Problem $2\mathrm{.}1,20$ points$)$

Using dimensional analysis, derive a dimensionless relationship that explains how the pressure drop $(P)$ in a system depends on the following variables:

Pipe diameter $($d$),$

Pipe length $(L),$

Fluid density $(),$

Fluid viscosity $(),$ and

Average fluid velocity $(v).$

Clearly outline the steps of the Buckingham $-$theorem to identify the relevant dimensionless groups.