Phototactic concentration in a vertical channel ($n(x)=n_{0}a\beta e^{\beta x}/\sinh (a\beta )$)

In a long channel $-a<x<a$, uniform horizontal mean swimming $V_s{K}_0{\mathbf{e}}_x$ and isotropic translational diffusion coefficient $D$ give zero transverse cell flux when $n^{\prime }=\beta n$, $\beta =V_s{K}_0/D$. Normalizing the transverse mean concentration to $n_0$ gives the displayed exponential. For $\beta a\gg 1$, cells form a layer of thickness ${\beta }^{-1}$ at the illuminated wall. This is a bulk approximation; exact closed-end cell conservation equation also requires the integrated axial cell flux to vanish.

In the Boussinesq approximation, downward excess cell weight gives $\mu w^{\prime \prime }=G+g^{\prime }n(x)$. The no-slip boundary condition at both walls and zero total fluid flux determine the excess axial pressure gradient $G$, with $b=a\beta$. In particular, $w=C[e^{\beta x}-\cosh b-(x/a)\sinh b]+G(x^2-a^2)/(2\mu )$, $C=g^{\prime }{n}_{0}a\beta /(\mu {\beta }^2\sinh b)$. For large $b$, writing $\xi =x/a$ and $W=g^{\prime }{n}_0{a}^2/(\mu b)$ gives the outer profile $w/W=(1+\xi )(1-3\xi )/2$. A thin layer restores no slip at the illuminated wall. Cells descend there and fluid returns upward on the other side. At $b=0$ the limiting solution is $G=-g^{\prime }{n}_0$, $w=0$.