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## Q. 12.1

Find the sedimentation velocities of (i) a pollen grain, diameter 10 µm, density 0.8 g c$m^{−3}$, and (ii) a hailstone, diameter 6 mm, density 0.5 g c$m^{−3}$. (Hint: You will have to estimate the Reynolds number and decide whether to use the trial-and error method described in the text).

## Verified Solution

i. If the pollen grain obeys Stokes’ Law, then $V_{s} = 2\rho gr^{2} / 9\rho_{a} v$. Hence $V_{s} = 2\times 0.8\times 10^{6} \times 9.81\times (5\times 10^{-6} )^{2} / 9\times 1.29\times 10^{3} \times 15\times 10^{-6} = 2.4 \; mm \; s^{-1}$.

To determine whether using Stokes’ Law is appropriate, calculate the particle Reynolds number $Re_{p}$.

$Re_{p} = 2.4\times 10^{-3} \times 10\times 10^{-6} /(15\times 10^{-6} ) = 1.6 \times 10^{-3}$.

So Stokes’ Law is valid.

ii. In this case, Stokes’ Law predicts that $V_{s} = 545 \; m \; s^{-1}$, but then $Re_{p}$ = 109 × 10³, so Stokes’ Law clearly is not adequate for calculating $V_{s}$. Using the iterative method described in the text, with a first guess for drag coefficient of 0.44 eventually yields $V_{s} = 13 \; m \; s^{-1}$. There is an interesting discussion of drag coefficients at