\[\Pi_1 = f(\Pi_2, \Pi_3, …, \Pi_n)\]

This equation is know as the theory of models. It says that each  dimensionless group (\(\Pi\)-term) is a function of the other dimensionless groups. For example, the coefficient of drag of a golf ball is known to be a function of the Reynolds number and the relative roughness of the ball: 

\[C_D = f\left(Re,\frac{\epsilon}{D}\right)\]In this case there are three \(Pi\)-terms: the coefficient of drag, \(C_D\), the Reynolds number, \(Re\), and the relative roughness, \(\frac{\epsilon}{D}\). If we specify the Reynolds number and the relative roughness, then the coefficient of drag is uniquely defined.

The subscripts \(m\) and \(p\) represent “model” and “prototype”. For example, to maintain similitude, it is often important to keep the Reynolds number the same between the model and the prototype:

\[Re_m = Re_p\] This means

\[\frac{\rho_m\,V_m\,L_m}{\mu_m}\,=\,\frac{\rho_p\,V_p\,L_p}{\mu_p}\]

Suppose the model is half as long as the prototype:

\[L_m\,=\,\frac{1}{2}L_p\]

To maintain similitude, the speed of the fluid over the model must be twice as fast:

\[V_m\,=\,2V_p\]