Gyrotactic focusing in a downward pipe flow ($n_0(r)=\frac{a}{1-e^{-a}}{e}^{-a{r}^2},a=\lambda \chi$)

For a bottom-heavy spherical swimmer in axial flow $u_z=W_0(-1+r^2+w)$, the bottom-heavy spherical-cell orientation dynamics with aligning rate $1/B$ gives $\sin \theta =-\lambda (2r+w^{\prime })$, $\lambda =B{W}_0/(2R)$. On the upward-stable branch $\cos \theta >0$, the cells tilt inward in a downward Poiseuille flow. Zero radial cell flux with scalar diffusivity gives $n^{\prime }=\chi n\sin \theta$, $\chi =V_{c}R/D$. For $w=0$, integration and the average normalization $2{\int }_0^{1}nrdr=1$ give the displayed concentration. A steady deterministic orientation requires $|\lambda (2r+w^{\prime })|<1$; beyond this shear threshold the stationary orientation closure fails.

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.