A Venturi flow meter is a pipe with a smooth constriction

that is used to measure flow rate as a function of the

pressure difference between the gauges installed at the

wide and narrow sections of the meter $($as shown in the

figure$).$ Derive the formula for the flow rate if you know the

pressure measurements $P_1$ and $P_2,$ the diameters of the

pipe $D_1$ and the throat of the constricted section $D_2$ and the

density of the fluid.

This is not a dimensional analysis problem.