Navier-Stokes equation in a triangular coordinate system

[MasterOgon]

TL;DR Summary

Is it possible to solve the Navier-Stokes equation in a triangular coordinate system and wouldn't this be more accurate?

The Navier-Stokes equation is solved in a vector grid in a Cartesian coordinate system. That is, rectangular. But does a rectangular mesh relate to what happens in a gas or liquid, and is it better to use a triangular mesh?

Undoubtedly, it is incredibly difficult to take into account all the factors even in a triangular or tetrider coordinate system, which is difficult even for visual perception. And in its direct form such a solution is impossible. But it is precisely this system that allows the logical formation of figures that we can see in water - a ring vortex or torus, similar to a figure eight (infinity) and a hexagon, similar to a snowflake or polar vortexes of gas giants.

Let's imagine a homogeneous medium that consists of individual particles. The only possible position of the particles relative to each other, at which absolute homogeneity is achieved, is a tetrider, or for simplicity, a triangular lattice in one plane, at the intersections of which the particles are located. Thus, all distances between particles are the same. Particles interact with each other by being attracted at a distance and repelled upon collision, which is caused by the forces of molecular attraction and repulsion.

Now suppose one particle received an impulse and moved from its place in the direction of the other two. If we considered particles as billiard balls, then we could assume that the momentum would be divided into two. But in this case we have forces of molecular attraction and repulsion, which allow us to regard further interaction as a chain reaction similar to the domino principle, where momentum is transmitted indefinitely due to the force of gravity. Having logically followed the trajectory of the particles, we will see that the impulse in a circle on both sides, forming a figure eight, returned to the first part, which caused the action, which will lead to an endless repetition of the process. It is precisely this mechanism that underlies the ring vortex, which under ideal conditions, according to viscous friction, can exist endlessly dissipating its energy.

 

[Arjan82]

Several remarks:

• The Navier-Stokes equations themselves are derived for a continuum (in which all particles are smeared out over space) and thus have nothing to do with molecular interactions what so ever.
• I've never seen or heard of the application of a triangular coordinate system, but I don't know that much about Mathematics, however, what I do know is that any coordinate system which is not orthogonal has as major disadvantage that movement in one direction is never independent from the other directions. Navier-Stokes are actually 3 equations. One equation for one dimension, this will then not work anymore.
• The nice ordered hexagonal structure in the drawings you provide simply don't exist in a fluid, certainly not in a gas but also not in a liquid. So this molecular interaction you hypothesize has no influence whatsoever on the shapes you see on macro scale in a fluid.
• You suggest we should solve for every fluid particle (molecule?). This is not even in the slightest way possible for any real application whatsoever. Even the tiniest of droplets contain way to many molecules for that to be feasible, have a look at Avogadro's constant...

... where momentum is transmitted indefinitely due to the force of gravity.

Gravity on molecular scale is way too small to even be measurable. It has no effect on the local motion of molecules.

Having logically followed the trajectory of the particles, we will see that the impulse in a circle on both sides, forming a figure eight, returned to the first part, which caused the action, which will lead to an endless repetition of the process. It is precisely this mechanism that underlies the ring vortex,

Absolutely not true. Where did you get this from?!?

which under ideal conditions, according to viscous friction, can exist endlessly dissipating its energy.

This is an oxymoron...

 

[Arjan82]

Let's imagine a homogeneous medium that consists of individual particles. The only possible position of the particles relative to each other, at which absolute homogeneity is achieved, is a tetrider, ...

This is by the way also very much not true. Homogeneity has nothing to do with crystal structure. It just means that you have the same stuff everywhere, not how this stuff is organized in space (glass is homogeneous but amorphous, i.e. has no crystal structure). Note that density in a gas or to much lesser extent in liquid may vary (sound waves can only travel in an elastic medium, which by definition means variable density, sound also travels in water). Also, the reason that a liquid behaves as a liquid is because the molecules can move relatively freely in the medium.

 

[MasterOgon]

Thank you. My question was not corrected regarding the grid. This is a liquid model and if not every separate particle in reality, then their simplified generalization. I needed this in order to explain the impulse that the attached mass of air tells the plate after it stopped at the top point (video). I could not find this explanation anywhere.
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