Correction to: Gyrotactic cluster formation of bottom-heavy squirmers

Abstract

Correction to:The European Physical Journal E (2022) 45:1-14 10.1140/epje/s10189-022-00183-5 Equation (9) should read the balance between the angular velocities from the external bottom-heavy torque (Eq. (5)) and from the Stokeslet vorticity (Eq. (6)). However, a factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3}{4}$$\end{document} 3 4 was erroneously omitted. The corrected equation reads 9 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{3}{4}\frac{{v_{0} }}{R}\frac{{r_{0} }}{R\alpha }\sin \vartheta = \frac{3}{4}\frac{{v_{0} }}{\alpha }\frac{R}{{r^{2} }}. $$\end{document} 3 4 v 0 R r 0 R α sin ϑ = 3 4 v 0 α R r 2 . Consequently, the two subsequent equations, where r = 2R, should read 10 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sin \vartheta = \frac{{1{/(4}\alpha {)}}}{{r_{0} {/(}R\alpha {)}}}. $$\end{document} sin ϑ = 1 / ( 4 α ) r 0 / ( R α ) . and 11 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{0} {/(}R\alpha {)} \ge {(4}\alpha {)}^{ - 1} .$$\end{document} r 0 / ( R α ) ≥ ( 4 α ) - 1 . The corrected balance of angular velocities requires Fig. 5 to be updated. We have plotted black dotted vertical lines for the corrected value of the equality condition from Eq. (11) and show the incorrect line from the original manuscript in red. Fig. 5 a Mean cluster radius \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle \vert {\mathbf {r}}-{\mathbf {\overline{r}}}\vert \rangle _{\mathrm {cl}}$$\end{document} ⟨ | r - r ¯ | ⟩ cl in units of R plotted versus torque value \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_0/R\alpha $$\end{document} r 0 / R α for different squirmer numbers N. b Normalized standard deviation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varDelta N_{\mathrm {cl}}/\langle N_{\mathrm {cl}} \rangle $$\end{document} Δ N cl / ⟨ N cl ⟩ and inset: mean number of squirmers in a cluster \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle N_{\mathrm {cl}} \rangle $$\end{document} ⟨ N cl ⟩ . The dotted vertical lines show the equality condition of Eq. (11) for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =0.8$$\end{document} α = 0.8 . The red dotted line shows the erroneous condition Conclusion The angular velocity balance between Stokeslet vorticity and bottom-heaviness results in a lower bound for the rescaled torque \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r_{0} {/(}R\alpha {)}$$\end{document} r 0 / ( R α ) that is lower than originally presented by a factor of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{3}{4}$$\end{document} 3 4 . The conclusions of the original paper are unaffected.