Cell conservation in a swimming suspension

If $C$ is the cell number density, $\mathbf{u}$ the fluid velocity, $V_s⟨\mathbf{p}⟩$ the mean swimming velocity, and $D$ an isotropic translational diffusion coefficient, conservation is ${\partial }_{t}C+\nabla \cdot [C(\mathbf{u}+V_s⟨\mathbf{p}⟩)-D\nabla C]=0$. Impermeable boundaries have zero normal total cell flux. For constant cell volume $v$, the volume fraction is $vC$ and obeys the same equation, but its buoyancy coefficient has no additional factor $v$.

A genuinely closed steady channel with impermeable sidewalls and ends must have zero integrated axial cell flux through every section, not only zero fluid flux. In the bulk phototactic ansatz $n^{\prime }=\beta n$, ${\omega }_y=-w^{\prime }$, and $⟨p_z⟩=-K_1{w}^{\prime }/(2D_r)$, the axial flux is $[1+V_s\beta K_1/(2D_r)]\int nwdx$. Meanwhile the momentum equation, zero fluid flux and integration by parts give $g^{\prime }\int nwdx=-\mu \int (w^{\prime }{)}^{2}dx$. Thus a nontrivial bulk profile is generally incompatible with exact closed-end cell conservation unless the prefactor vanishes. Additional axial concentration variation is needed; neglecting end regions does not itself remove this global constraint.