Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 355 / 1 / Solution

For each Volvox colony, the excess gravitational force isFor a normal Stokeslet a distance $R$ below a flat stress-free interface, the image is an oppositely directed Stokeslet a distance $R$ above it. The real force creates no lateral velocity at the other colony because both centres have the same height. Their separation from the image is $(x,-2R)$, however, so each colony moves toward the other with speedBoth colonies move, and therefore

DefineThe dimensionless dynamics areThis gradient flow descends an even attractive potential with minimum $V(0)=-1$ and $V\to 0^{-}$ as $|\xi |\to \infty$.

For $\xi \gg 1$, $d\xi /d\tau \simeq -1/{\xi }^2$, so the approach from ${\xi }_0=x_0/(2R)$ takes ${\tau }_c\simeq {\xi }_0^3/3$. Restoring dimensions,With water viscosity $\mu \simeq 10^{-3}\mathrm{Pa}\mathrm{s}$, ${\rho }_w\simeq 10^3\mathrm{kg}{\mathrm{m}}^{-3}$, $ϵ=0.03$, $R=200\mu \mathrm{m}$, and $x_0=5R$, this givesBecause $x_0/(2R)=2.5$ is only moderately large, this is a far-field estimate rather than the exact collision time.