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Vortex line entanglement in active Beltrami flows

Published online by Cambridge University Press:  01 March 2024

Nicolas Romeo Open the ORCID record for Nicolas Romeo [Opens in a new window] ,
Jonasz Słomka Open the ORCID record for Jonasz Słomka [Opens in a new window] ,
Jörn Dunkel Open the ORCID record for Jörn Dunkel [Opens in a new window]  and
Keaton J. Burns Open the ORCID record for Keaton J. Burns [Opens in a new window]
Nicolas Romeo
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Jonasz Słomka
Affiliation:
Institute of Environmental Engineering, ETH Zürich, Zurich 8092, Switzerland
Jörn Dunkel
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Keaton J. Burns*
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA
*
†Email address for correspondence: kjburns@mit.edu
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Abstract

Over the last decade, substantial progress has been made in understanding the topology of quasi-two-dimensional (2-D) non-equilibrium fluid flows driven by ATP-powered microtubules and microorganisms. By contrast, the topology of three-dimensional (3-D) active fluid flows still poses interesting open questions. Here, we study the topology of a spherically confined active flow using 3-D direct numerical simulations of generalized Navier–Stokes (GNS) equations at the scale of typical microfluidic experiments. Consistent with earlier results for unbounded periodic domains, our simulations confirm the formation of Beltrami-like bulk flows with spontaneously broken chiral symmetry in this model. Furthermore, by leveraging fast methods to compute linking numbers, we explicitly connect this chiral symmetry breaking to the entanglement statistics of vortex lines. We observe that the mean of linking number distribution converges to the global helicity, consistent with the asymptotic result by Arnold [In Vladimir I. Arnold – Collected Works (ed. A.B. Givental, B.A. Khesin, A.N. Varchenko, V.A. Vassiliev & O.Y. Viro), pp. 357–375. Springer]. Additionally, we characterize the rate of convergence of this measure with respect to the number and length of observed vortex lines, and examine higher moments of the distribution. We find that the full distribution is well described by a k-Gamma distribution, in agreement with an entropic argument. Beyond active suspensions, the tools for the topological characterization of 3-D vector fields developed here are applicable to any solenoidal field whose curl is tangent to or cancels at the boundaries of a simply connected domain.

JFM classification

Complex Fluids: Active matter Mathematical Foundations: Topological fluid dynamics Vortex Flows: Vortex interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 982 , 10 March 2024 , A12
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© The Author(s), 2024. Published by Cambridge University Press

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