Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-27T23:25:45.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Incompressible active phases at an interface. Part 1. Formulation and axisymmetric odd flows

Published online by Cambridge University Press:  08 November 2022

Leroy L. Jia Open the ORCID record for Leroy L. Jia [Opens in a new window] ,
William T.M. Irvine Open the ORCID record for William T.M. Irvine [Opens in a new window]  and
Michael J. Shelley
Leroy L. Jia*
Affiliation:
Center for Computational Biology, Flatiron Institute, New York, NY 10010, USA
William T.M. Irvine
Affiliation:
James Franck Institute, Enrico Fermi Institute, and Department of Physics, University of Chicago, Chicago, IL 60637, USA
Michael J. Shelley
Affiliation:
Center for Computational Biology, Flatiron Institute, New York, NY 10010, USA Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA
*
†Email address for correspondence: ljia@flatironinstitute.org
Get access Rights & Permissions [Opens in a new window]

Abstract

Inspired by the recent realization of a two-dimensional (2-D) chiral fluid as an active monolayer droplet moving atop a 3-D Stokesian fluid, we formulate mathematically its free-boundary dynamics. The surface droplet is described as a general 2-D linear, incompressible and isotropic fluid, having a viscous shear stress, an active chiral driving stress and a Hall stress allowed by the lack of time-reversal symmetry. The droplet interacts with itself through its driven internal mechanics and by driving flows in the underlying 3-D Stokes phase. We pose the dynamics as the solution to a singular integral–differential equation, over the droplet surface, using the mapping from surface stress to surface velocity for the 3-D Stokes equations. Specializing to the case of axisymmetric droplets, exact representations for the chiral surface flow are given in terms of solutions to a singular integral equation, solved using both analytical and numerical techniques. For a disc-shaped monolayer, we additionally employ a semi-analytical solution that hinges on an orthogonal basis of Bessel functions and allows for efficient computation of the monolayer velocity field, which ranges from a nearly solid-body rotation to a unidirectional edge current, depending on the subphase depth and the Saffman–Delbrück length. Except in the near-wall limit, these solutions have divergent surface shear stresses at droplet boundaries, a signature of systems with codimension-one domains embedded in a 3-D medium. We further investigate the effect of a Hall viscosity, which couples radial and transverse surface velocity components, on the dynamics of a closing cavity. Hall stresses are seen to drive inward radial motion, even in the absence of edge tension.

