Buckingham Pi Theorem

The Buckingham Pi Theorem is the basic theory of dimensional analysis.  It states the following.  “If an equation involving k variables is dimensionally homogeneous, it can be reduced to a relation among k – r independent dimensionless products, where r is the minimum number of reference dimension required to describe the variables.” This will allow us to determine number of dimensionless products needed to replace the original list of variables.  In turn, these dimensional products are called pi terms which are represented by $Π$.

The Buckingham Pi Theorem creates dimensional homogeneity.  For example lets take a look at the following equation.

(Eq 1) $u_1=f(u_2,u_3,…,u_k)$

This equation represents any physical meaningful equation with k variables. When applying the Buckingham Pi Theorem the dimensions on the left side of equation must be equal to any stand alone term on the right side of the equation.  As a result, the equation 1 will have to rearranged into a set of dimensional products.

(Eq 2) $Π_1=Φ(Π_2,Π_3,…,Π_{k-r})$

In the above equation $Φ(Π_2,Π_3,…,Π_{k-r})$ is a function of $Π_2$ through $Π_{k-r}$.

When you are applying the Buckingham Pi Theorem the number of pi terms will always be fewer than the original variables by $r$.  In turn, $r$ found by determining the number of reference dimensions needed to to describe the original list of variables.  This is done by using basic dimensions.  For example mass $M$, length $L$, and time $T$ are basic dimensions.  In addition, you can also use Force $F$, Length $L$, and time $T$ for the reference dimensions since $F=MLT^{-2}$.  Depending on the variable of interest you may need to use all three basic dimensions, but there are also cases when you may only need to use two.  If this is the case than $r$ will equal two instead of three.

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