Answer
$\dot{m} \propto \rho V D^{2}$
See proof in the step by step solution.
Work Step by Step
The question requires us to show the proportionality on the basis of units:
The mass flow rate is a measure of an amount of fluid flowing through an area in a period of time.
If two fluids at the same volumetric flow rates flow through two distinct areas, we can see that the one with greater density has a higher mass flow rate, due to the fluid carrying more mass per unit of volume:
$\dot{m} (kg/s) \propto \rho (kg/m³) $
If two fluids of same density flow through two distinct areas of same magnitude, we can expect the one with higher velocity to have a higher mass flowrate due to the higher amount of fluid flowing per unit of time.
$\dot{m} (kg/s) \propto V(m/s) $
If two fluids of same density and velocity flow through two distinct areas of different magnitudes, we should predict that the one flowing through the bigger area will have a higher flowrate due to carrying a bigger volume of fluid in the same period of time, and since in this case the cross section of the flow has a circular profile:
$\dot{m} (kg/s) \propto D(m)^{2} $
Putting it together:
$\dot{m}(kg/s) \propto \rho(kg/m³) V (m/s) D(m)^{2}$
