumption is that the atoms/molecules/particles ( although fundamental particles are modernly understood as point masses, let’s ignore this issue for now because most of the reaction in propulsion and fluid dynamics is chemical and not Atomic/Nuclear) cannot be subdivided, How and where to start the math for this kind of assumption to speak of the properties like density, pressure and velocity?.

 

Please go through the pictures/attachments for a little bit of math around this “continuum” assumption. If you don’t understand the question please ask.

Thanks in Advance.

[Moontanman]

3 hours ago, Rao said:

The “Continuum” assumption in propulsion and fluid dynamics is that their atomic structure will be ignored and they will be considered as capable of being subdivided into infinitesimal pieces of identical structure. In this way it is legitimate to speak of the properties of a continuum – e.g., density, pressure, velocity – as point properties. Now, this assumption needs to be modified, for example if the “Non-Continuum” assumption is that the atoms/molecules/particles ( although fundamental particles are modernly understood as point masses, let’s ignore this issue for now because most of the reaction in propulsion and fluid dynamics is chemical and not Atomic/Nuclear) cannot be subdivided, How and where to start the math for this kind of assumption to speak of the properties like density, pressure and velocity?.

 

Please go through the pictures/attachments for a little bit of math around this “continuum” assumption. If you don’t understand the question please ask.

Thanks in Advance.

What is the point you are trying to make here? 

[OceanBreeze]

On 12/18/2023 at 8:56 PM, Moontanman said:

What is the point you are trying to make here? 

As far as I can tell, the OP is asking what sort of mathematical analysis should be used on functions that are discontinuous. The simple answer is to use numerical methods employing approximate quadrature formulas such as the midpoint, trapezoid and Simpson's rules for integration. For differentiation, if we don’t have an exact relation such as y = f(x), we can use interpolation to derive approximate polynomial forms and then differentiate that expression.

I can remember spending many hours learning all these methods and they are all very ingenious, having come from the minds of people such as Newton, Bessel, Lagrange and of course Simpson, among many others.

Using these numerical methods is actually very interesting and can even be fun if you are at all mathematically inclined.

These days numerical integration and differentiation is somewhat a lost art as it can be easily handled by computer programs.

For anyone who is interested in learning more about these numerical methods, I recommend reading this paper.