First slide

Fluid dynamics

Question

A tank is filled with water of density 103 kg/ $m^{3}$ and oil of density 9 x 103 kg/ $m^{3}$ . The height of water layer is 1 m and that of the oil layer is 4 m. The velocity of efflux from an opening in the bottom of the tank is

Moderate

A

$\sqrt{85} m / s$
B

$\sqrt{88} m / s$
C

$\sqrt{92} m / s$
D

$\sqrt{98} m / s$

Solution

Let dw and $d_{0}$ be the densities of water and oil, respectively.
Then the pressure at the bottom of the tank is $h_{w} d_{w}$ g + $h_{0} d_{0}$ g. Let this pressure be equivalent to pressure due to water of height h. Then,
h $d_{w}$ g = $h_{w} d_{w} g + h_{0} d_{0} g$
$h = h_{w} + \frac{h_{0} d_{0}}{d_{w}} = 100 + \frac{400 \times 0.9}{1}$ = 100+ 360 = 460 = 4.6 m
According to Toricelli's theorem,
$v = \sqrt{2 g h} = \sqrt{2 \times 10 \times 4.6} m / s = \sqrt{92} m / s$

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