= Gyrotactic focusing in a downward pipe flow
{title2=$n_0(r)=\frac{a}{1-e^{-a}}e^{-ar^2},\quad a=\lambda\chi$}
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 $\sin\theta=-\lambda(2r+w')$, $\lambda=BW_0/(2R)$. 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 $n'=\chi n\sin\theta$, $\chi=V_cR/D$. For $w=0$, integration and the average normalization $2\int_0^1nr\,dr=1$ give the displayed concentration. A steady deterministic orientation requires $|\lambda(2r+w')|<1$; beyond this shear threshold the stationary orientation closure fails.
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