Rotlet dipole (${\mathbf{u}}_{RD}=-3D(\mathbf{e}\cdot \mathbf{r})(\mathbf{e}\times \mathbf{r})/(8\pi \mu r^5)$)

A rotlet dipole is the derivative of a rotlet along the separation of two opposed torques. A torque-free bacterium can have this far field because its rotating flagella and counterrotating body exert spatially separated internal torque reactions. The displayed axial formula uses the observation-coordinate derivative convention and signed torque moment $D$.

At a flat shear-free impermeable plane, a tangential torque is reflected with opposite sign because torque is an axial vector. A parallel axial rotlet dipole therefore has an opposite signed moment at the reflected source. Its velocity vanishes at the point directly beneath the image, but its velocity gradient can turn the swimmer.

For an axial rotlet dipole parallel to a flat free surface, its reflected image gives the displayed normal vorticity at height $h$. A spherical torque-free body responds with angular velocity ${\omega }_z/2$ and an elongated body also responds to strain. At constant height, self-propulsion plus yaw gives a circular path; the signed moment sets its handedness.