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Harnessing elasticity to generate self-oscillation via an electrohydrodynamic instability

Published online by Cambridge University Press:  12 February 2020

Lailai Zhu Open the ORCID record for Lailai Zhu [Opens in a new window]  and
Howard A. Stone Open the ORCID record for Howard A. Stone [Opens in a new window]
Lailai Zhu
Affiliation:
Department of Mechanical Engineering, National University of Singapore, 117575, Singapore Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA Linné Flow Centre and Swedish e-Science Research Centre (SeRC), KTH Mechanics, Stockholm, SE-10044, Sweden
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ08544, USA
*
†Email address for correspondence: hastone@princeton.edu
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Abstract

Under a steady DC electric field of sufficient strength, a weakly conducting dielectric sphere in a dielectric solvent with higher conductivity can undergo spontaneous spinning (Quincke rotation) through a pitchfork bifurcation. We design an object composed of a dielectric sphere and an elastic filament. By solving an elasto-electro-hydrodynamic (EEH) problem numerically, we uncover an EEH instability exhibiting diverse dynamic responses. Varying the bending stiffness of the filament, the composite object displays three behaviours: a stationary state, undulatory swimming and steady spinning, where the swimming results from a self-oscillatory instability through a Hopf bifurcation. By conducting a linear stability analysis incorporating an elastohydrodynamic model, we theoretically predict the growth rates and critical conditions, which agree well with the numerical counterparts. We also propose a reduced model system consisting of a minimal elastic structure which reproduces the EEH instability. The elasto-viscous response of the composite structure is able to transform the pitchfork bifurcation into a Hopf bifurcation, leading to self-oscillation. Our results imply a new way of harnessing elastic media to engineer self-oscillations, and more generally, to manipulate and diversify the bifurcations and the corresponding instabilities. These ideas will be useful in designing soft, environmentally adaptive machines.

JFM classification

Biological Fluid Dynamics: Swimming/flying MHD and Electrohydrodynamics: MHD and Electrohydrodynamics Low-Reynolds-number flows: Low-Reynolds-number flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 888 , 10 April 2020 , A31
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Alapan, Y., Yasa, O., Yigit, B., Yasa, I. C., Erkoc, P. & Sitti, M. 2019 Microrobotics and microorganisms: biohybrid autonomous cellular robots. Annu. Rev. Control Rob. Auton. Syst. 2, 205–230.CrossRefGoogle Scholar
Batchelor, G. K. 1970 Slender-body theory for particles of arbitrary cross-section in Stokes flow. J. Fluid Mech. 44 (3), 419–440.CrossRefGoogle Scholar
Bayly, P. V. & Dutcher, S. K. 2016 Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella. J. R. Soc. Interface 13 (123), 20160523.CrossRefGoogle ScholarPubMed
Bigoni, D., Kirillov, O. N., Misseroni, D., Noselli, G. & Tommasini, M. 2018 Flutter and divergence instability in the Pflüger column: experimental evidence of the Ziegler destabilization paradox. J. Mech. Phys. Solids 116, 99–116.CrossRefGoogle Scholar
