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Let $\hat{u}(\hat{r})=-W_0(1-{\hat{r}}^2/R^2)+\hat{w}(\hat{r})$ and $\mathbf{p}=\sin \theta {\mathbf{e}}_r+\cos \theta \mathbf{k}$, with $\cos \theta >0$ for the upward-stable orientation. Cell conservation in a swimming suspension gives radial fluxRegularity at the axis and impermeability at the wall imply ${\hat{j}}_r=0$, hence $D{\hat{n}}^{\prime }=V_C\hat{n}\sin \theta$. Axial advection has no divergence in this fully developed, axially uniform state.

The fluid is incompressible. Use dynamic stresslet strength $S_d$ and dynamic viscosity ${\mu }_d$ first, so the swimming contribution to its stress is $S_d\hat{n}(\mathbf{pp}-I/3)$. The isotropic part can be absorbed into the effective pressure. With the ordinary fluid hydrostatic pressure removed, the axial momentum equation isThe final term is the downward excess weight of the cells. Sedimentation is neglected in their swimming flux, not in this buoyancy force. Fluid convective inertia vanishes for the specified fully developed velocity even before a low-Reynolds-number approximation is invoked.

There is a dimensional convention to make explicit. The printed definition of $\gamma$ contains the extra fluid density $\rho$. All the printed coefficients are consistent if this question uses density-normalized pressure, kinematic viscosity and stresslet strength:Then ${\hat{p}}^{\prime }=4\mu W_0/R^2$ generates the stated background Poiseuille flow. If $\mu$ and $S$ instead mean dynamic viscosity and a physical force-dipole strength, the correct buoyancy parameter is $g\Delta \rho \nu N_0{R}^2/({\mu }_d{W}_0)$, without a further density denominator. The following derivation retains the consistent density-normalized interpretation of the printed parameters.

The bottom-heavy spherical-cell orientation dynamics isIts scalar tilt equation is $\dot{\theta }=-\sin \theta /B-{\hat{u}}^{\prime }/2$. Steady gravitational-viscous torque balance therefore gives $\sin \theta =-B{\hat{u}}^{\prime }/2$. This equilibrium is stable on the $\cos \theta >0$ branch and exists only when $|B{\hat{u}}^{\prime }/2|<1$.

Put $r=\hat{r}/R$, $w=\hat{w}/W_0$, $n=\hat{n}/N_0$. Subtract the background pressure-driven balance and divide by $\mu W_0/R^2$. The governing system iswhereThe last parameter contains $B$, as in the PDF, not the diffusivity substituted by the converted TeX. Boundary and normalization conditions areThe wall cell-flux condition is already incorporated in $n^{\prime }=\chi n\sin \theta$; there is no prescribed wall concentration or additional fixed-volume-flux condition. The radial divergence of the stresslet stress can be balanced by a radial pressure depending only on $r$. It neither produces radial flow in this ansatz nor changes the imposed axial pressure gradient, so it is eliminated from the axial problem, not assumed to vanish.

For $\gamma \ll 1$, $\sigma =\gamma {\sigma }_1$, let $w=\gamma w_1+O({\gamma }^2)$, $n=n_0+\gamma n_1+O({\gamma }^2)$. The leading tilt hasThe supplied $B{W}_0/R<1$ ensures this leading orientation remains on the stable branch. Put $a=\lambda \chi$. Integrating the concentration equation and imposing the average concentration gives the gyrotactic focusing in a downward pipe flow profileAt $a=0$ the continuous limit is $n_0=1$.

At first order the axial equation isIntegrate from the regular axis, where both boundary terms vanish:Since $\sin 2{\theta }_0=-4\lambda r\sqrt{1-4{\lambda }^2{r}^2}$, the stresslet correction to gyrotactic pipe flow iswith its regular limiting value at $r=0$, andThe orientation and concentration corrections, if needed to complete the expansion, follow without another differential boundary problem:These enforce $2{\int }_0^1{n}_{1}rdr=0$.

For $\lambda >0$, write the two positive integrand coefficients as $F(s)=A(1-e^{-a{s}^2})/(2as)$ and $H(s)=4\lambda s{n}_0(s)\sqrt{1-4{\lambda }^2{s}^2}$. ThenThus $w_1<0$ throughout the interior for ${\sigma }_1\ge 0$, and at any given radius a positive correction is possible only ifA simple sufficient condition for negativity is ${\sigma }_1\ge -1/(8\lambda )$: indeedWhen $\lambda =0$, the stresslet contribution vanishes and the buoyancy correction is negative regardless of ${\sigma }_1$.

Physically, downward shear tilts upward swimmers toward the axis, concentrating excess weight there and enhancing the downward flow. Positive $S$ in the specified $+Sn\mathbf{pp}$ stress convention reinforces this effect; sufficiently negative $S$ opposes it. A positive $w$ means reduced downward velocity, not necessarily reversal of the total pipe flow. The deterministic orientation closure fails at excessive shear, and scalar diffusivity neglects directional and shear-dependent dispersion.