= Gyrotactic concentration layer at a rotating-cylinder wall
{title2=$n-1\sim-\varepsilon\operatorname{Re}[e^{i\theta'}e^{-q(1-r)}/q]$}
With $\beta^2=\Omega R^2/D$ and $\varepsilon=2V_sR/(3B\Omega D)$, the first angular harmonic of the steady <cell conservation equation> satisfies $a''+a'/r-a/r^2-i\beta^2a=0$ and $a'(1)=-1$. The regular solution is $a=-I_1(qr)/[qI_1'(q)]$, $q=e^{i\pi/4}\beta$. 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.
Back to article page
