 Stresslet correction to gyrotactic pipe flow

Let weak excess-cell buoyancy be $\gamma \ll 1$ and the swimming stresslet coefficient be $\sigma =\gamma {\sigma }_1$. Set $w=\gamma w_1$ and use the leading gyrotactic focusing in a downward pipe flow density $n_0=A{e}^{-a{r}^2}$. With active stress entering as $+Sn\mathbf{pp}$, the axial momentum equation integrates from the regular axis to the displayed derivative. The wall condition gives $w_1(r)=-{\int }_r^1{w}_1^{\prime }(s)ds$. Both terms of the integrand are positive for ${\sigma }_1\ge 0$, so the correction increases downward flow. A negative stresslet strength can oppose this effect; reversal at a given radius requires ${\sigma }_1$ smaller than minus the ratio of the integrated buoyancy and stresslet coefficients. This convention must be specified before attaching extensile or contractile sign labels.