Bricard, A., Caussin, J. B., Desreumaux, N., Dauchot, O. & Bartolo, D. 2013 Emergence of macroscopic directed motion in populations of motile colloids. Nature 503 (7474), 95–98.CrossRefGoogle ScholarPubMed
Brokaw, C. J. 1971 Bend propagation by a sliding filament model for flagella. J. Expl Biol. 55 (2), 289–304.Google ScholarPubMed
Brokaw, C. J. 2009 Thinking about flagellar oscillation. Cell Motil. Cytoskel. 66 (8), 425–436.CrossRefGoogle ScholarPubMed
Brosseau, Q., Hickey, G. & Vlahovska, P. M. 2017 Electrohydrodynamic Quincke rotation of a prolate ellipsoid. Phys. Rev. Fluids 2 (1), 014101.CrossRefGoogle Scholar
Cates, M. E. & MacKintosh, F. C. 2011 Active soft matter. Soft Matt. 7 (7), 3050–3051.CrossRefGoogle Scholar
Cēbers, A., Lemaire, E. & Lobry, L. 2000 Electrohydrodynamic instabilities and orientation of dielectric ellipsoids in low-conducting fluids. Phys. Rev. E 63 (1), 016301.Google ScholarPubMed
Coq, N., du Roure, O., Marthelot, J., Bartolo, D. & Fermigier, M. 2008 Rotational dynamics of a soft filament: wrapping transition and propulsive forces. Phys. Fluids 20 (5), 051703.CrossRefGoogle Scholar
Das, D. & Lauga, E. 2019 Active particles powered by Quincke rotation in a bulk fluid. Phys. Rev. Lett. 122 (19), 194503.CrossRefGoogle Scholar
Das, D. & Saintillan, D. 2013 Electrohydrodynamic interaction of spherical particles under Quincke rotation. Phys. Rev. E 87 (4), 043014.Google ScholarPubMed
De Canio, G., Lauga, E. & Goldstein, R. E. 2017 Spontaneous oscillations of elastic filaments induced by molecular motors. J. R. Soc. Interface 14 (136), 20170491.CrossRefGoogle ScholarPubMed
Dreyfus, R., Baudry, J., Roper, M. L., Fermigier, M., Stone, H. A. & Bibette, J. 2005 Microscopic artificial swimmers. Nature 437 (7060), 862.CrossRefGoogle ScholarPubMed
Evans, B. A., Shields, A. R., Carroll, R. L., Washburn, S., Falvo, M. R. & Superfine, R. 2007 Magnetically actuated nanorod arrays as biomimetic cilia. Nano Lett. 7 (5), 1428–1434.CrossRefGoogle ScholarPubMed
Fatehiboroujeni, S., Gopinath, A. & Goyal, S. 2018 Nonlinear oscillations induced by follower forces in prestressed clamped rods subjected to drag. J. Comput. Nonlinear Dyn. 13 (12), 121005.Google Scholar
Fawcett, D. 1961 Cilia and flagella. In The Cell: Biochemistry, Physiology, Morphology (ed. Brachet, J. & Mirsky, A. E.), vol. 2, pp. 217–297. Elsevier.CrossRefGoogle Scholar
Gold, T. 1948 Hearing. II. The physical basis of the action of the cochlea. Proc. R. Soc. Lond. B 135 (881), 492–498.CrossRefGoogle Scholar
Guglielmini, L., Kushwaha, A., Shaqfeh, E. S. G. & Stone, H. A. 2012 Buckling transitions of an elastic filament in a viscous stagnation point flow. Phys. Fluids 24 (12), 123601.CrossRefGoogle Scholar
Hanasoge, S., Ballard, M., Hesketh, P. J. & Alexeev, A. 2017 Asymmetric motion of magnetically actuated artificial cilia. Lab on a Chip 17 (18), 3138–3145.CrossRefGoogle ScholarPubMed
Herrmann, G. & Bungay, R. W. 1964 On the stability of elastic systems subjected to nonconservative forces. Trans. ASME J. Appl. Mech. 31 (3), 435–440.CrossRefGoogle Scholar
Hilfinger, A., Chattopadhyay, A. K. & Jülicher, F. 2009 Nonlinear dynamics of cilia and flagella. Phys. Rev. E 79 (5), 051918.Google ScholarPubMed
Hines, M. & Blum, J. J. 1983 Three-dimensional mechanics of eukaryotic flagella. Biophys. J. 41 (1), 67.CrossRefGoogle ScholarPubMed
Hu, T. & Bayly, P. V. 2018 Finite element models of flagella with sliding radial spokes and interdoublet links exhibit propagating waves under steady dynein loading. Cytoskeleton 75 (5), 185–200.CrossRefGoogle ScholarPubMed
Huang, H.-W., Uslu, F. E., Katsamba, P., Lauga, E., Sakar, M. S. & Nelson, B. J. 2019 Adaptive locomotion of artificial microswimmers. Sci. Adv. 5 (1), eaau1532.CrossRefGoogle ScholarPubMed
Jenkins, A. 2013 Self-oscillation. Phys. Rep. 525 (2), 167–222.CrossRefGoogle Scholar
Jones, T. B. 1984 Quincke rotation of spheres. IEEE Trans. Ind. Applics. IA‐20 (4), 845–849.CrossRefGoogle Scholar
Kemp, D. T. 1979 Evidence of mechanical nonlinearity and frequency selective wave amplification in the cochlea. Arch. Otorhinolaryngol. 224 (1–2), 37–45.CrossRefGoogle ScholarPubMed
Kieseok, O., Chung, J.-H., Devasia, S. & Riley, J. J. 2009 Bio-mimetic silicone cilia for microfluidic manipulation. Lab on a Chip 9 (11), 1561–1566.Google Scholar
Koiter, W. T. 1996 Unrealistic follower forces. J. Sound Vib. 194, 636.CrossRefGoogle Scholar
Lauga, E. & Powers, T. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.CrossRefGoogle Scholar
Li, L., Manikantan, H., Saintillan, D. & Spagnolie, S. E. 2013 The sedimentation of flexible filaments. J. Fluid Mech. 735, 705–736.CrossRefGoogle Scholar
Ling, F., Guo, H. & Kanso, E. 2018 Instability-driven oscillations of elastic microfilaments. J. R. Soc. Interface 15 (149), 20180594.CrossRefGoogle ScholarPubMed
Livanovičs, R. & Cēbers, A. 2012 Magnetic dipole with a flexible tail as a self-propelling microdevice. Phys. Rev. E 85 (4), 041502.Google ScholarPubMed
Manghi, M., Schlagberger, X. & Netz, R. R. 2006 Propulsion with a rotating elastic nanorod. Phys. Rev. Lett. 96 (6), 068101.CrossRefGoogle ScholarPubMed
Marchetti, M. C., Joanny, J. F., Ramaswamy, S., Liverpool, T. B., Prost, J., Rao, M. & Simha, R. A. 2013 Hydrodynamics of soft active matter. Rev. Mod. Phys. 85 (3), 1143.CrossRefGoogle Scholar
Masuda, T., Hidaka, M., Murase, Y., Akimoto, A. M., Nagase, K., Okano, T. & Yoshida, R. 2013 Self-oscillating polymer brushes. Angew. Chem. 125 (29), 7616–7619.CrossRefGoogle Scholar
Needleman, D. & Dogic, Z. 2017 Active matter at the interface between materials science and cell biology. Nat. Rev. Mater. 2 (9), 17048.CrossRefGoogle Scholar
van Oosten, C. L., Bastiaansen, C. W. M. & Broer, D. J. 2009 Printed artificial cilia from liquid-crystal network actuators modularly driven by light. Nat. Mater. 8 (8), 677.CrossRefGoogle Scholar
Otto, J., Forbes, A. & Verschelde, J.2019 Solving polynomial systems with phcpy. Preprint, arXiv:1907.00096.Google Scholar
Peters, F., Lobry, L. & Lemaire, E. 2005 Experimental observation of Lorenz chaos in the Quincke rotor dynamics. Chaos 15 (1), 013102.CrossRefGoogle ScholarPubMed
Pflüger, A. 1950 Stabilitätsprobleme der Elastostatik. Springer.Google Scholar
Qian, B., Powers, T. R. & Breuer, K. S. 2008 Shape transition and propulsive force of an elastic rod rotating in a viscous fluid. Phys. Rev. Lett. 100 (7), 078101.CrossRefGoogle Scholar
Quincke, G. 1896 Ueber rotationen im constanten electrischen felde. Ann. Phys. 295 (11), 417–486.CrossRefGoogle Scholar
Ramaswamy, S. 2010 The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys. 1 (1), 323–345.CrossRefGoogle Scholar
Riedel-Kruse, I. H., Müller, C. & Oates, A. C. 2007 Synchrony dynamics during initiation, failure, and rescue of the segmentation clock. Science 317 (5846), 1911–1915.CrossRefGoogle ScholarPubMed
Sartori, P., Geyer, V. F., Scholich, A., Jülicher, F. & Howard, J. 2016 Dynamic curvature regulation accounts for the symmetric and asymmetric beats of chlamydomonas flagella. Elife 5, e13258.CrossRefGoogle ScholarPubMed
Sel’kov, E. E. 1968 Self-oscillations in glycolysis 1. A simple kinetic model. Eur. J. Biochem. 4 (1), 79–86.CrossRefGoogle ScholarPubMed
Sidorenko, A., Krupenkin, T., Taylor, A., Fratzl, P. & Aizenberg, J. 2007 Reversible switching of hydrogel-actuated nanostructures into complex micropatterns. Science 315 (5811), 487–490.CrossRefGoogle ScholarPubMed
Singh, H., Laibinis, P. E. & Hatton, T. A. 2005 Synthesis of flexible magnetic nanowires of permanently linked core-shell magnetic beads tethered to a glass surface patterned by microcontact printing. Nano Lett. 5 (11), 2149–2154.CrossRefGoogle ScholarPubMed
den Toonder, J., Bos, F., Broer, D., Filippini, L., Gillies, M., de Goede, J., Mol, T., Reijme, M., Talen, W., Wilderbeek, H., Khatavkar, V. & Anderson, P. 2008 Artificial cilia for active micro-fluidic mixing. Lab on a Chip 8 (4), 533–541.CrossRefGoogle Scholar
Tornberg, A. K. & Shelley, M. J. 2004 Simulating the dynamics and interactions of flexible fibers in Stokes flows. J. Comput. Phys. 196 (1), 8–40.CrossRefGoogle Scholar
Tsebers, A. O. 1980a Electrohydrodynamic instabilities in a weakly conducting suspension of ellipsoidal particles. Magnetohydrodynamics 16 (2), 175–180.Google Scholar
Tsebers, A. O. 1980b Internal rotation in the hydrodynamics of weakly conducting dielectric suspensions. Fluid Dyn. 15 (2), 245–251.CrossRefGoogle Scholar
Tsebers, A. O. 1991 Chaotic solutions for the relaxation equations of electrical polarization. Magnetohydrodynamics 27 (3), 251–258.Google Scholar
Turcu, I. 1987 Electric field induced rotation of spheres. J. Phys. A: Math. Gen. 20 (11), 3301–3307.CrossRefGoogle Scholar
Verschelde, J.1997 PHCPACK: A general-purpose solver for polynomial systems by homotopy continuation, Tech. Rep. TW265, Department of Computer Science, Katholieke Universiteit Leuven.Google Scholar
Verschelde, J.2013 Modernizing PHCpack through phcpy. Preprint, arXiv:1310.0056.Google Scholar
Wiggins, C. H. & Goldstein, R. E. 1998 Flexive and propulsive dynamics of elastica at low Reynolds number. Phys. Rev. Lett. 80 (17), 3879–3882.CrossRefGoogle Scholar
Wiggins, C. H., Riveline, D., Ott, A. & Goldstein, R. E. 1998 Trapping and wiggling: elastohydrodynamics of driven microfilaments. Biophys. J. 74 (2), 1043–1060.CrossRefGoogle ScholarPubMed
Zaks, M. A. & Shliomis, M. I.2014 Onset and breakdown of relaxation oscillations in the torsional Quincke pendulum. Preprint on webpage at https://www.researchgate.net/publication/267410780_Onset_and_breakdown_of_relaxation_oscillations_in_the_torsional_Quincke_pendulum.Google Scholar
Zhu, L. & Stone, H. A. 2019 Propulsion driven by self-oscillation via an electrohydrodynamic instability. Phys. Rev. Fluids 4 (6), 061701.CrossRefGoogle Scholar
Ziegler, H. 1952 Die stabilitätskriterien der elastomechanik. Ing.-Arch. 20 (1), 49–56.CrossRefGoogle Scholar