Explain the Methods of Dimensional Analysis.

Based on the Fouriers principle of homogeneity, some of the methods of dimensional analysis are:

• Rayleighs Method

• Buckinghams `pi` Method.

Rayleighs Method:

Rayleigh method of dimensional analysis is a method that gives a special form of relationship among the dimensionless groups, which can be used for determining the expression of a variable which depends upon maximum three or four variables only. This method has an inherent drawback that it does not provide any information regarding the number of dimensionless groups to be obtained as a result of dimensionless analysis.

This method employs the following procedures:

• First of all, gather all the independent variables which influence the dependent variable.

• Write down the functional relationship.

• Change the functional relationship to the equation in exponential terms.

• Express each of the quantities (variables) on  b....

Based on the Fouriers principle of homogeneity, some of the methods of dimensional analysis are:

• Rayleighs Method

• Buckinghams `pi` Method.

Rayleighs Method:

Rayleigh method of dimensional analysis is a method that gives a special form of relationship among the dimensionless groups, which can be used for determining the expression of a variable which depends upon maximum three or four variables only. This method has an inherent drawback that it does not provide any information regarding the number of dimensionless groups to be obtained as a result of dimensionless analysis.

This method employs the following procedures:

• First of all, gather all the independent variables which influence the dependent variable.

• Write down the functional relationship.

• Change the functional relationship to the equation in exponential terms.

• Express each of the quantities (variables) on  both sides of the equation in terms of fundamental dimensions.

• Solve the equation simultaneously and find the exponents.

Let X is a variable , which depends on `X_1,X_2,...X_n` independent variables. Mathematically, it can be written as,

`X=f (X_1,X_2, X_3,...X_n)`...(i)

Now, by Rayleigh method, it can be written as,

`X=K(X_1)^a*(X_2)^b*...(X_n)^z`...(ii)

Where, `K` is a dimensionless constant and the exponents a, b, ..z are found by comparing the power of fundamental dimensions on both sides of the equation and thus the equation is obtained for the dependent variables.

Buckingham `pi` Theorem:

Buckinghams method of dimensional analysis is an improvement over Rayleigh method, which let us know in advance analysis as to how ,any dimensionless groups are to be expected. Buckingham designated these dimensionless group by a greek letter `pi`. It is therefore called as Buckingham`pi` theorem.

It states, �if there are n variables (dependent and independent variables) in a dimensionally homogeneous equation and if these variables contains m fundamental dimensions (such as M, L, T, etc) them the variables are arranged into (n-m) dimensionless terms. These dimensionless terms are called pi terms.

Let any variable `X_1` depends on independent variables `X_2`, `X_3`,...`X_n`. Mathematically, it can be written as,

`X_1=f (X_2, X_3,...X_n)`

or,`F(X_1,X_2, X_3,...X_n)=0`.....(i)

It is a dimensionally homogeneous equations and contains `n` variables. If this equation contains `m` fundamental dimensions, then by buckinghams `pi` theorem,  this equation can be written in terms of `n-m` `pi` terms. Thus,

`F_1(pi_1,pi_2,....pi_(n-m))=0`...(ii)

Where each `pi` terms contains `m+1` variables where `m=3, M L T` are fundamental variables. The `m` variables which appears repeatedly in each of the `pi` terms are called repeating variables.

Selection of Repeating variables:

The following points should be kept in view while selecting `m` repeating variables:

1. The repeating variables must not form the non-dimensional parameters among themselves.

2. As far as possible, the dependent variables should not be selected as repeating variables.

3. No two repeating variables should have the same dimensions.

4. `m` repeating variables must contain jointly all the fundamental dimensions involved in the phenomenon.

5.  The repeating variables should be chosen in such a way that one variable contains geometric property (length, diameter, height), the other variables should contains flow property (velocity, acceleration) and the third variables should contain fluid property (mass density `rho`, weight density `w`, dynamic viscosity `mu`). The choice of repeating variables in most of the fluid mechanics problems may be,

1. `L,V,rho`

2. `D, V, rho`

3. `L, V, mu`

4. `D, V, mu`