= Zero-flux pressure gradient in a cell-driven channel
{title2=$G=3g'n_0[1-b\coth b]/b^2$}
In the <Boussinesq approximation>, downward excess cell weight gives $\mu w''=G+g'n(x)$. 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, $w=C[e^{\beta x}-\cosh b-(x/a)\sinh b]+G(x^2-a^2)/(2\mu)$, $C=g'n_0a\beta/(\mu\beta^2\sinh b)$. For large $b$, writing $\xi=x/a$ and $W=g'n_0a^2/(\mu b)$ 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 $G=-g'n_0$, $w=0$.
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