# Query regarding a published description of a mechanical hydraulic system analogous to $V = IR$ in an electrical system

The screenshot is taken from the wikipedia article for a resistor - https://en.wikipedia.org/wiki/Resistor

I understand that a larger head results in greater hydrostatic pressure, therefore a greater pressure gradient (analogous to potential difference).

I take it that a clogged sink increases the "resistance" to flow due to less cross-sectional area for the fluid to move through, like how the cross-sectional area of a wire is inversely proportional to electrical resistance.

But what I don't understand is, in the hydraulic analogy, what they actually mean by "flow".

"When the pipe is clogged, it takes a larger pressure to achieve the same flow of water”.. - What does "flow" actually mean in this context?

I'm assuming it's not flow rate as I believe, in accordance with the continuity equation, that that quantity remains constant for the given system regardless of any area increase/decrease. And it can't be flow velocity as, for the clogged sink, that will surely increase. So how does the flow actually decrease?

• If you have difficulties with the hydraulic system, what is the point in using it to model the electric phenomena? The fluid mechanics is at least as complicated as electrical circuits phenomena. If not more complicated.
– nasu
Sep 10, 2021 at 20:32

## 2 Answers

Flow is flow rate. The continuity equation doesn't mean what you think. The continuity equation says all the cross-sections of a system have the same flow rate as each other.

By adding the hair we have changed the system and made a different system, and all the cross-sections of this system also have the same flow rate as each other, but (if the pressure is the same as the first system) less than the cross-sections of the first system.

• Do you mind posting the equations to prove this? Sep 10, 2021 at 15:34
• Equations mostly just express the things we have already proved. Does it matter whether I say $A_1v_1=A_2v_2$ or whether I say "all cross-sections of a system have the same flow rate as each other"? Because how do you know what $A_1$ and $A_2$ are? I'd have to explain it in words anyway. Sep 10, 2021 at 15:36
• I understand that that equation shows that all cross-sections of a system has the same flow rate as each other. What I don't quite understand is why, given the two different systems with their only difference being one the sink is more clogged than the other, why exactly an increase in head is needed for the clogged-sink system to achieve the original system's flow-rate. Additionally, will the flow velocity through the clogged sink in system 2 be faster than that through the unclogged sink in the orginal system? If so, why? How do Bernouli and continuity prove this? Sep 10, 2021 at 15:41
• I don't doubt what you're saying but I'd rather fully understand the reason why by breaking it down thoroughly equation-wise rather than just memorise the "correct answer"... Sep 10, 2021 at 15:42

The kind of situation this equation refers to is one in which the viscous frictional pressure drop dominates over gravitational and kinetic energy effects, so that the latter are negligible. In this equation, the flow refers to the mass flow rate and the R refers to the viscous frictional resistance. The V is analogous to the pressure drop.