# Parallel pipes in fluid dynamic

For example, if I have three parallel pipes.

Resistance of whole system and three pipes are R, R1, R2, R3 respectively

Pressure drop of whole system and two ends of three pipes are ∆P, ∆P1, ∆P2, ∆P3 respectively

Flow of whole system and three pipes are Q, Q1, Q2, Q3 respectively

By Darcy's law (also, Ohm's law), we have ∆P=QxR

And, since this is parallel configuration, we also have

Q=Q1+Q2+Q3

1/R=1/R1+1/R2+1/R3

So, my question is as follow

Is ∆P=∆P1=∆P2=∆P3?

I'm asking this question because in electricity, V=V1=V2=V3 for three parallel wires. So I wonder if this is true for fluid dynamic as well? Thank you

• Short answer: yes. Bonus: try driving $1/R=1/R_1+1/R_2+1/R_3$ from considering that $∆P=∆P_1=∆P_2=∆P_3$ – Eagle Apr 27 at 14:28
• The pipes A & B are not included in the analysis? – user207455 Apr 27 at 14:35
• @SolarMike yes, ∆P is the pressure difference between A and B, R is the total resistance – nthntn Apr 27 at 14:41
• @Eagle But I saw this video, it seems a little bit confusing youtube.com/watch?v=N_0-H8nxQdo – nthntn Apr 27 at 14:43

In one word, Yes, the pressure difference will be same across the parallel combination, analogous to the same potential difference across the resistors in parallel combination. This is the beauty of physics. These types of analogies help in better understanding, help to correlate concepts.

The resistance approach is used for fluid flow and current flow and heat flow. See this reference for a table. For systems where a difference in potential $$\Delta P$$ causes a flow of something $$\dot{s}$$ that is mediated by a resistance $$R_T$$, we can write an Ohm's law analogy.

$$\Delta P = \dot{s} R_T$$

## Series

For a set of pipes, resistors, and walls in series, the flow of fluid, current, or heat is the same across all components. Each component will have a different potential (pressure drop, voltage drop, or temperature drop).

$$\dot{s}_j = \dot{s}\ \ \ \ \Delta P_T = \sum \Delta P_j \ \ \ \ R_T = \sum R_j$$

## Parallel

For a set of pipes, resistors, and walls in parallel, the flow of fluid, current, or heat is different in each components. Each component will have the same potential across it (pressure drop, voltage drop, or temperature drop).

$$\Delta P_j = \Delta P_T\ \ \ \ \dot{s}_T = \sum \dot{s}_j \ \ \ \ R_T^{-1} = \sum R_j^{-1}$$