How to use this calculator
List each variable that affects the problem with its units (kg/m^3, Pa*s, N, m/s) or its dimensions (M L^-3). Put the quantity you want to predict first. The calculator finds the rank r of the dimensional matrix, picks r repeating variables (or uses yours), and forms one Pi group for each other variable.
Formulas
Worked example
Drag F on a sphere: F, ρ, V, D, μ
- n = 5 variables in M, L, T, so r = 3 and k = 2 groups. Repeating: ρ, V, D.
- Π₁ = FρaVbDc: M: 1 + a = 0, L: 1 − 3a + b + c = 0, T: −2 − b = 0, so a = −1, b = −2, c = −2.
- Π₁ = F/(ρV²D²), a drag coefficient; similarly Π₂ = μ/(ρVD), the inverse of the Reynolds number.
- So F/(ρV²D²) = φ(Re).
Frequently asked questions
How do I choose repeating variables?
Together they must contain every basic dimension, they must not form a dimensionless group by themselves, and the dependent variable must not be one of them. A density, a velocity and a length is the usual choice in fluid mechanics.
My group is the inverse of the textbook one. Is it wrong?
No. Any power or product of Pi groups is also dimensionless, so μ/(ρVD) and ρVD/μ are equally valid.
What if the rank is less than the number of dimensions?
Then the dimension rows are not independent (for example, every variable has M and T only in the combination M T⁻²). The number of groups is still n minus the rank.