To what extent are hosepipes linear systems?

[Dave2002]

I was once picked up by an electronics engineer. I had been talking to some people at NASA about modelling, and they'd noticed that in the computer models they had, that several components could be swapped over, and the results were exacty the same. I mean - really exactly the same. As the original scientists were oriented towards biology they didn't know what some mathematicians or engineers would know - and I realised since. When I mentioned this to the EE scientist his comment was - "What would you expect - they were modelling a linear system?". Indeed that was the case, and since their model was very accurate, and they were effectively modelling components connected in series, the numbers, even using floating point arithmetic, came out precisely the same even down to the nth decimal point.

However, not all systems are linear, so that property wouldn't necessarily hold for all physical systems.

In electrical engineering and electronics it's quite common to draw an analogy between water flowing through a pipe and current through a wire, or network of components. I wondered if the analogy always works in reverse. I have two hosepipes. One is narrow bore, and one is wide bore, and I am using them to water plants in the garden. In particular we now have a small cheappie greenhouse with a plastic cover, and I've been delegated the task of watering seed trays in there. [a previous "delegated" task was making the greenhouse ... and subsequently re-erecting it whan it blew over in high winds ....]

It seems intuitively obvious to me to use the wider bore tubing connected to the mains tap, then the new narrow bore tubing with the sprinkler on, to water the seed trays, but it did occur to me that if the pipes work as linear systems it will make no difference at all. So - are pipes essentially linear systems, or is there sufficient non-linearity (particularly under flow conditions) that there would be a difference in behaviour? Probably makes no difference at all in our garden though, and won't make any difference to how well the seeds come up!

[Belgrove]

The fundamental governing equations of electromagnetism in vacuum, Maxwell’s equations, are linear. When the fields interact with matter, linear approximations to nonlinear interactions are assumed, and these are frequently adequate for modelling of many devices. The governing equations of fluid mechanics, the Navier-Stokes equations, are fundamentally nonlinear, and linearising assumptions are used to estimate behaviours in the first instance. If, for example, the pressure difference between the ends of a pipe is too large, the flow becomes nonlinear, which manifests itself as turbulence. In 1964 Feynman singled out turbulence as being the outstanding unsolved problem in classical physics, and that is still so.

[Dave2002]

There are very strange effects which can be discovered and indeed observed in fluids. However, I did suggest low flow levels, rather than high speed jets or turbulent flows.

I liked that pecular behaviour I think mentioned in Bachelor's book on fluid dynamics in which a fluid in rotation .... oh dear - I'm going to have to find my copy and look at it again, to remember it. It's a very strange behaviour - I'm not sure which came first - the observations or the mathematics, but in that case I think it was the mathematics, and once it had been noted as a possiblity, experimenters went and showed that it actually happens.