GATE 2021 CH – Question 51

Fluid Mechanics and Mechanical Operations · Fluid properties, fluid statics and fluid kinematics · 2 marks · Numerical answer

Consider a tank filled with 3 immiscible liquids A, B and C at static equilibrium, as shown in the figure. At 2 cm below the liquid A–liquid B interface, a tube is connected from the side of the tank. Both the tank and the tube are open to the atmosphere. At the operating temperature and pressure, the specific gravities of liquids A, B and C are 1, 2 and 4, respectively. Neglect any surface tension effects in the calculations. The length of the tube L that is wetted by liquid B is _____ cm.

A is 4 cm deep, B 4 cm deep and C 2 cm deep. The connection is halfway down B. The tube slopes upwards at 60° to the vertical.

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Correct answer: 8

Explanation

At the tube connection, the gauge pressure is the hydrostatic contribution of 4 cm of A and 2 cm of B:
$$p-p_{atm}=\rho_Ag(4\ \mathrm{cm})+\rho_Bg(2\ \mathrm{cm}).$$

In the tube, liquid B rises to a vertical height h above that connection, giving $\rho_Bgh=p-p_{atm}$. With $\rho_A/\rho_B=1/2$, $h=(1/2)(4)+2=4$ cm.

The tube makes 60° with the **vertical**, so $h=L\cos60^\circ=L/2$. Hence **L = 8 cm**. The lower C layer does not contribute to pressure above its interface.