Is it possible to solve the maximum flow problem in the real world, ie. using water running through physical pipes? So you would have a tap at the source node, pipes of various diameter (depending on their capacity) connecting nodes and a sink at the target node. You would turn on the tap on full and then try to measure how much water is collected at the sink (say per minute) - this should give you maximum flow of the corresponding network. Of course there will be some issues like water leakage, tricky pipe connections etc., but in theory it should be possible to construct this in the real world. The big question for me: would the water actually flow optimally and produce the theoretical maximum flow rate?
If this setup works, you could probably have an ingenious method for automatically adjusting the diameter of the pipes. This way you will be able to solve a variety of networks.