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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 $u\left ( r \right ) = V \dfrac{\text{ln}\left ( r/R \right )}{\text{ln}\left ( a \right )}$, where $r$ is the radial coordinate. Ignoring end effects, the magnitude of the pulling force per unit rod length is

  1. $\pi \mu V$
  2. $-\frac{2\pi \mu V} {\text{ln}(a)}$
  3. $0$
  4. $-\frac{\pi \mu V} {\text{ln}(a)}$
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