Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 355 / 3

Let the unit normal $\mathbf{n}$ point from the fluid toward the free surface, let $\mathbf{t}$ lie along the surface, and write the swimmer direction asReflecting the swimmer position across the plane and reflecting its orientation toconstructs the free-surface image of a force dipole. At the surface the two dipoles have equal tangential velocity and opposite normal velocity. Their sum therefore satisfies the no-penetration boundary condition; tangential velocity is even across the plane and normal velocity is odd, so the tangential traction vanishes and the stress-free boundary condition is also satisfied.

The vector from the image to the swimmer is $\mathbf{r}=-2h\mathbf{n}$, for which $({\mathbf{p}}^{\ast }\cdot \mathbf{r}{)}^2=4h^2{\sin }^2\theta$. Evaluating the image force-dipole flow at the swimmer givesThere is no tangential image velocity at the swimmer in this point-dipole approximation. With the convention that $\mathcal{P}>0$ is an extensile pusher microswimmer, a nearly parallel pusher is attracted toward the surface, whereas a nearly parallel contractile puller microswimmer is repelled. For $|\sin \theta |>1/\sqrt{3}$ the normal drift reverses because the image samples the axial rather than equatorial part of the dipolar flow.

For a fieldonly the gradient of the scalar prefactor contributes to its vorticity. At the swimmer,The prescribed rotation law has ${\mathbf{\Omega }}_f={\mathbf{\omega }}^{\ast }/2$. Since $\mathbf{b}\times \mathbf{p}=-\partial \mathbf{p}/\partial \theta$, comparison with $\dot{\mathbf{p}}=(\partial \mathbf{p}/\partial \theta )\dot{\theta }$ givesA pusher has $\mathcal{P}>0$ and rotates toward the stable parallel orientation $\theta =0$; there its image flow attracts it to the free surface. A puller has $\mathcal{P}<0$ and rotates toward a normal orientation $|\theta |=\pi /2$. Depending on which way its polar swimming direction points, it then swims away from the surface or meets it head-on; the dipolar image drift at the exactly normal orientation is toward the surface for this puller sign convention.