Buckingham Pi repeating variable rules

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 $\pi_1 = \phi(\pi_2, \pi_3, \dots)$.