Buckingham's Pi Theorem

Assume, a physical phenomenon is described by m number of independent variables like x1 , x2 , x3 , ..., xm The phenomenon may be expressed analytically by an implicit functional relationship of the controlling variables as (19.2)           Now if n be the number of fundamental dimensions like mass, length, time, temperature etc ., involved in these m variables, then according to Buckingham's p theorem - The phenomenon can be described in terms of (m - n) independent dimensionless groups like π1 ,π2 , ..., πm-n , where p terms, represent the dimensionless parameters and consist of different combinations of a number of dimensional variables out of the m independent variables defining the problem. Therefore. the analytical version of the phenomenon given by Eq. (19.2) can be reduced to (19.3)   according to Buckingham's pi theorem This physically implies that the phenomenon which is basically described by m independent dimensional variables, is ultimately controlled by (m-n) independent dimensionless parameters known as π terms.