In hydraulic analogy one compares electrical circuits with water circuits. For the electric case the formula $P = U \cdot I$ for the electric power holds. The analogous formula for water flow would be $P = \Delta p \cdot I_W$ where $\Delta p$ ist the pressure difference and $I_W$ the flow rate of the water through the pipe. I have some questions about this:
- under what circumstances/assumptions does this analogous formula hold
- $P$ in the electric case can be interpreted as the energy per second which is dissipated for example in a resistor. Is there a similar interpretation in the water case and why does it hold?
- with the assumptions from above, how can one derive the formula from first principles (e.g. from Bernoulli-equation or even from Navier-Stokes)?
- with the assumptions from above, is there a nice conceptual argument, why the formula holds in the water case?