= Solution
With zero fluid <velocity> and all mean orientations vertical, the steady <cell conservation equation> is
$$
\frac{d}{d\hat z}\left(V_s\hat n-D\frac{d\hat n}{d\hat z}\right)=0.
$$
The plates are impermeable to cells, so the constant vertical flux is zero. Hence $D\hat n_{\hat z}=V_s\hat n$, giving $\hat n=C\exp(V_s\hat z/D)$. Determine $C$ by the prescribed depth average:
$$
\frac1H\int_{-H}^0\hat n\,d\hat z=\hat n_0.
$$
With $z=\hat z/H$ and $h=V_sH/D$, the result is
$$
\boxed{\hat n=\hat n_0\bar n(z),\qquad\bar n(z)=\frac{he^{hz}}{1-e^{-h}}.}
$$
The same normalized profile describes cell number density or cell volume fraction, because cell volume is constant. Its weak-stratification expansion is
$$
\bar n=1+h\left(z+\frac12\right)+h^2\left(\frac{z^2}2+\frac z2+\frac1{12}\right)+O(h^3),
$$
which is depth-normalized at each displayed order and will be used in part (c). Upward swimming makes concentration larger near the upper plate, creating the density stratification responsible for <bioconvection> when cells are heavier than the fluid.
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