 Closed-channel cell-flux constraint

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.