Let $W_f$ be the vertical velocity of the carrier fluid. Each species has its isolated <Stokes settling velocity> relative to that fluid, so $W_i=W_f+\widehat W_i$. The mixture has no imposed vertical <volumetric flow rate>. Using the <particle volume fractions>, its zero-volume-flux condition is
$$
(1-\phi_1-\phi_2)W_f+\phi_1W_1+\phi_2W_2=0.
$$
Substitution of $W_i=W_f+\widehat W_i$ gives the common carrier-fluid backflow
$$
W_f=-\phi_1\widehat W_1-\phi_2\widehat W_2.
$$
Consequently the <hindered settling> velocities are
$$
W_1=(1-\phi_1)\widehat W_1-\phi_2\widehat W_2,
\qquad
W_2=(1-\phi_2)\widehat W_2-\phi_1\widehat W_1.
$$
The same $W_f$ appears for both populations precisely because every particle is assumed to feel the same volume-averaged backflow.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 1 / a / Solution

Hindered settling is the reduction or modification of particle settling by the displaced carrier-fluid backflow and by interactions with other particles. A simple dilute mixture model gives every species the same volume-averaged backflow.


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