---
title: "Buckingham Pi repeating variable rules"
subject: "Fluid Mechanics"
chapter: 10
key: "semester_3/fluid_mechanics:buckingham-repeating-variables-rules"
source: https://www.bctengineer.com/q/semester_3/fluid_mechanics/buckingham-repeating-variables-rules
site: BCT Engineering
---

# Buckingham Pi repeating variable rules

Fluid Mechanics · Chapter 10: Similitude and physical modeling

- Asked in: 2078 Bhadra, 2+2 marks; 2073 Shrawan, 2 marks (asked 2 times)
- Video: [Buckingham's Pie Theorem](https://www.youtube.com/watch?v=6VeCQeb7TZk) — TutorialsPoint
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/fluid_mechanics/buckingham-repeating-variables-rules
- Whole chapter: https://www.bctengineer.com/md/semester_3/fluid_mechanics/chapter-10

## Question

List the guiding rules for the choice of repeating variables in the Buckingham $\pi$ method. Also state the rules that apply to form the groups of dimensionless $\pi$-terms.

## Short answer

Choose m repeating variables (typically length, velocity, density) that are dimensionally independent and contain all fundamental dimensions. The dependent variable is excluded. Form n−m π terms by combining repeating variables with each remaining variable, solving exponents for dimensionless groups. The final relation is π₁ = φ(π₂, π₃, …).

## Answer

#### Rules for choosing the repeating variables

1. The number of repeating variables equals the number of fundamental dimensions in the problem (usually 3: M, L, T), $m = 3$.
2. Together they must contain all the fundamental dimensions in the problem, and they must be **dimensionally independent** (no repeating variable, and no product of powers of them, can be dimensionless).
3. The **dependent variable** must not be chosen as a repeating variable.
4. The repeating variables should not themselves form a dimensionless $\pi$-group.
5. Pick variables that are easy to measure and control, one from each class: a **geometric** one (e.g. $D$ or $L$), a **flow** property (e.g. $V$), and a **fluid** property (e.g. $\rho$). Typical choice: $D, V, \rho$.
6. Avoid choosing two variables with the same dimensions (like $D$ and $L$).

#### Rules for forming the $\pi$ terms

1. Number of $\pi$ terms $= n - m$, where $n$ is the total number of variables.
2. Each $\pi$ term is formed from **all** repeating variables, each raised to an unknown power, times **one** non-repeating variable (raised to power 1).
3. The powers are found by equating the exponents of M, L and T to zero so that the group is dimensionless.
4. Each $\pi$ should contain only one non-repeating variable; the dependent variable appears in only one $\pi$ (called $\pi_1$).
5. Any $\pi$ may be inverted, raised to a power, or multiplied by a constant or another $\pi$ without changing its dimensionless nature. Dimensionless variables given in the problem (angles, ratios like $\epsilon/D$) are already $\pi$ terms.
6. The final relation is $\pi_1 = \phi(\pi_2, \pi_3, \dots)$.

## Check yourself

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

1. How many repeating variables are chosen in Buckingham's method?
   A. Equal to the number of variables, n
   B. Equal to n - m
   C. Equal to the number of fundamental dimensions, m
   D. Always one
   Answer: C. The number of repeating variables equals m, usually 3 (M, L, T). (Key point: Number of repeating variables)

2. Which variable must NOT be chosen as a repeating variable?
   A. The fluid density
   B. The velocity
   C. The geometric length
   D. The dependent variable
   Answer: D. The dependent variable should appear in only one pi group, so it cannot be repeating. (Key point: Dependent variable)

3. How many pi terms are formed?
   A. n - m
   B. n + m
   C. m
   D. n x m
   Answer: A. The number of independent dimensionless groups is the number of variables minus the number of fundamental dimensions. (Key point: Number of pi terms)

4. Which set is a typical good choice of repeating variables?
   A. D, L, V (two lengths)
   B. D (geometric), V (flow), rho (fluid)
   C. mu, rho, nu (all fluid properties)
   D. The dependent variable, D, V
   Answer: B. Choose one geometric, one flow and one fluid property, and avoid two with the same dimensions. (Key point: Choice of repeating variables)

## Related in Fluid Mechanics

- [Pressure drop in pipe by Buckingham-Pi](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Apipe-pressure-drop-buckingham) — Fluid Mechanics, chapter 10
- [Wall shear stress as dimensionless function](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Awall-shear-stress-dimensionless-pi) — Fluid Mechanics, chapter 10
- [Distorted vs undistorted model; Pi theorem](https://www.bctengineer.com/question.md?key=semester_3%2Ffluid_mechanics%3Adistorted-vs-undistorted-buckingham-principle) — Fluid Mechanics, chapter 10

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