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The cell conservation equation equates the change of cell number density $n$ to minus the divergence of its total flux. The flux $n\mathbf{u}$ is advection by the water, $n{V}_s⟨\mathbf{p}⟩$ is the mean swimming flux, and $-\mathbf{D}\cdot \nabla n$ is the dispersal flux. The tensor $\mathbf{D}$ describes effective translational diffusion, whereas $D_r$ in part (i) describes rotational diffusion; they have different dimensions. This conservation law neglects birth, death and sources at the walls.

The equation $\nabla \cdot \mathbf{u}=0$ expresses incompressible flow. In the momentum equation, $\rho$ is the reference suspension density, $\rho [{\partial }_t\mathbf{u}+(\mathbf{u}\cdot \nabla )\mathbf{u}]$ is material acceleration times density, $-\nabla p_e$ is the pressure force after subtracting the pure-water hydrostatic pressure, and $\mu {\nabla }^2\mathbf{u}$ is the viscous force for constant dynamic viscosity $\mu$ in a Newtonian fluid. Since $n$ counts cells per volume, $g^{\prime }$ in $-g^{\prime }n{\mathbf{e}}_z$ is excess downward weight per cell; it is not an acceleration. The small cell-fluid density difference permits the Boussinesq approximation: retain that difference in buoyancy while using constant $\rho$ elsewhere. Diluteness permits neglect of cell-cell interactions, changes of viscosity, and active cell stresses. Any separate settling speed is assumed negligible or already incorporated in the specified mean drift. The light strength and swimming speed are uniform.

Consider the intended long-channel bulk approximation. At the sidewalls impose zero normal cell flux and the no-slip boundary condition. The closed ends require zero total fluid flux through each horizontal section, so ${\int }_{-a}^{a}wdx=0$; localized turning regions are omitted from the bulk equations. The mean concentration is ${\int }_{-a}^{a}ndx=2a{n}_0$. The final paragraph below distinguishes these bulk conditions from exact closed-end cell conservation.

For $\mathbf{u}=w(x){\mathbf{e}}_z$, the vorticity is $-w^{\prime }(x){\mathbf{e}}_y$. Hence $\epsilon =-w^{\prime }/(2D_r)$, and the leading horizontal mean swimming direction is $K_0{\mathbf{e}}_x$. With $\mathbf{D}=DI$ and $n=n(x)$, steady cell conservation equation says that $J_x=V_s{K}_{0}n-D{n}^{\prime }$ is constant. Its wall value is zero, so $n^{\prime }=\beta n$ with $\beta =V_s{K}_0/D$. The phototactic concentration in a vertical channel is thereforeThis result neglects the $O({\epsilon }^2)$ correction to horizontal swimming. For positive phototaxis, $\beta >0$ and cells accumulate towards the illuminated wall $x=a$.

The steady inertial terms vanish for $w=w(x)$, since $\mathbf{u}\cdot \nabla =w{\partial }_z$. The horizontal momentum equation gives ${\partial }_x{p}_e=0$. The vertical momentum equation then requires a constant excess pressure gradient $G={\partial }_z{p}_e$ and gives $\mu w^{\prime \prime }=G+g^{\prime }{n}_1{e}^{\beta x}$. Twice integrating producesSet $b=a\beta$. The two no-slip boundary conditions give $A=-C\sinh b/a$ and $B=-C\cosh b-G{a}^2/(2\mu )$. The zero fluid-flux condition follows by integrating this expression:Consequently the remaining constants areThe resulting velocity can also be written without separate polynomial constants asHere $G$ is the zero-flux pressure gradient in a cell-driven channel. It is negative for $b>0$: pressure decreases upwards, creating the upward pressure force that balances the excess cell weight and drives the return flow. Its magnitude is determined by zero net fluid transport, not by prescribing pressure independently. In the zero-light limit, the continuous result is $n=n_0$, $G=-g^{\prime }{n}_0$ and $w=0$.

For the requested large-$b$ sketch, cells occupy a boundary layer of thickness ${\beta }^{-1}$ near $x=a$. Put $\xi =x/a$ and $W=g^{\prime }{n}_0{a}^2/(\mu b)$. The exact dimensionless profile above isAway from the illuminated-wall layer it tends to $(1+\xi )(1-3\xi )/2$. There is broad upward return flow for $-1<\xi <1/3$ and downward flow on the illuminated side. The outer maximum is $2W/3$ at $\xi =-1/3$. Near $\xi =1$, the exponential term is approximately $2e^{-b(1-\xi )}$ and restores the no-slip boundary condition from the negative outer velocity to zero at the wall. The finite-$b$ profile shown below retains that layer. The small-vorticity assumption must still be enforced, for example by sufficiently weak excess weight or a sufficiently dilute suspension; $b\gg 1$ alone does not ensure $|w^{\prime }|/(2D_r)\ll 1$.

