= Closed-channel cell-flux constraint
{title2=$\int_{-a}^a J_z\,dx=0$}
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 $n'=\beta n$, $\omega_y=-w'$, and $\langle p_z\rangle=-K_1w'/(2D_r)$, the axial flux is $[1+V_s\beta K_1/(2D_r)]\int nw\,dx$. Meanwhile the momentum equation, zero fluid flux and <integration by parts> give $g'\int nw\,dx=-\mu\int(w')^2dx$. 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.
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