Classical Approach to Dimensional Analysis

The pi theorem is a generalized method of dimensional analysis and detailed discussions can be found in [1-7]. Below is a brief review of the pi theorem. 

Consider the magnitude/j of some physical quantity depending on other, independent magnitudes g, g2, , gm then 

Equation 8.1 is required to be dimensionally homogeneous. The pi theorem says that if the number of distinct reference quantities required to express the dimensional formula of all n magnitudes is r, then the n magnitudes may be grouped into n — r independent dimensionless IT terms, resulting in the relation 

An immediate advantage is demonstrated if Equation 8.1 is compared with Equation 8.2. That is, the number of independent variables to be studied is reduced to n — r. 

The next step is to define the dimensional matrix whose elements are the exponents of the fundamental dimensions M, L, and t appearing in Table 8.3. 

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