Discussion
A very small particle of diameter \(d_{p}\) and density \(\rho_{p}\) freely settles at constant velocity in a tank of depth L containing liquid of viscosity \(\mu_{l}\). The density of the liquid is \(\rho_{l}\) where \(\rho_{l} \lt \rho_{p}\). The velocity of particle in the liquid can be expressed as
Answer: C
The settling velocity is \(\frac{g(\rho_{p} - \rho_{l})d_{p}^{2}}{18\mu_{l}}\), Stokes' law.
A very small particle settles in the laminar (Stokes) regime. At terminal velocity the net weight balances the Stokes drag \(3\pi\mu_{l} d_{p} v\):
\[ \begin{aligned} \frac{\pi d_{p}^{3}}{6}(\rho_{p} - \rho_{l})g &= 3\pi\mu_{l} d_{p} v \\ v &= \frac{g(\rho_{p} - \rho_{l})d_{p}^{2}}{18\mu_{l}} \end{aligned} \]
The tank depth \(L\) plays no part in a terminal velocity, which rules out every option containing \(L\). Those options also fail dimensionally: only \(d_{p}^{2}\) gives a velocity with the other factors.
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