Rayleigh's Indicial Method

This alternative method is also based on the fundamental principle of dimensional homogeneity of physical variables involved in a problem.

Procedure-

1. The dependent variable is identified and expressed as a product of all the independent variables raised to an unknown integer exponent.
2. Equating the indices of n fundamental dimensions of the variables involved, n independent equations are obtained .
3. These n equations are solved to  obtain the dimensionless groups.

Example

Let us illustrate this method by solving the pipe flow problem

. Step 1 - ----- Here, the dependent variable Δp/l can be written as

(where, A is a dimensionless constant.)

           

Step 2 -----Inserting the dimensions of each variable in the above equation, we obtain,

Equating the indices of M, L, and T on both sides, we get ,

c + d = 1                             a + b - 3c - d = -2 -a - d = -2

Step 3 -----There are three equations and four unknowns. Solving these equations in terms of the unknown d, we have

a = 2- d b = -d - 1 c = 1- d

Hence , we can be written

or,

Therefore we see that there are two independent dimensionless terms of the problem, namely,

• Both Buckingham's method and Rayleigh's method of dimensional analysis determine only the relevant independent dimensionless parameters of a problem, but not the exact relationship between them.

 For example, the numerical values of A and d   can never be known from dimensional analysis. They are found out from experiments.

 If the system of equations is solved for the unknown c, it results,

Therefore different interdependent sets of dimensionless terms are obtained with the change of unknown indices in terms of which the set of indicial equations are solved. This is similar to the situations arising with different possible choices of repeating variables in Buckingham's Pi theorem.