 Gyrotactic concentration layer at a rotating-cylinder wall

With ${\beta }^2=\Omega R^2/D$ and $\epsilon =2V_{s}R/(3B\Omega D)$, the first angular harmonic of the steady cell conservation equation satisfies $a^{\prime \prime }+a^{\prime }/r-a/r^2-i{\beta }^{2}a=0$ and $a^{\prime }(1)=-1$. The regular solution is $a=-I_1(qr)/[q{I}_1^{\prime }(q)]$, $q=e^{i\pi /4}\beta$. At large $\beta$ it gives the displayed layer with width $1/\beta$. The exact Modified Bessel function of the first kind expression remains regular at the axis, unlike extrapolation of the boundary-layer formula there.