Question

A tank with a small orifice contains oil on top of water. It is immersed in a large tank of the same oil. Water flows through the hole. Assuming the level of oil outside the tank above orifice does not change. a. What is the velocity of this flow initially? b. When the flow stops, what would be the position of the oil-water interface in the tank? c. Determine the time at which the flow stops? Density of oil $=800kg/m^3$.

Answer 1

$\left(a\right)$ Applying Bernouli's equation outside and just inside the orifice.

$P_0+{\rho }_{oil}g\left(15\right)+\frac{1}{2}{\rho }_w{V}^2=P_0+{\rho }_{w}g\left(10\right)+{\rho }_{oil}g\left(5\right)\therefore V=\sqrt{40}=6.3m/s$

$\left(b\right)$ Let h be the light when flow through orifice stops.

$\therefore P_0+{\rho }_{oil}g\left(5\right)+{\rho }_{w}gh=P_0+{\rho }_{oil}g\left(15\right)\Rightarrow h=\frac{{\rho }_{oil}}{{\rho }_w}10=8m\left(c\right)P_0+{\rho }_{oil}g\left(5\right)+{\rho }_{w}gh=P_0+{\rho }_{oil}g\left(15\right)+\frac{1}{2}{\rho }_w{V}^2\Rightarrow V=\sqrt{20(h-8)}\Rightarrow -\frac{dh}{dt}=\sqrt{20(h-8)}$

$\Rightarrow -{\int }_{10}^8\frac{dh}{\sqrt{20(h-8)}}={\int }_0^{t}dt$

$\therefore t=\sqrt{\frac{2}{5}}s$