An oil drum with radius R=0.5m initially contains oil to a height H0=2.0m. A small circular hole of radius r=2.0mm is punched near the bottom of the drum at height h0=10cm above the ground. The oil has density ρ=850kg/m3 and dynamic viscosity η=0.15Pa⋅s. The hole creates a short cylindrical channel through the drum wall with length L=5.0mm (equal to the wall thickness). Initially, the oil level is much higher than the hole, so oil flows out under pressure. However, as the oil level drops and approaches the hole height, the flow becomes more complex.
Find the height h of oil remaining in the drum at the moment when the flow transitions from being dominated by the hydrostatic pressure difference to being dominated by viscous effects in the channel. This transition occurs when the Reynolds number of the flow through the hole equals Re=100.