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.
π1= ρx1Hy1 g z1Q
= (ML−3)x1(L)y1 (LT-2)Z1 (L3T-1); M: x1 = 0, L: -3x1 y1 z1 3 = 0, T: -2z1 – 1 = 0
Therefore: z1 = -½, y1 = -5/2
π = ρ0H-5/2g-1/2 Q, π1=, π 2=ρ X2 HY2gz2 µ
M: x2 1 = 0; L: -3x2 y2 z2 -1 = 0; T: -2z2 – 1 = 0
X2 = -1, z2 = -1/2, y2 = 3/2; π =ρ1 H 3/2g -1/2 µ