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For a flat impermeable free surface with zero tangential stress, the image of a parallel stresslet is a stresslet of the same strength and parallel orientation at the reflected point. Reflection makes the normal velocity odd and tangential velocity even across the plane, enforcing zero normal flow and zero tangential shear. This is the free-surface image of a force dipole; it is not the no-slip-wall image system.

The cell at $z=-h$ lies a distance $2h$ vertically below its image. Since $\mathbf{e}\cdot \mathbf{r}=0$ there, its induced vertical velocity is $S/(32\pi \mu h^2)$ towards the surface. There is no image-induced tangential velocity, vorticity or tangent-normal strain at this point, so a parallel orientation remains parallel in this idealization. Therefore the finite-time free-surface approach of a point stresslet isThe trajectory combines its parallel self-propulsion with this normal drift. Finite-time contact is the point-model prediction for $S>0$, carried over from part (b). A puller with $S<0$ moves away instead. For a finite cell, the far-field approximation fails before $h$ reaches zero; surface deformation and near-contact physics can change the final encounter.

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.