JFM classification

Complex Fluids: Active matter Low-Reynolds-number flows: Stokesian dynamics
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 951 , 25 November 2022 , A36
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexander, J.C., Bernoff, A.J., Mann, E.K., Mann, J.A. Jr. & Wintersmith, J.R. 2007 Domain relaxation in Langmuir flims. J. Fluid Mech. 571, 191–219.CrossRefGoogle Scholar
Alexander, J.C., Bernoff, A.J., Mann, E.K., Mann, J.A. Jr. & Zou, L. 2006 Hole dynamics in polymer langmuir films. Phys. Fluids 18, 062103.CrossRefGoogle Scholar
Avron, J.E. 1998 Odd viscosity. J. Stat. Phys. 92, 543–557.CrossRefGoogle Scholar
Avron, J.E., Seiler, R. & Zograf, P.G. 1995 Viscosity of quantum hall fluids. Phys. Rev. Lett. 75, 697–700.CrossRefGoogle ScholarPubMed
Barentin, C., Ybert, C., di Meglio, J.-M. & Joanny, J.-F. 1999 Surface shear viscosity of Gibbs and Langmuir monolayers. J. Fluid Mech. 397, 331–349.Google Scholar
Berdyugin, A.I., et al. 2019 Measuring hall viscosity of Graphene's electron fluid. Science 364, 162.CrossRefGoogle ScholarPubMed
Bililign, E.S, Balboa Usabiaga, F., Ganan, Y.A, Poncet, A., Soni, V., Magkiriadou, S., Shelley, M.J, Bartolo, D. & Irvine, W. 2021 Motile dislocations knead odd crystals into whorls. Nat. Phys. 18, 212–218.CrossRefGoogle Scholar
Busbridge, I.W. 1938 Dual integral equations. Proc. Lond. Math. Soc. 44, 115–129.CrossRefGoogle Scholar
Cooke, J.C. 1956 A solution of Tranter's dual integral equations problem. Q. J. Mech. Appl. Maths 9, 103–110.CrossRefGoogle Scholar
Cooke, J.C. 1963 Triple integral equations. Q. J. Mech. Appl. Maths 16, 193–203.Google Scholar
Cooke, J.C. 1965 The solution of triple integral equations in operational form. Q. J. Mech. Appl. Maths 18, 57–72.CrossRefGoogle Scholar
Cressman, J.R, Davoudi, J., Goldburg, W.I & Schumacher, J. 2004 Eulerian and lagrangian studies in surface flow turbulence. New J. Phys. 6 (1), 53.CrossRefGoogle Scholar
Elfring, G.J., Leal, L.G. & Squires, T.M. 2016 Surface viscosity and Marangoni stresses at surfactant laden interfaces. J. Fluid Mech. 792, 712–739.CrossRefGoogle Scholar
Evans, E. & Sackmann, E. 1988 Translational and rotational drag coefficients for a disk moving in a liquid membrane associated with a rigid substrate. J. Fluid Mech. 194, 553–561.CrossRefGoogle Scholar
Gao, T., Betterton, M.D, Jhang, A.-S. & Shelley, M.J. 2017 Analytical structure, dynamics, and coarse graining of a kinetic model of an active fluid. Phys. Rev. Fluids 2 (9), 093302.CrossRefGoogle Scholar
Gao, T., Blackwell, R., Glaser, M.A, Betterton, M.D. & Shelley, M.J. 2015 Multiscale polar theory of microtubule and motor-protein assemblies. Phys. Rev. Lett. 114 (4), 048101.CrossRefGoogle ScholarPubMed
Goldburg, W.I., Cressman, J.R., Vörös, Z, Eckhardt, B. & Schumacher, J. 2001 Turbulence in a free surface. Phys. Rev. E 63 (6), 065303.CrossRefGoogle Scholar
Goodrich, F.C. 1969 The theory of absolute surface shear viscosity. I. Proc. R. Soc. A 310, 359–372.Google Scholar
Gradshteyn, I.S. & Ryzhik, I.M. 2007 Table of Integrals, Series, and Products, 7th edn. Elsevier.Google Scholar
Held, I.M., Pierrehumbert, R.T., Garner, S.T. & Swanson, K.L. 1995 Surface quasi-geostrophic dynamics. J. Fluid Mech. 282, 1–20.CrossRefGoogle Scholar
Henle, M.L. & Levine, A.J. 2009 Effective viscosity of a dilute suspension of membrane-bound inclusions. Phys. Fluids 21, 033106.CrossRefGoogle Scholar
Jeffery, G.B. 1915 On the steady rotation of a solid of revolution in a viscous fluid. Proc. Lond. Math. Soc. 2, 327–338.Google Scholar
Jia, L.L. & Shelley, M.J. 2022 The role of monolayer viscosity in Langmuir film closure dynamics. J. Fluid Mech. 948, A1.CrossRefGoogle Scholar
Kokot, G., Das, S., Winkler, R.G., Gompper, G., Aranson, I.S. & Snezhko, A. 2017 Active turbulence in a gas of self-assembled spinners. Proc. Natl Acad. Sci. USA 114, 12870.CrossRefGoogle Scholar
Lubensky, D.K. & Goldstein, R.E. 1996 Hydrodynamics of monolayer domains at the air-water interface. Phys. Fluids 8, 843–854.CrossRefGoogle Scholar
Martin, P.A. & Smith, S.G.L. 2011 Generation of internal gravity waves by an oscillating horizontal disc. Proc. R. Soc. A 467, 3406–3423.CrossRefGoogle Scholar
