A circular duct carrying water gradually contracts from a diameter of $30 \: cm$ to $15 \: cm$. The figure (not drawn to scale) shows the arrangement to differential manometer attached to the duct.
When the water flows, the differential manometer shows a deflection of $8 \: cm$ of mercury (Hg). The values of specific gravity of mercury and water are $13.6$ and $1.0$, respectively. Consider the acceleration due to gravity, $g=9.81 \: m/s^2$. Assuming frictionless flow, the flow rate (in $m^3/s$, round off to $3$ decimal places) through the duct is ________