Orientation averaging of an axisymmetric stresslet ($⟨S(\mathbf{ee}-I/3)⟩=-\frac{S}{2}(\mathbf{nn}-I/3)$)

The angular factor of an axial stresslet is linear in the symmetric traceless rank-two tensor $\mathbf{ee}-I/3$. Hence averaging a collection of directors amounts to averaging this tensor. For uniformly distributed in-plane directions, $⟨\mathbf{ee}⟩=(I-\mathbf{nn})/2$, so $⟨S(\mathbf{ee}-I/3)⟩=-(S/2)(\mathbf{nn}-I/3)$. The averaged director is normal to the plane and its signed strength is reversed and halved.

A stresslet whose centre follows a circle and whose director stays tangent has a leading far-field mean stresslet centred at the circle centre. Orientation averaging of an axisymmetric stresslet gives normal-axis strength $-S/2$. Thus a pusher microswimmer becomes a leading mean puller microswimmer, and conversely. The radius and traversal frequency enter higher-order spatial corrections or averaging time, but not the leading dipole strength.

Opposite points on the orbit have opposite displacement and the same stresslet director up to sign. Their paired fields are $[G(\mathbf{r}-R\mathbf{p},\mathbf{e})+G(\mathbf{r}+R\mathbf{p},\mathbf{e})]/2$, so odd displacement powers cancel. Since a stresslet decays as $r^{-2}$, the first correction to its far-field orbit average of a tangent stresslet is $O(S{R}^2/(\mu r^4))$. The finite-radius mean need not be an exact point stresslet.