= Solution
For a field
$$
\mathbf u(\mathbf r)=\frac{\mathcal P}{8\pi\mu}
\left[\frac{3(\mathbf p^*\mathbin\cdot\mathbf r)^2}{r^5}
-\frac1{r^3}\right]\mathbf r,
$$
only the gradient of the scalar prefactor contributes to its <vorticity>. At the swimmer,
$$
\boldsymbol\omega^*
=\frac{3\mathcal P}{32\pi\mu h^3}
\sin\theta\cos\theta\,\mathbf b,
\qquad
\mathbf b=\mathbf n\mathbin\times\mathbf t.
$$
The prescribed rotation law has $\boldsymbol\Omega_f=\boldsymbol\omega^*/2$. Since $\mathbf b\times\mathbf p=-\partial\mathbf p/\partial\theta$, comparison with $\dot{\mathbf p}=(\partial\mathbf p/\partial\theta)\dot\theta$ gives
$$
\boxed{
\dot\theta=-\frac{3\mathcal P}{64\pi\mu h^3}
\sin\theta\cos\theta
=-\frac{3\mathcal P}{128\pi\mu h^3}\sin2\theta
}.
$$
A 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.
Back to article page
