First slide

Fluid dynamics

Question

A large open tank has two holes in the wall. One is a square hole of side L at a depth y from the top and the other is a circular hole of radius R at a depth 4y from the top. When the tank is completely filled with water the quantities of water flowing out per second from both the holes are the same. Then R is equal to

Moderate

A

$2 π L$
B

$\frac{L}{\sqrt{2 π}}$
C

$L$
D

$\frac{L}{2 π}$

Solution

Velocity of efflux when the hole is at depth h, $v = \sqrt{2 gh}$
Rate of flow of water from square hole
$Q_{1} = a_{1} v_{1}$ = $L^{2} \sqrt{2 gy}$
Rate of flow of water from circular hole
$Q_{2} = a_{2} v_{2}$ = ${πR}^{2} \sqrt{2 g (4 y)}$
According to problem $Q_{1} = Q_{2}$
$\Rightarrow L^{2} \sqrt{2 gy} = {πR}^{2} \sqrt{2 g (4 y)} \Rightarrow R = \frac{L}{\sqrt{2 π}}$

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