Could an NP-hard problem have a mechanical or physical solution method? Is there any NP-hard problem that we can find a mechanical "polynomial time" solution to? For example, suppose we construct a graph out of something physical, e.g. we have have pipes through which we can move water. If this pipe system was suitably built, could we solve say some routing problem by seeing if water flows from one node to another?

So by a mechanical solution, I mean a physical configuration and not a computer program, i.e. I am not looking for a computer algorithm. I also used "polynomial mechanical solution" to describe a solution on a physical device that runs in time polynomial in the order of the input. For example, a piping system could yield the answer in "number of nodes squared" seconds.
No.

Unless $P=NP$, it's unlikely that any mechanical process can solve an $NP$-hard problem efficiently. Given a mechanical approach to solving a problem, we could run a physics simulation on a computer to get the same result.

Of course this depends on being able to simulate what happens in the real world efficiently. In order for "mechanical polynomial time" to be stronger than "polynomial time" you'd need to find some physical process that can not be simulated efficiently. One possible candidate would be quantum mechanics but that doesn't seem to be what you're asking about (and it is speculated that even quantum computing doesn't allow one to solve $NP$-hard problems efficiently).