Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 74 / 3 / b / Solution

Let the particle and carrier-fluid mass densities be ${\rho }_p$ and ${\rho }_f$, with identical density contrast for both species. At small particle Reynolds number, balance Stokes drag law against submerged weight:ConsequentlyAt large particle Reynolds number, use approximately constant drag coefficient $C_D$ and quadratic drag. Balancing $\frac{1}{2}{C}_D{\rho }_f\pi r^2{W}_s^2$ against the same submerged weight gives $W_s^2=8({\rho }_p-{\rho }_f)gr/(3C_D{\rho }_f)$, soBoth particles must lie in the constant-$C_D$ regime for this simple square-root ratio. More generally it contains an additional factor $\sqrt{C_D(r_2)/C_D(r_1)}$. The speeds are positive downward magnitudes; the signs of the shock velocities below depend on the chosen spatial coordinate.