A reverse$-$flow cyclone device $($as shown in the figure$)$ is commonly used to clean particle$-$laden air. Dusty air with density $$ and volume flow rate $V^{}$ enters tangentially through an opening in the side and swirls in the cyclone. Dust particles are flung outward and fall out the bottom, while clean air is drawn out the top. The reverse$-$flow cyclones being studied are all geometrically similar; hence, diameter $D$ represents the only length scale required to fully specify the entire cyclone geometry. Engineers are concerned about the pressure drop $p$ through the cyclone.

a$.$ Generate a dimensionless relationship between the pressure drop$p$ and the given parameters.

b$.$ If the cyclone size is doubled, all else being equal, by what factor will the pressure drop change.

c$.$ If the volume flow rate is doubled, all else being equal, by what factor will the pressure drop change?