The Buckingham Pi Theorem A frequent strategy in engineering analysis is to group

together different characteristic properties $d_i($diffusivity$,$ density, heat capacity, viscosity, etc$)$

mass$/$length$/$time$)$ into dimensionless groups that give us some physical insight into the prob$-$

lem at hand.

Formally we can define a new dimensionless group $$ as the product of $d_i$ quantities raised to

some power $a_i$ or

$=d_1^{a_1}{d}_2^{a_2}{d}_3^{a_3}$cdots

because all units must cancel we can cast the above into a matrix of form

$[d_{ij}]\left[\begin{array}{c}{a}_1\\ a_2\\ \text{v}\text{d}\text{o}\text{t}\text{s}\\ a_M\end{array}\right]=[0]$

What are the elements of the matrix $d?$ What do the rows of matrix $d_{ij}$ represent? For

$m$ units and $n$ characteristic properties, how many independent dimensionless groups can we

write? Under what conditions are the groups unique?