First slide

Fluid dynamics

Question

A horizortal-oriented tube AB of length l :2.5 m rotates with a constant angular velocity $ω$ = 5 rad/s about a stationary vertical axis OO' passing through the end A
(figure). Initially the tube is filled with an ideal fluid. The end A of the tube is open, the closed end B has a very small orifice. Find the velocity of the fluid (in m/s) relative to the tube when the liquid column length in tube reduces to h=1 m.

Difficult

Solution

Using Bernoulli's theorem, we have
$P_{B} = P_{atm} + \frac{1}{2} ρ v^{2} . . . . . (i)$
$and P_{B} - P_{atm} = \int_{l - h}^{l} d r \cdot ρ \cdot ω^{2} x = \frac{ρ ω^{2}}{2} [l^{2} - (l - h)^{2}]$
$P_{B} - P_{atm} = \frac{1}{2} ρ ω^{2} (2 l - h^{2}) . . . . . (i i)$
$From Eqs. (i) and (ii), \frac{1}{2} ρ ω^{2} (2 l - h^{2}) = \frac{1}{2} ρ v^{2}$
$\Rightarrow v = ω h \sqrt{(\frac{2 l}{h} - 1)} = 5 \times 2.5 \sqrt{(\frac{2 \times 2.5}{1} - 1)} = 25 m / s$

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