Buckingham Pi-theorem states that, “If there are n variables (dependent and independent variables) in a dimensionally homogenous equation and if these variable contain m fundamental dimensionless terms are called Π terms.”

Step 1: Mathematically, if any variable X1 depends on independent variables X₂,X₃,X_4,.......,X_n’. Then, the functional relationship between dependent and independent variable is expressed as

X₁ = f (X₂,X₃.X₄,.........,X_n)

Equation can be written as,

f₁ (X₁,X₂,X₃,X₄,..........X_n)  ∫ = 0

It is dimensionally homogenous equation and contains n variables.

Step 2: The equation can be written in terms of number of dimensionless Π term which is equal to (n-m).

f (Π₁, Π_2,......Π_n ' _m)  = 0                   

Step 3: Each Π-term contains m + 1 variables, where m is the number of fundamental dimensions and it is also called as repeating variable.

Step 4: If X₂,X₃,X₄ are repeating variables, then each Π term is written as:

Π₁ =  X^a1 . X^b1 . X^c1.  X

Π₂  =  X₂^a2 . X₃^b2 . X₄^c2.  X₅

Π_n-m = X₁^{a n-m}   X₂^{b n-m}    X₃^{c n-m} . X _n-m

Where X_4,X₅..........X_n-m are non repeating variables.

Step 5: Where equation is solved by principle of dimensional homogeneity and the value of a, b, c are obtained.

Step 6: Substituting the value of a, b, c in their corresponding Π₁, Π₂, Π₃,......etc.

Step 7: The value of Π₁, Π₂, Π₃.........etc. are substitute in Equation (5.4.3)

Step 8: The required expression can obtained by expressing any one of the Π terms as a function of others.

Π₁ =  Φ [ Π_2, Π_3,........Π_n-m ]

The following points should be considered for selecting the repeated variables:

1.    The variables should not be dimensionless.
2.    No two variables should have same dimensions.
3.    As far as possible, the dependant variables should not be selected as repeated variables.
4.    The repeated variables is selected in such a way that is contains one from each.
        (i)    Geometric properties i.e. length, diameter, height etc.
        (ii)    Flow properties i.e. velocity, acceleration, rotational speed etc.
        (iii)    Fluid properties i.e. mass density, specific weight, viscosity etc.

The choice of repeating variable for most of fluid mechanics problems may be:

i) d, v, ρ     (ii) d, v, μ    (iii) d, N, ρ       (iv) L, v, μ        (v) H, g, ρ

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