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.

[Markus Hanke]

It’s possible to consider discrete models of fluid dynamics, but generally speaking these don’t take the form of neat, closed formulas that can be easily worked on paper - rather, you’ll be dealing with numerical models on powerful computers. An example of this is CFD-DEM. The standard Navier-Stokes equations are then just the continuum limit of such a model.

[studiot]

2 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.

Good morning and welcome, Rao.

Good first post.  +1

 

One way to view the equations of mathematical physics or engineering systems is to dvide them into

The equations of constitution or the constitutive equations, for a continuum also called 'the material description'.  For example the equations of motion of the particles.

The equations of compatibility.  These are usually geometric constraints for example the continuity equation in incompressible flow.

These are then solved as a set of simultaneous equations.

 

It is not necessary to discretise or digitise all these equations, a mixed set will also do.

 

If you would like to give us more details of you intended application we may be able to be more specific.

There are also discrete phenomena which appear in the continuum model, such as water hammer, hydraulic jumps, vortices, sonic booms, water bores, etc.

 

For example, fluid flows are often modelled by using finite element techniques.

The mesh is calibrated against known values at the nodes and the actual equations set is replaced by simplere ones for the internodal spaces.
So called hat functions (similar to dirac delta functions) are popular for this. the give 'pulses' which match the change of measured variable from one node to the next.

 

There is not enough computing power on our whole planet to track the constitutive equations at the molecular level.

For example there are about 10²⁸ molecules in the fuel tank of for a Saturn V rocket.

 

 

 

 

 

 

 

Edited December 18, 2023 by studiot

[sethoflagos]

7 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?.

Take viscosity as an example.

We can picture two molecules within a streamline, one with slightly less momentum than the average for that streamline a little ahead of one with slightly more. Clearly at some point there will be a collision. If both molecules are identical and have spherically symmetric electrical fields, we can come up with some simple distribution of momentum scattering angle and quantify the transfer of momentum into adjacent streamlines. The mass interchange between flowlines generates some incremental changes in density, the momentum interchange generates some incremental changes in pressure. The streamlines will undergo nett local acceleration/deceleration accordingly generating further collisions.

That's about as far as I ever took my perusal of the emergence of viscosity phenomena at the molecular level. I guess I could incrementally improve my model of molecular velocity distridution within the flowstream, integrating over all flowstreams and imposing whatever constraint (eg no-slip pipe wall boundary, steady state etc) seemed appropriate until the overall velocity and pressure profiles stabilised. Eventually, I'd come up with some number for pipe wall shear stress which is the usual parameter of interest to my colleagues in piping and mechanical sections.

Or I could just make use of Newton's cunning postulate that shear stress seemed to be directly proportional to shear rate for many fluids of interest. It saves an awful amount of time. 

Edited December 18, 2023 by sethoflagos
sp