List of top Fluid Mechanics and Thermal Sciences Questions asked in GATE PI

Temperature field inside a sphere of radius \( R = 1 \, \text{m} \) with origin at its center is \[ T(x, y, z) = 100 - 70x + 51y - 80z - 10x^2 - 20y^2 - 20z^2. \] If thermal conductivity of the sphere material is \( K = 50 \, \text{W/m.K} \) and Fourier law of heat conduction is valid, net heat leaving the sphere per unit time in W is _________. 
[round off to one decimal place]

A fluid with dynamic viscosity \( \mu = 1 \, \text{Pa.s} \) is flowing through a circular pipe with diameter 1 cm. If the flow rate (discharge) in the pipe is 0.2 liters/s, the maximum velocity in m/s of the fluid in the pipe is (assume fully developed flow and take fluid density \( \rho = 1000 \, \text{kg/m}^3 \)) _________.
A 3 mm thick steel sheet, kept at room temperature of 30ºC, is cut by a fiber laser beam. The laser spot diameter on the top surface of the sheet is 0.2 mm. The laser absorptivity of the sheet is 50%. The properties of steel are density = 8000 kg/m\(^3\), specific heat = 500 J/kg.°C, melting temperature = 1530ºC, and latent heat of fusion = \( 3 \times 10^5 \) J/kg. Assume that melting efficiency is 100% and that the kerf width is equal to the laser spot diameter. The maximum speed (in m/s) at which the sheet can be fully cut at 2 kW laser power is _________.
A circular tank of 4 m diameter is filled up to a height of 3 m. Assuming almost steady flow and neglecting losses, the time taken in seconds to empty the tank through a 5 cm diameter hole located at the center of the tank bottom (take acceleration due to gravity \( g = 9.81 \, \text{m/s}^2 \)) is [round off to the nearest integer].
Heat is being removed from a refrigerator at a rate of 300 kJ/min to maintain its inside temperature at \( 2^\circ \text{C} \). If the input power to the refrigerator is 2 kW, the coefficient of performance of the refrigerator is _________.
\text{[round off to one decimal place]}
In an ideal Otto cycle, 800 kJ/kg is transferred to air during the constant volume heat addition process and 381 kJ/kg is removed during the constant volume heat rejection process. The thermal efficiency in % of the cycle is _________.
The dimensionless number defined by the ratio of inertial force to viscous force is called
A small capillary tube of 3 mm inner diameter is inserted into a fluid having density 900 kg/m\(^3\), surface tension 0.1 N/m, and contact angle 30°. The rise in the height of fluid in the capillary tube due to surface tension is