Martínez-Prat, B., Ignés-Mullol, J., Casademunt, J. & Sagués, F. 2019 Selection mechanism at the onset of active turbulence. Nat. Phys. 15 (4), 362–366.CrossRefGoogle Scholar
Masoud, H. & Shelley, M.J. 2014 Collective surfing of chemically active particles. Phys. Rev. Lett. 112, 128304.CrossRefGoogle ScholarPubMed
Noble, B. 1958 Certain dual integral equations. J. Math. Phys. 37, 128–136.CrossRefGoogle Scholar
Oppenheimer, N., Stein, D.B. & Shelley, M.J. 2019 Rotating membrane inclusions crystallize through hydrodynamic and steric interactions. Phys. Rev. Lett. 123, 148101.CrossRefGoogle ScholarPubMed
Oppenheimer, N., Stein, D.B., Yah Ben Zion, M. & Shelley, M.J. 2022 Hyperuniformity and phase enrichment in vortex and rotor assemblies. Nat. Commun. 13, 804.CrossRefGoogle ScholarPubMed
Petroff, A.P., Wu, X.-L. & Libchaber, A. 2015 Fast-moving bacteria self-organize into active two-dimensional crystals of rotating cells. Phys. Rev. Lett. 114, 158102.CrossRefGoogle ScholarPubMed
Pullin, D.I. 1992 Contour dynamics methods. Annu. Rev. Fluid Mech. 24 (1), 89–115.CrossRefGoogle Scholar
Ratnanather, J.T., Kim, J.H., Zhang, S., Davis, A.M.J. & Lucas, S.K. 2014 Algorithm 935: IIPBF, a MATLAB toolbox for infinite integral of products of two Bessel functions. ACM Trans. Math. Softw. 40 (2), 1–12.Google Scholar
Rodrigo, J.L. & Fefferman, C.L. 2004 The vortex patch problem for the surface quasi-geostrophic equation. Proc. Natl Acad. Sci. USA 101 (9), 2684–2686.CrossRefGoogle ScholarPubMed
Saffman, P.G 1995 Vortex Dynamics. Cambridge University Press.Google Scholar
Saffman, P.G. & Delbrück, M. 1975 Brownian motion in biological membranes. Proc. Natl Acad. Sci. USA 72 (8), 3111–3113.CrossRefGoogle ScholarPubMed
Sanchez, T., Chen, D.T.N., DeCamp, S.J, Heymann, M. & Dogic, Z. 2012 Spontaneous motion in hierarchically assembled active matter. Nature 491 (7424), 431–434.CrossRefGoogle ScholarPubMed
Sherwood, J.D. 2013 Stokes drag on a disc with a Navier slip condition near a plane wall. Fluid Dyn. Res. 45, 055505.CrossRefGoogle Scholar
Sneddon, I.N. 1946 The distribution of stress in the neighbourhood of a crack in an elastic solid. Proc. R. Soc. Lond. Ser. A 187 (1009), 229–260.Google Scholar
Sneddon, I.N. 1966 Mixed Boundary Value Problems in Potential Theory, 1st edn. North-Holland Pub. Co.Google Scholar
Sneddon, I.N. 1975 The use in mathematical physics of Erdélyi-Kober operators and of some of their generalizations. In Fractional Calculus and Its Applications (ed. B. Ross). Lecture Notes in Mathematics, vol. 457, pp. 37–79. Springer.Google Scholar
Soni, V., Bililign, E., Magkiriadou, S., Sacanna, S., Bartolo, D., Shelley, M.J. & Irvine, W.T.M. 2019 The free surface of a colloidal chiral fluid: waves and instabilities from odd stress and Hall viscosity. Nat. Phys. 15, 1188–1194.Google Scholar
Souslov, A., Dasbiswas, K., Fruchart, M., Vaikuntanathan, S. & Vitelli, V. 2019 Topological waves in fluids with odd viscosity. Phys. Rev. Lett. 122, 128001.CrossRefGoogle ScholarPubMed
Stone, H.A. 1995 Fluid motion of monomolecular films in a channel flow geometry. Phys. Fluids 7, 2931–2937.CrossRefGoogle Scholar
Stone, H.A. & McConnell, H.M. 1995 Hydrodynamics of quantized shape transitions of lipid domains. Proc. R. Soc. Lond. A 448, 97–111.Google Scholar
Tranter, C.J. 1954 A further note on dual integral equations and an application to the diffraction of electromagnetic waves. Q. J. Mech. Appl. Maths 7, 317–325.CrossRefGoogle Scholar
Wiegmann, P. & Abanov, A.G. 2014 Anomalous hydrodynamics of two-dimensional vortex fluids. Phys. Rev. Lett. 113, 034501.CrossRefGoogle ScholarPubMed
Yan, W., Corona, E., Malhotra, D., Veerapaneni, S. & Shelley, M.J. 2020 A scalable computational platform for particulate Stokes suspensions. J. Comput. Phys. 416, 109524.CrossRefGoogle Scholar
Yan, Y. & Sloan, I.H. 1988 On integral equations of the first kind with logarithmic kernels. J. Integral Equ. Appl. 1 (4), 549–580.CrossRefGoogle Scholar
Yeo, K., Lushi, E. & Vlahovska, P.M. 2015 Collective dynamics in a binary mixture of hydrodynamically coupled microrotors. Phys. Rev. Lett. 114, 188301.CrossRefGoogle Scholar