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GATE 2022 CH – Question 39

Fluid Mechanics and Mechanical Operations · Fluid properties, fluid statics and fluid kinematics · 2 marks · Multiple choice

Consider a horizontal rod of radius $aR$ ($a < 1$) in a stationary pipe of radius $R$. The rod is pulled coaxially at a constant velocity $V$ as shown in the figure. The annular region is filled with a Newtonian incompressible fluid of viscosity $\mu$. The steady state fully developed axial velocity profile in the fluid is given by $u(r) = V\frac{\ln(r/R)}{\ln(a)}$, where $r$ is the radial coordinate. Ignoring end effects, the magnitude of the pulling force per unit rod length is

a rod inside a pipe, pulled to the right at the velocity $V$, with the radial coordinate $r$ marked.
  1. $\pi\mu V$
  2. $-\frac{2\pi\mu V}{\ln(a)}$
  3. 0
  4. $-\frac{\pi\mu V}{\ln(a)}$

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Correct answer: (B) $-\frac{2\pi\mu V}{\ln(a)}$

Explanation

The shear stress on the rod surface ($r = aR$) is $\tau = \mu\frac{du}{dr} = \frac{\mu V}{r\ln a} = \frac{\mu V}{aR\ln a}$. The force per unit length is the stress times the perimeter $2\pi aR$: $\frac{2\pi\mu V}{\ln a}$, with the magnitude $-\frac{2\pi\mu V}{\ln a}$ (since $\ln a < 0$).