---
title: "Pressure drop in pipe by Buckingham-Pi"
subject: "Fluid Mechanics"
chapter: 10
key: "semester_3/fluid_mechanics:pipe-pressure-drop-buckingham"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/pipe-pressure-drop-buckingham
site: BCT Engineering
---

# Pressure drop in pipe by Buckingham-Pi

Fluid Mechanics · Chapter 10: Similitude and physical modeling

- Asked in: 2078 Kartik, 8 marks
- Video: [Fluid Mechanics - dimensional analysis pipe flow - Example 8-4](https://www.youtube.com/watch?v=iUKbw7qjYeE) — Dr Karl's engineering channel
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/pipe-pressure-drop-buckingham
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-10

## Question

A pressure drop $\Delta P$ provides a measure of the frictional losses of a fluid as it flows through a pipe. Determine how $\Delta P$ is related to the variables that influence it, namely, pipe dia. D, its length L, fluid density $\rho$, viscosity $\mu$, velocity V and the relative roughness factor $\frac{\epsilon}{D}$, which is ratio of average size of surface irregularities to the pipe diameter. Use Buckingham-$\pi$ method.

## Short answer

Using Buckingham π with repeating variables D, V, ρ yields ΔP/(ρV²) = φ(ρVD/μ, L/D, ε/D). For pipe flow this reduces to ΔP = (L/D)ρV² φ₁(Re, ε/D), the Darcy–Weisbach form.

## Answer

Use the Buckingham $\pi$ method with the variables $\Delta P, D, L, \rho, \mu, V$ and $\epsilon/D$.

#### Variables and dimensions

| Variable | Symbol | Dimensions |
|---|---|---|
| Pressure drop | $\Delta P$ | $ML^{-1}T^{-2}$ |
| Diameter | $D$ | $L$ |
| Length | $L$ | $L$ |
| Density | $\rho$ | $ML^{-3}$ |
| Viscosity | $\mu$ | $ML^{-1}T^{-1}$ |
| Velocity | $V$ | $LT^{-1}$ |
| Roughness ratio | $\epsilon/D$ | dimensionless |

$\epsilon/D$ is already dimensionless, so it is a $\pi$ term by itself. For the other six variables: $n = 6$, fundamental dimensions $m = 3$ (M, L, T), so $n - m = 3$ more $\pi$ terms. Total $= 4$.

Repeating variables: $D$ (geometric), $V$ (kinematic), $\rho$ (fluid property).

#### $\pi_1$ (with $\Delta P$)

$$
\pi_1 = \Delta P\,D^{a}V^{b}\rho^{c}
$$

M: $1 + c = 0 \Rightarrow c = -1$. T: $-2 - b = 0 \Rightarrow b = -2$. L: $-1 + a + b - 3c = 0 \Rightarrow a = 0$.

$$
\pi_1 = \frac{\Delta P}{\rho V^2}
$$

#### $\pi_2$ (with $L$)

Same dimension as $D$, so $\pi_2 = \dfrac{L}{D}$.

#### $\pi_3$ (with $\mu$)

$$
\pi_3 = \mu\,D^{a}V^{b}\rho^{c}:\quad c = -1,\; b = -1,\; a = -1
\;\Rightarrow\; \pi_3 = \frac{\mu}{\rho V D} = \frac1{Re}
$$

#### Result

$$
\frac{\Delta P}{\rho V^2} = \phi\left(\frac{\rho V D}{\mu},\ \frac LD,\ \frac\epsilon D\right)
$$

Experiments show $\Delta P \propto L/D$, so

$$
\Delta P = \frac{L}{D}\,\rho V^2\,\phi_1\left(Re,\ \frac\epsilon D\right)
$$

This is the Darcy-Weisbach form, with friction factor $f$ related to $\phi_1$ by $\phi_1 = f/2$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. How many pi terms are in the answer?
   A. 3 only
   B. 6
   C. 4 in total, including epsilon/D
   D. 7
   Answer: C. The six variables with m = 3 give 3 pi terms, and epsilon/D is already dimensionless, giving 4. (Key point: Number of pi terms)

2. Which repeating variables are chosen?
   A. DeltaP, D, V
   B. L, D, mu
   C. mu, V, epsilon
   D. D, V, rho
   Answer: D. One geometric (D), one flow (V) and one fluid (rho) property, with no dependent variable. (Key point: Repeating variables)

3. What is pi1, the group containing DeltaP?
   A. DeltaP/(rho V^2)
   B. DeltaP D/(mu V)
   C. DeltaP/(rho V D)
   D. DeltaP rho V^2
   Answer: A. M: c = -1, T: b = -2, L: a = 0 gives DeltaP/(rho V^2). This is Euler-number-like. (Key point: pi1)

4. What are pi2 (with L) and pi3 (with mu)?
   A. L x D and rho V D/mu
   B. L/D and mu/(rho V D) = 1/Re
   C. D/L and mu V
   D. L/D and rho V^2
   Answer: B. L has the dimension of D, and mu gives the inverse Reynolds number. (Key point: pi2 and pi3)

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## Related in other subjects

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- [Pressure drop in a pipe network](https://www.bctengineer.com/question.md?key=semester_7%2Fapplied_computational_fluid_dynamics%3Apipe-network-pressure-drop-numerical) — Applied Computational Fluid Dynamics, chapter 6