 Solution

Let $W_f$ be the vertical velocity of the carrier fluid. Each species has its isolated Stokes settling velocity relative to that fluid, so $W_i=W_f+{\hat{W}}_i$. The mixture has no imposed vertical volumetric flow rate. Using the particle volume fractions, its zero-volume-flux condition isSubstitution of $W_i=W_f+{\hat{W}}_i$ gives the common carrier-fluid backflowConsequently the hindered settling velocities areThe same $W_f$ appears for both populations precisely because every particle is assumed to feel the same volume-averaged backflow.