In the steady flow of a Newtonian fluid through a long uniform circular tube, if $N_{R e}<2000$, the flow is laminar and the fluid elements move in smooth straight parallel lines. Under these conditions, it is known that the relationship between the flow rate and the pressure drop in the pipe does not depend upon the fluid density or the pipe wall material.

(a) Perform a dimensional analysis of this system to determine the dimensionless groups that apply. Express your result in a form in which the Reynolds number can be identified.

(b) If water is flowing at a rate of $0.5 \mathrm{gpm}$ through a pipe with an ID of $1 \mathrm{in}$. , what is the value of the Reynolds number? If the diameter is doubled at the same flow rate, what will be the effect on the Reynolds number and on the pressure drop?