Creating the list of variables and parameters to include in a dimensional analysis effort is the most important step. The parameter list should usually contain only one unknown variable, the solution variable
The rest of the variables and parameters should come from the problem’s geometry, boundary conditions, initial conditions, and material parameters. Physical constants and other fundamental limits may also be included
However, shorter parameter lists tend to produce the most powerful dimensional analysis results; expansive lists commonly produce less useful results
For the round-pipe pressure drop example, select \(\Delta p\) as the solution variable, and then choose as additional parameters: the distance \(\Delta x\) between the pressure measurement locations, the inside diameter \(d\) of the pipe, the average height \(\varepsilon\) of the pipe’s wall roughness, the average flow velocity \(U\), the fluid density \(\rho\), and the fluid viscosity \(\mu\)
The resulting functional dependence between these seven parameters can be stated as
\[f(\Delta p, \Delta x, d, \varepsilon, U, \rho, \mu) = 0\]
This parameter listing embodies the assumptions of isothermal and constant density flow