First slide

Fluid dynamics

Question

A cylindrical container of cross sectional area A is filled with water. There is a small orifice at the bottom of the container and cross sectional area of the orifice is ‘a’ = nA (n < 1). When depth of water in the container is 0.8 m, the velocity of water emerging from the orifice is (Take $g = 10 m / s^{2}$ )

Moderate

Solution

By equation of continuity,
$\begin{array}{l} Q = A u = a v \Rightarrow u = \frac{a}{A} v \\ ∴ \Rightarrow u = n v \end{array}$
Applying Bernoulli's equation between A and B,
$p_{a} + \frac{1}{2} ρ u^{2} + ρ g h = p_{a} + \frac{1}{2} ρ v^{2}$
$\Rightarrow v^{2} - u^{2} = 2 g h \Rightarrow v^{2} (1 - n^{2}) = 2 g h \Rightarrow v = \sqrt{\frac{2 g h}{(1 - n^{2})}}$
$∴ v = \frac{\sqrt{2 \times 10 \times 0.8}}{\sqrt{1 - n^{2}}} = \frac{4}{\sqrt{1 - n^{2}}} m / s$

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