Determination of π terms

• A group of n (n = number of fundamental dimensions) variables out of m (m = total number of independent variables defining the problem) variables is first chosen to form a basis so that all n dimensions are represented . These n variables are referred to as repeating variables.

• Then the p terms are formed by the product of these repeating variables raised to arbitrary unknown integer exponents and anyone of the excluded (m -n) variables.

For example , if x₁ x₂ ...x_n are taken as the repeating variables. Then

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• The sets of integer exponents a₁, a₂ . . . a_n are different for each p term.
• Since p terms are dimensionless, it requires that when  all the variables in any p term are expressed in terms of their fundamental dimensions, the exponent of all the fundamental dimensions must be zero.
•  This leads to a system of n linear equations in a, a₂ . . . a_n which gives a unique solution for the exponents. This gives  the values of a₁ a₂ . . . a_n for each p term  and hence the p terms are uniquely defined.

In selecting the repeating variables, the following points have to be considered:

1. The repeating variables must include among them all the n fundamental dimensions, not necessarily in each one but collectively.
2. The dependent variable or the output parameter of the physical phenomenon should not be included in the repeating variables.

No physical phenomena is represented when -

• m < n    because there is no solution   and

•  m = n   because there is a unique solution of the variables involved and hence all the parameters have fixed values.

. Therefore all feasible phenomena are defined with m > n .

• When m = n + 1, then, according to the Pi theorem, the number of pi term is one and the phenomenon can be expressed as

where, the non-dimensional term π₁ is some specific combination of n + 1 variables involved in the problem.

When m > n+ 1 ,

1.  the number of π terms are more than one.
2.  A number of choices regarding the repeating variables arise in this case.

Again, it is true that if one of the repeating variables is changed, it results in a different set of π terms. Therefore the interesting question is which set of repeating variables is to be chosen , to arrive at the correct set of π terms to describe the problem. The answer to this question lies in the fact that different sets of π terms resulting from the use of different sets of repeating variables are not independent. Thus, anyone of such interdependent sets is meaningful in describing the same physical phenomenon.

From any set of such π terms, one can obtain the other meaningful sets from some combination of the π terms of the existing set without altering their total numbers (m-n) as fixed by the Pi theorem.

See the Example