Consider a water distribution setup consisting of a feeder line blocked at one end but having 100 needle outlets all the same size. The feeder line cross sectional area is much greater than a needle cross sectional area so the flow from any needle is very close to one hundredth the overall flow into the feeder line. I think the feeder line would act like a massive reservoir which is being topped up at the entry point. Lets say that the flow into the feeder line is 1L/sec. Then the flow out through any needle is 10mL/sec. Assume that the system is in steady state with all needles happily delivering their 10mL/sec.
At the entry point of the feeder line we introduce 100mL of red dyed water. Assume that this does not affect the water characteristics other than it's now red. How is this red water distributed through the needles? Intuitively, I'm thinking that the first needles (the lower numbered ones) will get more of the red water than the later needles. But is this true? Does anyone have any mathematics that could describe this? I briefly considered that it was like an impulse function; but this would only be so if the steady state of the system changed. It doesn't. It's still water with the same characteristics, just red. It's like being able to tag electric current with a colour - it's still electric current...
At the entry point to the feeder line we install a water switch. The water can be switched to either blue water or red water. If we have the switch set to BLUE and we have all needles delivering their 10mL/sec of blue water and then switch to RED. How long does it take the system to be delivering 10mL/sec of red water from all of the nozzles? Assume that the switching action has no effect on the system itself other than changing the colour of the water. I'm hoping to find some sort of mathematics to describe this.
Otherwise I might have to build the thing and see what happens!