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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.