There is a genuine global qualification to the bulk construction. For an exactly steady channel with impermeable ends as well as impermeable sidewalls, the closed-channel cell-flux constraint also requires $\int J_{z}dx=0$ at every height. In this bulk ansatz the first-order orientation closure givessince integration by parts, $n^{\prime }=\beta n$ and $w(\pm a)=0$ imply $\int n{w}^{\prime }=-\beta \int nw$. Moreover, multiply $\mu w^{\prime \prime }=G+g^{\prime }n$ by $w$ and integrate. The wall values and zero fluid flux givefor a nontrivial profile. Thus zero axial cell flux generally fails unless the extra factor $1+V_s\beta K_1/(2D_r)$ vanishes, or the flow is trivial. A complete steady closed-channel solution generically needs axial concentration variation. The explicit profile solves the stated bulk equations, sidewall conditions, concentration normalization and zero fluid-flux condition; it is not, for arbitrary parameters, an exact globally closed steady cell distribution. Ignoring end regions gives the intended bulk approximation, but does not by itself prove the stronger global existence claim.

Neglecting cell rotational inertia, the light and viscous torque sum to zero. Therefore the angular velocity is $\mathbf{\Omega }=\mathbf{\omega }/2+(\mathbf{p}\times \mathbf{L})/\alpha$. A material orientation evolves by $\dot{\mathbf{p}}=\mathbf{\Omega }\times \mathbf{p}$. The vector triple-product identity givesThe last term is the projection of $\mathbf{L}/\alpha$ onto the tangent plane to the unit sphere. In particular, $\mathbf{p}\cdot \dot{\mathbf{p}}=0$, as required to preserve the unit orientation. It aligns the swimmer towards the light; the vorticity rotates that orientation.

For rotational diffusion coefficient $D_r$, the phototactic orientation Fokker-Planck equation balances deterministic orientation drift with diffusion on the sphere. Its density is normalized with respect to solid angle, not $d\theta d\varphi$. When $\epsilon =0$, there is no azimuthal drift and the zero-current condition is $f_{0,\theta }=-\lambda \sin \theta f_0$. Thus $f_0=\eta e^{\lambda \cos \theta }$, and normalization givesThe limiting density at $\lambda =0$ is $1/(4\pi )$.

To justify the angular dependence of the first correction, define the operatorsThe steady Fokker-Planck equation is $\mathcal{A}f=-\epsilon \mathcal{R}f$. Its regular expansion $f=f_0+\epsilon f^{(1)}+O({\epsilon }^2)$ therefore givesThe coefficients of $\mathcal{A}$ are independent of $\varphi$, so each azimuthal Fourier harmonic is preserved. The forcing has only the sine harmonic with azimuthal number one. There can be no extra unforced harmonic in a smooth normalized first correction: writing a homogeneous solution as $f_{0}h$ gives $\mathcal{A}(f_{0}h)={\nabla }_p\cdot (f_0{\nabla }_{p}h)$. Multiplication by $h$ and integration by parts on the sphere gives $\int f_0|{\nabla }_{p}h{|}^{2}d\Omega =0$, hence $h$ is constant. Normalization $\int f^{(1)}d\Omega =0$ removes that multiple of $f_0$. It follows thatNo explicit calculation of $g$ is needed. Its dependence is on $\theta$ and the parameter $\lambda$, and its behavior at the poles must make the density smooth.

In the stated spherical coordinates, $\mathbf{p}=\cos \theta {\mathbf{e}}_x+\sin \theta \cos \varphi {\mathbf{e}}_y+\sin \theta \sin \varphi {\mathbf{e}}_z$. Azimuthal integration makes the leading $y,z$ components and the first-order $x,y$ corrections vanish. The surviving coefficients areThus the small-vorticity phototactic orientation response isBoth coefficients depend only on $\lambda$. In particular, $K_0$ is the Langevin function of $\lambda$ and is positive for positive light strength; its integral representation above suffices.

For isotropic rotational diffusion, orientation drift and probability density obey the displayed Fokker-Planck equation, normalized with respect to solid angle. Without flow, constant light $L{\mathbf{e}}_x$ gives the zero-current density $f_0=\lambda e^{\lambda p_x}/(4\pi \sinh \lambda )$, $\lambda =L/(\alpha D_r)$. Integration over the sphere gives unit mass; its mean projection is the Langevin function. The zero-light limit is the uniform density $1/(4\pi )$.