---
title: "Buckingham pi theorem for pipe pressure drop"
subject: "Fluid Mechanics"
chapter: 5
key: "semester_4/fluid_mechanics_mechanical:buckingham-pi-pipe-pressure-drop"
source: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/buckingham-pi-pipe-pressure-drop
site: BCT Engineering
---

# Buckingham pi theorem for pipe pressure drop

Fluid Mechanics · Chapter 5: Dimensional Analysis and Dynamic Similitude

- Asked in: year not given, 8 marks
- Video: [Fluid Mechanics 10.5 - Solved Example Problem, Determining Pi Terms, Turbulent Flow in a Pipe](https://www.youtube.com/watch?v=ZQFyFKjNygE) — Rasim Guldiken
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/fluid_mechanics_mechanical/buckingham-pi-pipe-pressure-drop
- Whole chapter: https://www.bctengineer.com/md/semester_4/fluid_mechanics_mechanical/chapter-5

## Question

State Buckingham's pi theorem. The pressure drop $\Delta p$ in a pipe of diameter $D$ and length $L$ depends on the mean velocity $V$, the fluid density $\rho$, the dynamic viscosity $\mu$ and the roughness height $\varepsilon$ of the pipe wall. Using Buckingham's method, obtain a dimensionless relation for $\Delta p$.

## Short answer

Buckingham's pi theorem states a problem with n variables and m dimensions reduces to n-m dimensionless groups. For pipe pressure drop, using D, V, ρ as repeating variables yields Δp/(ρV²) = φ(L/D, Re, ε/D), leading to the Darcy-Weisbach equation.

## Answer

#### Buckingham's pi theorem
If a physical phenomenon involves $n$ variables and these contain $m$ fundamental dimensions (M, L, T), then the relation among them can be expressed in terms of $(n - m)$ independent dimensionless groups ($\pi$-terms).

#### Step 1: Variables
$\Delta p = f(D, L, V, \rho, \mu, \varepsilon)$, so $n = 7$.

| Variable | Dimensions |
|---|---|
| $\Delta p$ | $ML^{-1}T^{-2}$ |
| $D$, $L$, $\varepsilon$ | $L$ |
| $V$ | $LT^{-1}$ |
| $\rho$ | $ML^{-3}$ |
| $\mu$ | $ML^{-1}T^{-1}$ |

#### Step 2: Number of pi terms
$m = 3$ (M, L, T), so the number of $\pi$ terms $= 7 - 3 = 4$.

#### Step 3: Repeating variables
Choose 3 repeating variables that together contain M, L and T and cannot form a dimensionless group among themselves: $D$ (geometric), $V$ (kinematic) and $\rho$ (dynamic).

#### Step 4: Form the pi terms
**$\pi_1$ 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 $L$:** length has the same dimension as $D$, so $\pi_2 = L/D$.

**$\pi_3$ with $\mu$:** $\pi_3 = \mu D^a V^b \rho^c$

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

$$
\pi_3 = \frac{\mu}{\rho V D} = \frac{1}{Re}
$$

**$\pi_4$ with $\varepsilon$:** $\pi_4 = \varepsilon/D$ (relative roughness).

#### Step 5: Final relation
$$
\frac{\Delta p}{\rho V^2} = \phi\left(\frac{L}{D}, Re, \frac{\varepsilon}{D}\right)
$$

Experiments show that $\Delta p$ is directly proportional to $L$. Therefore

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

where $f = 2\phi_1(Re, \varepsilon/D)$ is the Darcy friction factor. This is the Darcy-Weisbach equation, $h_f = f L V^2/(2 g D)$.

## Check yourself

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

1. A phenomenon involves n variables containing m fundamental dimensions. How many independent dimensionless pi terms describe it?
   A. n + m
   B. n - m
   C. n x m
   D. m - n
   Answer: B. Buckingham's theorem states the relation can be expressed with (n - m) independent dimensionless groups. (Key point: Buckingham pi theorem)

2. For Delta p = f(D, L, V, rho, mu, epsilon) with M, L, T as the dimensions, how many pi terms are formed?
   A. 3
   B. 7
   C. 10
   D. 4
   Answer: D. n = 7 variables and m = 3 dimensions, so 7 - 3 = 4 pi terms. (Key point: Number of pi terms here)

3. Which repeating variables are chosen, and why?
   A. D, V, rho: one geometric, one kinematic, one dynamic, together containing M, L and T
   B. Delta p, mu, epsilon: the most important variables
   C. L, D, epsilon: all lengths so that L is covered
   D. V, mu, Delta p: variables with the most dimensions
   Answer: A. The repeating variables must together contain M, L and T and not form a dimensionless group; D, V and rho do. (Key point: Choice of repeating variables)

4. After finding pi terms (Delta p/(rho V^2), L/D, Re, epsilon/D), how is the Darcy-Weisbach equation reached?
   A. Because Delta p is inversely proportional to L
   B. Because Re is dropped for turbulent flow
   C. Because experiment shows Delta p is proportional to L, so the L/D factor is taken out and f = 2*phi1(Re, epsilon/D)
   D. Because epsilon/D is always zero
   Answer: C. Delta p proportional to L gives Delta p = (L/D) rho V^2 phi1(Re, epsilon/D) = f (L/D)(rho V^2/2). (Key point: Final relation and Darcy-Weisbach)

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