= Stresslet correction to gyrotactic pipe flow
{title2=$w_1'=\frac{A(1-e^{-ar^2})}{2ar}+4\sigma_1\lambda r n_0\sqrt{1-4\lambda^2r^2}$}
Let weak excess-cell buoyancy be $\gamma\ll1$ and the swimming <stresslet> coefficient be $\sigma=\gamma\sigma_1$. Set $w=\gamma w_1$ and use the leading <gyrotactic focusing in a downward pipe flow> density $n_0=Ae^{-ar^2}$. With active stress entering as $+Sn\mathbf p\mathbf p$, the axial momentum equation integrates from the regular axis to the displayed derivative. The wall condition gives $w_1(r)=-\int_r^1w_1'(s)ds$. Both terms of the integrand are positive for $\sigma_1\geq0$, so the correction increases downward flow. A negative stresslet strength can oppose this effect; reversal at a given radius requires $\sigma_1$ smaller than minus the ratio of the integrated buoyancy and stresslet coefficients. This convention must be specified before attaching extensile or contractile sign labels.
Back to article page
