## Buckingham’s π theorem

**Introduction: **This is the most well-known method and is the basis for the matrix and other methods. “If there are n variables in a problem and these variables contain m primary dimensions (e.g. M, L, T), the equation relating the variables will contain n-m dimensionless groups”. Buckingham referred to these dimensionless groups as π1, π2, …, πn-m, and the final equation obtained is: π1 = φ (π2, π3,…, πn-m).

E.g. consider the previous problem with 5 variables, hence there will be 5 – 3 = 2 dimensionless groups.

**The dimensionless groups can be formed as follows: **

1) Select 3 repeating variables from the list of variables which together must contain M, L, and T. eg. ρ (ML−3), H (L), g (LT-2)

2) Combine repeaters with one other variable in turn,

3) Never pick the dependent variable as a repeater.

Consider: Q = f(ρ, µ, H, g) = 5 variables Pick repeaters: ρ, H, g

Combine with one other variable, say, Q.

π_{1}= ρ^{x1}H^{y1} g ^{z1}Q

= (ML^{−3})^{x1}(L)^{y1} (LT^{-2})^{Z1} (L^{3}T^{-1}); M: x1 = 0, L: -3x1 y1 z1 3 = 0, T: -2z1 – 1 = 0

Therefore: z1 = -½, y1 = -5/2

π = ρ^{0}H^{-5/2}g^{-1/2} Q, π_{1}=**, **π _{2=}ρ ^{X2} H^{Y}2gz2 ^{µ}

M: x2 1 = 0; L: -3x2 y2 z2 -1 = 0; T: -2z2 – 1 = 0

X_{2} = -1, z2 = -1/2, y_{2} = 3/2; π =ρ^{1} H ^{3/2}g ^{-1/2} µ

Π2=

That is: