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  • Open Access

Instability and self-propulsion of flexible autophoretic filaments

Ursy Makanga*,†,‡ and Akhil Varma*

Panayiota Katsamba

  • Department of Chemical Engineering, Cyprus University of Technology, 30 Archbishop Kyprianou Str., Limassol 3036, Cyprus and Computational-based Science and Technology Research Center (CaSToRC), The Cyprus Institute, Nicosia 2121, Cyprus

  • *These authors contributed equally to this work.
  • †Contact author: umakanga@flatironinstitute.org
  • ‡Present address: Center for Computational Biology, Flatiron Institute, Simons Foundation, New York, New York 10010, USA.
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Phys. Rev. Fluids 11, 053101 – Published 11 May, 2026

DOI: https://doi.org/10.1103/51vg-yb2b

Abstract

Over the past decade, autophoretic colloids have emerged as a prototypical system for studying self-propelled motion at microscopic scales, with promising applications in microfluidics, micromachinery, and therapeutics. Their motion in a viscous fluid hinges on their ability to induce surface slip flows that are spatially asymmetric from self-generated solute gradients. Here we demonstrate theoretically that a straight elastic filament with homogeneous surface chemical properties—which is otherwise immotile—can spontaneously achieve self-propulsion by experiencing a buckling instability that serves as the symmetry-breaking mechanism. Using efficient numerical simulations, we characterize the nonlinear dynamics of the elastic filament and show that, over time, it attains distinct swimming modes such as a steadily translating “U” shape and a metastable rotating “S” shape when semiflexible, and an oscillatory state when highly flexible. Our findings provide physical insight into future experiments and the design of reconfigurable synthetic active colloids.

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References (129)

  1. R. Feynman, There's plenty of room at the bottom, Science 254, 1300 (1991).
  2. J. Katuri, K. D. Seo, D. S. Kim, and S. Sánchez, Artificial micro-swimmers in simulated natural environments, Lab Chip 16, 1101 (2016).
  3. M. Guix, S. M. Weiz, O. G. Schmidt, and M. Medina-Sánchez, Self-propelled micro/nanoparticle motors, Part. Part. Syst. Char. 35, 1700382 (2018).
  4. A. Nsamela, A. I. Garcia Zintzun, T. D. Montenegro-Johnson, and J. Simmchen, Colloidal active matter mimics the behavior of biological microorganisms—An overview, Small 19, 2202685 (2023).
  5. B. J. Nelson, I. K. Kaliakatsos, and J. J. Abbott, Microrobots for minimally invasive medicine, Annu. Rev. Biomed. Eng. 12, 55 (2010).
  6. Y. Alapan, O. Yasa, B. Yigit, I. C. Yasa, P. Erkoc, and M. Sitti, Microrobotics and microorganisms: Biohybrid autonomous cellular robots, Annu. Rev. Control Rob. Auton. Syst. 2, 205 (2019).
  7. Z. Wu, Y. Chen, D. Mukasa, O. S. Pak, and W. Gao, Medical micro/nanorobots in complex media, Chem. Soc. Rev. 49, 8088 (2020).
  8. C. Maggi, J. Simmchen, F. Saglimbeni, J. Katuri, M. Dipalo, F. De Angelis, S. Sanchez, and R. Di Leonardo, Self-assembly of micromachining systems powered by Janus micromotors, Small 12, 446 (2016).
  9. A. Aubret, M. Youssef, S. Sacanna, and J. Palacci, Targeted assembly and synchronization of self-spinning microgears, Nat. Phys. 14, 1114 (2018).
  10. J. Simmchen, J. Katuri, W. E. Uspal, M. N. Popescu, M. Tasinkevych, and S. Sánchez, Topographical pathways guide chemical microswimmers, Nat. Commun. 7, 10598 (2016).
  11. R. Dreyfus, J. Baudry, M. L. Roper, M. Fermigier, H. A. Stone, and J. Bibette, Microscopic artificial swimmers, Nature (London) 437, 862 (2005).
  12. S. Sundararajan, P. E. Lammert, A. W. Zudans, V. H. Crespi, and A. Sen, Catalytic motors for transport of colloidal cargo, Nano Lett. 8, 1271 (2008).
  13. L. Baraban, M. Tasinkevych, M. N. Popescu, S. Sanchez, S. Dietrich, and O. G. Schmidt, Transport of cargo by catalytic Janus micro-motors, Soft Matter 8, 48 (2012).
  14. A. Zöttl and H. Stark, Emergent behavior in active colloids, J. Phys.: Condens. Matter 28, 253001 (2016).
  15. P. Illien, R. Golestanian, and A. Sen, ‘fuelled’ motion: Phoretic motility and collective behaviour of active colloids, Chem. Soc. Rev. 46, 5508 (2017).
  16. J. L. Anderson, Colloid transport by interfacial forces, Annu. Rev. Fluid Mech. 21, 61 (1989).
  17. B. V. Derjaguin, G. Sidorenkov, E. Zubashchenko, and E. Kiseleva, Kinetic phenomena in the boundary layers of liquids 1. The capillary osmosis, Prog. Surf. Sci. 43, 138 (1993).
  18. W. F. Paxton, K. C. Kistler, C. C. Olmeda, A. Sen, S. K. St. Angelo, Y. Cao, T. E. Mallouk, P. E. Lammert, and V. H. Crespi, Catalytic nanomotors: Autonomous movement of striped nanorods, J. Am. Chem. Soc. 126, 13424 (2004).
  19. J. R. Howse, R. A. L. Jones, A. J. Ryan, T. Gough, R. Vafabakhsh, and R. Golestanian, Self-motile colloidal particles: From directed propulsion to random walk, Phys. Rev. Lett. 99, 048102 (2007).
  20. R. Golestanian, T. B. Liverpool, and A. Ajdari, Propulsion of a molecular machine by asymmetric distribution of reaction products, Phys. Rev. Lett. 94, 220801 (2005).
  21. R. Golestanian, T. B. Liverpool, and A. Ajdari, Designing phoretic micro- and nano-swimmers, New J. Phys. 9, 126 (2007).
  22. J. L. Moran and J. D. Posner, Phoretic self-propulsion, Annu. Rev. Fluid Mech. 49, 511 (2017).
  23. S. Michelin, E. Lauga, and D. Bartolo, Spontaneous autophoretic motion of isotropic particles, Phys. Fluids 25, 061701 (2013).
  24. M. Morozov and S. Michelin, Nonlinear dynamics of a chemically-active drop: From steady to chaotic self-propulsion, J. Chem. Phys. 150, 044110 (2019).
  25. C. C. Maass, C. Krüger, S. Herminghaus, and C. Bahr, Swimming droplets, Annu. Rev. Condens. Matter Phys. 7, 171 (2016).
  26. S. Michelin, Self-propulsion of chemically active droplets, Annu. Rev. Fluid Mech. 55, 77 (2023).
  27. A. Varma, T. D. Montenegro-Johnson, and S. Michelin, Clustering-induced self-propulsion of isotropic autophoretic particles, Soft Matter 14, 7155 (2018).
  28. S. Shklyaev, J. F. Brady, and U. M. Córdova-Figueroa, Non-spherical osmotic motor: Chemical sailing, J. Fluid Mech. 748, 488 (2014).
  29. S. Michelin and E. Lauga, Autophoretic locomotion from geometric asymmetry, Eur. Phys. J. E 38, 7 (2015).
  30. M. N. Popescu, S. Dietrich, M. Tasinkevych, and J. Ralston, Phoretic motion of spheroidal particles due to self-generated solute gradients., Eur. Phys. J. E 31, 351 (2010).
  31. A. Nourhani, V. H. Crespi, P. E. Lammert, and A. Borhan, Self-electrophoresis of spheroidal electrocatalytic swimmers, Phys. Fluids 27, 092002 (2015).
  32. S. Michelin and E. Lauga, Geometric tuning of self-propulsion for Janus catalytic particles, Sci. Rep. 7, 42264 (2017).
  33. F. Kümmel, B. Ten Hagen, R. Wittkowski, I. Buttinoni, R. Eichhorn, G. Volpe, H. Löwen, and C. Bechinger, Circular motion of asymmetric self-propelling particles, Phys. Rev. Lett. 110, 198302 (2013).
  34. A. Ganguly and A. Gupta, Going in circles: Slender body analysis of a self-propelling bent rod, Phys. Rev. Fluids 8, 014103 (2023).
  35. L. C. Schmieding, E. Lauga, and T. D. Montenegro-Johnson, Autophoretic flow on a torus, Phys. Rev. Fluids 2, 034201 (2017).
  36. R. D. Baker, T. Montenegro-Johnson, A. D. Sediako, M. J. Thomson, A. Sen, E. Lauga, and I. S. Aranson, Shape-programmed 3D printed swimming microtori for the transport of passive and active agents, Nat. Commun. 10, 4932 (2019).
  37. P. Sharan, C. Maslen, B. Altunkeyik, I. Rehor, J. Simmchen, and T. D. Montenegro-Johnson, Fundamental modes of swimming correspond to fundamental modes of shape: Engineering I-, U-, and S-shaped swimmers, Adv. Intell. Syst. 3, 2100068 (2021).
  38. S. Riedel, L. A. Hoffmann, L. Giomi, and D. J. Kraft, Designing highly efficient interlocking interactions in anisotropic active particles, Nat. Commun. 15, 5692 (2024).
  39. B. Delmotte and F. B. Usabiaga, A scalable method to model large suspensions of colloidal phoretic particles with arbitrary shapes, J. Comput. Phys. 518, 113321 (2024).
  40. Y. Solomentsev and J. L. Anderson, Electrophoresis of slender particles, J. Fluid Mech. 279, 197 (1994).
  41. E. Yariv, Slender-body approximations for electro-phoresis and electro-rotation of polarizable particles, J. Fluid Mech. 613, 85 (2008).
  42. O. Schnitzer and E. Yariv, Osmotic self-propulsion of slender particles, Phys. Fluids 27, 031701 (2015).
  43. E. Yariv, Self-diffusiophoresis of slender catalytic colloids, Langmuir 36, 6903 (2020).
  44. D. Saintillan, E. Darve, and E. S. G. Shaqfeh, Hydrodynamic interactions in the induced-charge electrophoresis of colloidal rod dispersions, J. Fluid Mech. 563, 223 (2006).
  45. P. Katsamba, S. Michelin, and T. D. Montenegro-Johnson, Slender phoretic theory of chemically active filaments, J. Fluid Mech. 898, A24 (2020).
  46. P. Katsamba, M. D. Butler, L. Koens, and T. D. Montenegro-Johnson, Chemically active filaments: Analysis and extensions of slender phoretic theory, Soft Matter 18, 7051 (2022).
  47. P. Katsamba, M. D. Butler, L. Koens, and T. D. Montenegro-Johnson, Active colloidal loops and knots, Phys. Rev. Fluids 9, 054201 (2024).
  48. P. Katsamba and T. D. Montenegro-Johnson, Introduction to the theories and modelling of active colloids, in Active Colloids: From Fundamentals to Frontiers, edited by W. Wang, J. Simmchen, and W. Uspal (Royal Society of Chemistry, London, 2024), Vol. 20, Chap. 10, pp. 315–375.
  49. R. Poehnl and W. Uspal, Phoretic self-propulsion of helical active particles, J. Fluid Mech. 927, A46 (2021).
  50. L. Guglielmini, A. Kushwaha, E. S. G. Shaqfeh, and H. A. Stone, Buckling transitions of an elastic filament in a viscous stagnation point flow, Phys. Fluids 24, 123601 (2012).
  51. L. Li, H. Manikantan, D. Saintillan, and S. E. Spagnolie, The sedimentation of flexible filaments, J. Fluid Mech. 735, 705 (2013).
  52. A. A. Evans, S. E. Spagnolie, D. Bartolo, and E. Lauga, Elastocapillary self-folding: Buckling, wrinkling and collapse of floating filaments, Soft Matter 9, 1711 (2013).
  53. Y.-N. Young and M. J. Shelley, Stretch-coil transition and transport of fibers in cellular flows, Phys. Rev. Lett. 99, 058303 (2007).
  54. V. Kantsler and R. E. Goldstein, Fluctuations, dynamics, and the stretch-coil transition of single actin filaments in extensional flows, Phys. Rev. Lett. 108, 038103 (2012).
  55. R. Chelakkot, R. G. Winkler, and G. Gompper, Flow-induced helical coiling of semiflexible polymers in structured microchannels, Phys. Rev. Lett. 109, 178101 (2012).
  56. B. Chakrabarti, Y. Liu, J. LaGrone, R. Cortez, L. Fauci, O. Du Roure, D. Saintillan, and A. Lindner, Flexible filaments buckle into helicoidal shapes in strong compressional flows, Nat. Phys. 16, 689 (2020).
  57. C. Duprat and H. Stone, Fluid–Structure Interactions in Low-Reynolds-Number Flows (The Royal Society of Chemistry, London, 2015).
  58. O. du Roure, A. Lindner, E. N. Nazockdast, and M. J. Shelley, Dynamics of flexible fibers in viscous flows and fluids, Annu. Rev. Fluid Mech. 51, 539 (2019).
  59. L. Bourdieu, M. O. Magnasco, D. A. Winkelmann, and A. Libchaber, Actin filaments on myosin beds: The velocity distribution, Phys. Rev. E 52, 6573 (1995).
  60. B. Alberts, D. Bray, K. Hopkin, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter, Essential Cell Biology, 4th ed. (Garland Science, New York, 2014).
  61. A. Vilfan, S. Subramani, E. Bodenschatz, R. Golestanian, and I. Guido, Flagella-like beating of a single microtubule, Nano Lett. 19, 3359 (2019).
  62. X. Shi, A.-K. Pumm, J. Isensee, W. Zhao, D. Verschueren, A. Martin-Gonzalez, R. Golestanian, H. Dietz, and C. Dekker, Sustained unidirectional rotation of a self-organized DNA rotor on a nanopore, Nat. Phys. 18, 1105 (2022).
  63. G. Jayaraman, S. Ramachandran, S. Ghose, A. Laskar, M. S. Bhamla, P. B. S. Kumar, and R. Adhikari, Autonomous motility of active filaments due to spontaneous flow-symmetry breaking, Phys. Rev. Lett. 109, 158302 (2012).
  64. A. Laskar, R. Singh, S. Ghose, G. Jayaraman, P. B. S. Kumar, and R. Adhikari, Hydrodynamic instabilities provide a generic route to spontaneous biomimetic oscillations in chemomechanically active filaments, Sci. Rep. 3, 1964 (2013).
  65. R. Chelakkot, A. Gopinath, L. Mahadevan, and M. F. Hagan, Flagellar dynamics of a connected chain of active, polar, Brownian particles, J. R. Soc. Interface. 11, 20130884 (2014).
  66. A. Laskar and R. Adhikari, Brownian microhydrodynamics of active filaments, Soft Matter 11, 9073 (2015).
  67. A. Martín-Gómez, T. Eisenstecken, G. Gompper, and R. G. Winkler, Active Brownian filaments with hydrodynamic interactions: Conformations and dynamics, Soft Matter 15, 3957 (2019).
  68. R. K. Manna, P. B. S. Kumar, and R. Adhikari, Colloidal transport by active filaments, J. Chem. Phys. 146, 024901 (2017).
  69. B. Németh, T. Kobayashi, and R. Adhikari, Nonreciprocal constitutive laws for oriented active solids, arXiv:2509.11430.
  70. S. C. Al-Izzi, Y. Du, J. Veenstra, R. G. Morris, A. Souslov, A. Carlson, C. Coulais, and J. Binysh, Nonreciprocal buckling makes active filaments polyfunctional, Proc. Natl. Acad. Sci. USA 123, e2531723123 (2026).
  71. G. De Canio, E. Lauga, and R. E. Goldstein, Spontaneous oscillations of elastic filaments induced by molecular motors, J. R. Soc., Interface 14, 20170491 (2017).
  72. F. Ling, H. Guo, and E. Kanso, Instability-driven oscillations of elastic microfilaments, J. R. Soc. Interface. 15, 20180594 (2018).
  73. Y. Man and E. Kanso, Morphological transitions of axially-driven microfilaments, Soft Matter 15, 5163 (2019).
  74. D. B. Stein, G. De Canio, E. Lauga, M. J. Shelley, and R. E. Goldstein, Swirling instability of the microtubule cytoskeleton, Phys. Rev. Lett. 126, 028103 (2021).
  75. B. Clarke, Y. Hwang, and E. E. Keaveny, Bifurcations and nonlinear dynamics of the follower force model for active filaments, Phys. Rev. Fluids 9, 073101 (2024).
  76. O. Schnitzer, Onset of spontaneous beating and whirling in the follower force model of an active filament, J. Fluid Mech. 1007, A65 (2025).
  77. Y.-N. Young, Dynamics of a semiflexible polar filament in Stokes flow, Phys. Rev. E 82, 016309 (2010).
  78. W. Lough, D. B. Weibel, and S. E. Spagnolie, Self-buckling and self-writhing of semi-flexible microorganisms, Soft Matter 19, 7349 (2023).
  79. S. Camalet and F. Jülicher, Generic aspects of axonemal beating, New J. Phys. 2, 324 (2000).
  80. B. Chakrabarti and D. Saintillan, Spontaneous oscillations, beating patterns, and hydrodynamics of active microfilaments, Phys. Rev. Fluids 4, 043102 (2019).
  81. D. Nishiguchi, J. Iwasawa, H.-R. Jiang, and M. Sano, Flagellar dynamics of chains of active Janus particles fueled by an ac electric field, New J. Phys. 20, 015002 (2018).
  82. H.-W. Huang, F. E. Uslu, P. Katsamba, E. Lauga, M. S. Sakar, and B. J. Nelson, Adaptive locomotion of artificial microswimmers, Sci. Adv. 5, eaau1532 (2019).
  83. T. Yang, B. Sprinkle, Y. Guo, J. Qian, D. Hua, A. Donev, D. W. M. Marr, and N. Wu, Reconfigurable microbots folded from simple colloidal chains, Proc. Natl. Acad. Sci. USA 117, 18186 (2020).
  84. B. Biswas, P. More, and H. N. Kandula, Emergent softening and stiffening dictate transport of active filaments, arXiv:2507.08535.
  85. M. Wei and D. J. Kraft, Autonomous life-like behavior emerging in active and flexible microstructures, Proc. Natl. Acad. Sci. USA 123, e2531743123 (2026).
  86. B. Biswas, R. K. Manna, A. Laskar, P. B. S. Kumar, R. Adhikari, and G. Kumaraswamy, Linking catalyst-coated isotropic colloids into “active” flexible chains enhances their diffusivity, ACS Nano 11, 10025 (2017).
  87. H. R. Vutukuri, B. Bet, R. van Roij, M. Dijkstra, and W. T. S. Huck, Rational design and dynamics of self-propelled colloidal bead chains: From rotators to flagella, Sci. Rep. 7, 16758 (2017).
  88. M. Kumar, A. Murali, A. G. Subramaniam, R. Singh, and S. Thutupalli, Emergent dynamics due to chemo-hydrodynamic self-interactions in active polymers, Nat. Commun. 15, 4903 (2023).
  89. T. D. Montenegro-Johnson, Microtransformers: Controlled microscale navigation with flexible robots, Phys. Rev. Fluids 3, 062201(R) (2018).
  90. R. K. Manna, A. Laskar, O. Shklyaev, and A. Balazs, Harnessing the power of chemically active sheets in solution., Nat. Rev. Phys. 4, 125 (2022).
  91. L. Qiao and R. Kapral, Control of active polymeric filaments by chemically powered nanomotors, Phys. Rev. Appl. 18, 024051 (2022).
  92. B. Altunkeyik, A. Rahmat, and T. Montenegro-Johnson, Dynamics of active poroelastic filaments in Stokes flow, Phys. Rev. E 111, 025421 (2025).
  93. M. B. Butler, B. J. Walker, T. D. Montenegro-Johnson, and P. Katsamba, Elastohydrodynamics of three-dimensional chemically active filaments, J. Fluid Mech. 1029, A42 (2026).
  94. B. J. Walker, K. Ishimoto, and E. A. Gaffney, Efficient simulation of filament elastohydrodynamics in three dimensions, Phys. Rev. Fluids 5, 123103 (2020).
  95. B. Audoly and Y. Pomeau, Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells (Oxford University Press, Oxford, 2010).
  96. L. D. Landau, L. P. Pitaevskii, A. M. Kosevich, and E. M. Lifshitz, Theory of Elasticity: Volume 7, 3rd ed. (Butterworth-Heinemann, Amsterdam, 1986).
  97. G. Hancock, The self-propulsion of microscopic organisms through liquids, Proc. R. Soc. Lond. A 217, 96 (1953).
  98. R. G. Cox, The motion of long slender bodies in a viscous fluid Part 1. General theory, J. Fluid Mech. 44, 791 (1970).
  99. G. K. Batchelor, Slender-body theory for particles of arbitrary cross-section in Stokes flow, J. Fluid Mech. 44, 419 (1970).
  100. J. Lighthill, Flagellar hydrodynamics, SIAM Rev. 18, 161 (1976).
  101. J. B. Keller and S. I. Rubinow, Slender-body theory for slow viscous flow, J. Fluid Mech. 75, 705 (1976).
  102. R. E. Johnson, An improved slender-body theory for Stokes flow, J. Fluid Mech. 99, 411 (1980).
  103. T. Götz, Interactions of fibers and flow: Asymptotics, theory and numerics, dissertation, Technische Universität Kaiserslautern, 2001.
  104. R. Cortez and M. Nicholas, Slender body theory for Stokes flows with regularized forces, Commun. Appl. Math. Comput. Sci. 7, 33 (2012).
  105. L. Koens and E. Lauga, The boundary integral formulation of Stokes flows includes slender-body theory, J. Fluid Mech. 850, R1 (2018).
  106. L. A. Ohm, Mathematical foundations of slender body theory, Ph.D. thesis, University of Minnesota, 2020.
  107. O. Maxian, A. Mogilner, and A. Donev, An integral-based spectral method for inextensible slender fibers in Stokes flow, Phys. Rev. Fluids 6, 014102 (2021).
  108. A.-K. Tornberg and M. J. Shelley, Simulating the dynamics and interactions of flexible fibers in Stokes flows, J. Comput. Phys. 196, 8 (2004).
  109. H. Manikantan and D. Saintillan, Buckling transition of a semiflexible filament in extensional flow, Phys. Rev. E 92, 041002(R) (2015).
  110. E. Nazockdast, A. Rahimian, D. Zorin, and M. Shelley, A fast platform for simulating semi-flexible fiber suspensions applied to cell mechanics, J. Comput. Phys. 329, 173 (2017).
  111. O. Maxian, Hydrodynamics of transiently cross-linked actin networks: Theory, numerics, and emergent behaviors, Ph.D. thesis, New York University, 2023.
  112. C. Moreau, L. Giraldi, and H. Gadêlha, The asymptotic coarse-graining formulation of slender-rods, bio-filaments and flagella, J. R. Soc. Interface. 15, 20180235 (2018).
  113. A. L. Hall-McNair, T. D. Montenegro-Johnson, H. Gadêlha, D. J. Smith, and M. T. Gallagher, Efficient implementation of elastohydrodynamics via integral operators, Phys. Rev. Fluids 4, 113101 (2019).
  114. B. J. Walker, K. Ishimoto, H. Gadêlha, and E. A. Gaffney, Filament mechanics in a half-space via regularised Stokeslet segments, J. Fluid Mech. 879, 808 (2019).
  115. M. Jabbarzadeh and H. C. Fu, A numerical method for inextensible elastic filaments in viscous fluids, J. Comput. Phys. 418, 109643 (2020).
  116. O. Maxian and A. Donev, A simulation platform for slender, semiflexible, and inextensible fibers with Brownian hydrodynamics and steric repulsion, Phys. Fluids 36, 123320 (2024).
  117. L. N. Trefethen, Spectral Methods in Matlab (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2000).
  118. J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., revised ed. (Dover, Mineola, NY, 2001).
  119. K. Shoemake, Animating rotation with quaternion curves, in Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH '85, Vol. 19 (Association for Computing Machinery, New York, 1985), pp. 245–254.
  120. L. E. Becker and M. J. Shelley, Instability of elastic filaments in shear flow yields first-normal-stress differences, Phys. Rev. Lett. 87, 198301 (2001).
  121. C. H. Wiggins, D. Riveline, A. Ott, and R. E. Goldstein, Trapping and wiggling: Elastohydrodynamics of driven microfilaments, Biophys. J. 74, 1043 (1998).
  122. See Supplemental Material at http://link.aps.org/supplemental/10.1103/51vg-yb2b for movies and their description.
  123. P. Glendinning, Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations, Cambridge Texts in Applied Mathematics Vol. 11 (Cambridge University Press, Cambridge, UK, 1994).
  124. F. Charru, Instabilités Hydrodynamiques (EDP Sciences, Les Ulis, France, 2007).
  125. W.-F. Hu, T.-S. Lin, S. Rafai, and C. Misbah, Chaotic swimming of phoretic particles, Phys. Rev. Lett. 123, 238004 (2019).
  126. G. Li, Swimming dynamics of a self-propelled droplet, J. Fluid Mech. 934, A20 (2022).
  127. G. Zhu and L. Zhu, Self-propulsion of an elliptical phoretic disk emitting solute uniformly, J. Fluid Mech. 974, A57 (2023).
  128. R. Cobos, A. S. Khair, and O. Schnitzer, Spontaneous locomotion of a symmetric squirmer, J. Fluid Mech. 983, R3 (2024).
  129. B. Marchetti, V. Raspa, A. Lindner, O. du Roure, L. Bergougnoux, É. Guazzelli, and C. Duprat, Deformation of a flexible fiber settling in a quiescent viscous fluid, Phys. Rev. Fluids 3, 104102 (2018).

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Phys. Rev. Fluids 11, 053101– Published 11 May, 2026
  • Received 22 December 2025
  • Accepted 10 April 2026
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