Dynamics and convergence of a general flow network

[SteveMaryland]

Sorry if this is the wrong place to post, but my inquiry spans so many STEM disciplines I figured I would post it here. Also, I have really looked for papers which address this issue and hope someone on PF can advise.

Given a flow network, which could be any connected set of N resistors, or water pipes, etc. of any finite ohmage, diameter etc. and connected in any sort of parallel, series, delta-wye combinations. Under an applied energy gradient (voltage, gravity etc.), a flow will occur through this network, and each branch of the network will exhibit a non-zero flux.

Hypothesis: Upon gradient application, this network + fluid system will spontaneously converge to a specific set of flux allocations for each branch, and the sum of all branch fluxes will be a maximum possible for the given system metrics. True?

Why would the flux converge to a "max" flux? And, by what means (selection, trial/error, filtering, sortation) do flow systems in general converge to a "solution"? The convergence (to steady-state flow) cannot be instantaneous, but what does actually go on in the process? (by what physics does Nature solve such an N X N matrix "automatically"?)

(The above system is in steady-state flow, but is not in equilibrium.)

Thanks for your wisdom. And, for my further reading, please advise what branch of physics would study this general phenomenon!

 

[Baluncore]

Under an applied energy gradient (voltage, gravity etc.), a flow will occur through this network, and each branch of the network will exhibit a non-zero flux.

There are some branches that may have zero flux, for example, a balanced bridge network.

 

[DaveE]

and each branch of the network will exhibit a non-zero flux.

Not always. for example, a balanced bridge network:

by what means (selection, trial/error, filtering, sortation) do flow systems in general converge to a "solution"?

A nearly impossible question to answer, in general (non-linear networks, for example). Linear networks will have solution(s) as in a set of linear algebra equations.

 

[DaveE]

what branch of physics would study this general phenomenon?

For linear networks, it's mostly Linear Algebra and EE.
For non-linear networks look for Control Systems and Non-linear Dynamics.
But honestly, it's really mostly Math.

Steve Strogatz at MIT has some stuff you'll probably like. Some very accessible, some free online MIT courses. His pop-science book "Sync" is quite good, I think.

 

[SteveMaryland]

I understand that Man has math methods to calculate flow networks, but real material systems don't know any math yet they get the right answer anyway! How? What deeper laws of thermo-physics governs this behavior? It is like a Maxwells Demon is operating here... tuning each and every branch flow simultaneously (?) such that the total flow is maximized. And what is optimized here? Min energy? Max entropy? See https://en.wikipedia.org/wiki/Principle_of_minimum_energy

 

[Baluncore]

https://en.wikipedia.org/wiki/Network_theory

 

[DaveE]

I understand that Man has math methods to calculate flow networks, but real material systems don't know any math yet they get the right answer anyway! How? What deeper laws of thermo-physics governs this behavior? It is like a Maxwells Demon is operating here... tuning each and every branch flow simultaneously (?) such that the total flow is maximized. And what is optimized here? Min energy? Max entropy? See https://en.wikipedia.org/wiki/Principle_of_minimum_energy

Some very general questions just don't have simple answers. Networks could be as complicated as a human brain, or a collection of human brains interacting. In general, we just don't know (yet). I would have questions about the computability of generalized network solutions because of the huge number of degrees of freedom and their complex interactions.

real material systems don't know any math yet they get the right answer anyway!

This sounds like a post-hoc definition of "the right answer". There have been several examples of electrical distribution networks that did "the wrong thing" because of network stability issues. Epileptic seizures may also be a network doing "the wrong thing".
https://en.wikipedia.org/wiki/Northeast_blackout_of_1965

 

[Baluncore]

What deeper laws of thermo-physics governs this behavior?

Each network segment offers an impedance to flux.
Flux, is a rising function, of segment potential difference.
Power is dissipated in a segment, in proportion to flux and potential difference.

Segments exist in a network of other segments, each with an impedance to flux.
The flux, in one segment, passes through other segments.
The available potential difference is limited, and is shared by the segments.
Parallel segments share the flux, series segments share the potential.

An increase in segment flux, reduces the share of potential difference available from the network.
But segment flux was defined to rise with potential difference.
So, every segment in the network has a convergent, self-regulating flux.
Which leads to the concept of impedance matching and power transfer.


Mathematics is our symbolic analogue, of real world relationships.
The real world does not need mathematics, it is real.

 

[Borek]

Intuition tells me network built from linear elements will converge on some steady state solution, but if the elements are nonlinear (and all real elements are nonlinear) system can be chaotic (producing some randomly pulsing flow).

 

[SteveMaryland]

Thanks everyone for your contributions. My motivation for this enquiry:

[Personal Speculation has been removed from this reply]

 

[berkeman]

Thread is paused for Moderation...

 

[berkeman]

After some cleanup of Personal Speculation, this thread will remain closed. Thanks to everybody for trying to help the OP.