Chapter 6 : Principles of Physical Similarity and Dimensional Analysis
Lecture 19 :


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 π12 , ..., 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.