Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 81 / 4 / a / Solution

With zero fluid velocity and all mean orientations vertical, the steady cell conservation equation isThe 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:With $z=\hat{z}/H$ and $h=V_{s}H/D$, the result isThe same normalized profile describes cell number density or cell volume fraction, because cell volume is constant. Its weak-stratification expansion iswhich 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.