GATE 2022 AE – Question 52
Consider a solid cylinder housed inside another cylinder as shown in the figure. Radius of the inner cylinder is 1 m and its height is 2 m. The gap between the cylinders is 5 mm and is filled with a fluid of viscosity 10-4 Pa-s. The inner cylinder is rotating at a constant angular speed of 5 rad/s while the outer cylinder is stationary. Friction at the bottom surfaces can be ignored. Velocity profile in the vertical gap between the cylinders can be assumed to be linear. The driving moment required for the rotating motion of the inner cylinder is ____________ Nm (rounded off to two decimal places).
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 1.25 to 1.27
Explanation
**Couette flow in the gap.** The gap is thin ($5$ mm compared with the radius of 1 m), and the velocity profile is linear. The inner cylinder surface moves at $v=\omega r$ while the outer wall is stationary.
**Shear stress on the wall:**
$$\tau=\mu\frac{\omega r}{g}=10^{-4}\times\frac{5\times1}{0.005}=0.1\text{ Pa}.$$
**Torque.** The shear force acts on the lateral area $2\pi r h$ at radius $r$:
$$T=\tau\,(2\pi r h)\,r=0.1\times2\pi\times1\times2\times1=0.4\pi=1.2566\text{ N m}.$$
The driving moment is **1.26 N m**.