The Rayleigh's method of dimensional analysis becomes more laborious if the variables are more than the number of fundamental dimensions (M, L, T). This difficulty is overcame by using Buckingham's π-theorem, which states, "If there are n variables contain m fundamental dimensions (M, L, T), then the variables are arranged into (n - m) dimensionless terms. Each term is called π-term".
Let X1, X2, X3, X4, .......Xn are the variables involved in a physical problem. Let X1 be the dependent variable and X2, X3, .....Xn are the independent variables on which X1 depends. Then X1 is a function of X2, X3......Xn and mathematically it is expressed as
X1 = f (X2, X3, ......,Xn). Eq.(1)
Equation (1) can be also written as
f1( X1, X2, X3,..........,Xn)= 0. Eq.(2)
Equation (2) is a dimensionally homogenous equation. It contains n variables. If there are m fundamental dimensions then according to Bunkingham's π-theorem, equation (2) can be written in terms of number of dimensionless groups or π-terms in which number of π-terms is equal to (n - m). Hence equation (2) become as
f( π1, π2, π3,........π(n -m ) = 0. Eq.(3)
Each of the π-terms is dimensionless and is independent of the system. Division or multiplication by a constant does not change the character of the π-term. Each π-term contains m + 1 variables, where m is the number of fundamental dimensions and is also called repeating variables. Let in the above case X2, X3, X4 are repeating variables if the fundamental dimensions m (M, L, T) = 3. Then each π-terms is written as
{ π1= X2a1 . X3b1 . X4c1 . X1
π2= X2a2 . X3b2 . X4c2 . X5
:
:
π(n-m)= X2a(n-m) . X3b(n-m) . X4c(n-m) } Eq.(4)
Each equation is solved by the principle of dimensionless homogeneity and values of a1, b1, c1 etc., are obtained. These values are substituted in equation (4) and values of π1, π2, π3........,πn-m are obtained .
These values of π's are subtituted in equation (3). This final equation for the phenomenon is obtained by expressing any one of the π-terms as a function of others as
Eq.(5)
Let X1, X2, X3, X4, .......Xn are the variables involved in a physical problem. Let X1 be the dependent variable and X2, X3, .....Xn are the independent variables on which X1 depends. Then X1 is a function of X2, X3......Xn and mathematically it is expressed as
X1 = f (X2, X3, ......,Xn). Eq.(1)
Equation (1) can be also written as
f1( X1, X2, X3,..........,Xn)= 0. Eq.(2)
Equation (2) is a dimensionally homogenous equation. It contains n variables. If there are m fundamental dimensions then according to Bunkingham's π-theorem, equation (2) can be written in terms of number of dimensionless groups or π-terms in which number of π-terms is equal to (n - m). Hence equation (2) become as
f( π1, π2, π3,........π(n -m ) = 0. Eq.(3)
Each of the π-terms is dimensionless and is independent of the system. Division or multiplication by a constant does not change the character of the π-term. Each π-term contains m + 1 variables, where m is the number of fundamental dimensions and is also called repeating variables. Let in the above case X2, X3, X4 are repeating variables if the fundamental dimensions m (M, L, T) = 3. Then each π-terms is written as
{ π1= X2a1 . X3b1 . X4c1 . X1
π2= X2a2 . X3b2 . X4c2 . X5
:
:
π(n-m)= X2a(n-m) . X3b(n-m) . X4c(n-m) } Eq.(4)
Each equation is solved by the principle of dimensionless homogeneity and values of a1, b1, c1 etc., are obtained. These values are substituted in equation (4) and values of π1, π2, π3........,πn-m are obtained .
These values of π's are subtituted in equation (3). This final equation for the phenomenon is obtained by expressing any one of the π-terms as a function of others as
Eq.(5)
Credit:- Book Author- DR.R.K.Bansal
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