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From active stresses and forces to self-propulsion of droplets

Published online by Cambridge University Press:  25 May 2017

R. Kree*
Affiliation:
Georg-August-Universität Göttingen, Institut für Theoretische Physik, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
P. S. Burada
Affiliation:
Department of Physics, Indian Institute of Technology, Kharagpur – 721302, India
A. Zippelius
Affiliation:
Georg-August-Universität Göttingen, Institut für Theoretische Physik, Friedrich-Hund-Platz 1, 37077 Göttingen, Germany
*
Email address for correspondence: kree@theorie.physik.uni-goettingen.de

Abstract

We study the self-propulsion of spherical droplets as simplified hydrodynamic models of swimming micro-organisms or artificial micro-swimmers. In contrast to approaches that start from active velocity fields produced by the system, we consider active interface tractions, body force densities and active stresses as the origin of autonomous swimming. For negligible Reynolds number and given activity, we compute the external and internal flow fields as well as the centre of mass velocity and angular velocity of the droplet at fixed time. To construct trajectories from single time snapshots, the evolution of active forces or stresses must be determined in the laboratory frame. Here, we consider the case of active matter, which is carried by a continuously distributed rigid but sparse (cyto)-skeleton that is immersed in the droplet interior. We calculate examples of trajectories of a droplet and its skeleton from force densities or stresses, which may be explicitly time-dependent in a frame fixed within the skeleton.

Type
Papers
Copyright
© 2017 Cambridge University Press 

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References

Ateshian, G. A. & Humphrey, J. D. 2012 Continuum mixture models of biological growth and remodeling: past successes and future opportunities. Annu. Rev. Biomed. Engng 14, 97111.Google Scholar
Basset, A. B. 1888 On the motion of a sphere in a viscous fluid. Phil. Trans. R. Soc. Lond. 179, 4363.Google Scholar
Bechinger, C., Leonardo, R. D., Löwen, H., Reichhardt, C., Volpe, G. & Volpe, G. 2016 Active particles in complex and crowded environments. Rev. Mod. Phys. 88, 045006.CrossRefGoogle Scholar
Berg, H. C. 2003 The rotary motor of bacterial flagella. Annu. Rev. Biochem. 72 (1), 1954.Google Scholar
Biot, M. A. 1955 Theory of elasticity and consolidation for a porous anisotropic solid. J. Appl. Phys. 26, 182185.Google Scholar
Blake, J. R. 1971 A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199208.Google Scholar
Boussinesq, J. 1885 Sur la resistance qu’oppose un fluide indefini au repos au mouvement varie d’une sphere solide. C. R. Acad. Sci. Paris 100, 935937.Google Scholar
Callan, A. C. & Jülicher, F. 2011 Hydrodynamics of active permeating gels. New J. Phys. 13, 093027.Google Scholar
Dembo, M. & Harlow, F. 1986 Contractile networks, and the physics of interpenetrating reactive flow. Biophys. J. 50, 109.Google Scholar
Drescher, K., Goldstein, R. E., Michel, N., Polin, M. & Tuval, I. 2010 Direct measurement of the flow field around swimming microorganisms. Phys. Rev. Lett. 105, 168101.Google Scholar
Ebbens, S. J. & Howse, J. R. 2010 In pursuit of propulsion at the nanoscale. Soft Matt. 6, 726738.Google Scholar
Edmonds, A. R. 1957 Angular Momentum in Quantum Mechanics. Princeton University Press.Google Scholar
Ehlers, K. & Oster, G. 2012 On the mysterious propulsion of Synechococcus . PLoS ONE 7, e36081.Google Scholar
Einarsson, J. & Mehlig, B.2016 Spherical particle sedimenting in weakly viscoelastic shear flow. arXiv:1609.05531.Google Scholar
Elgeti, J., Winkler, R. G. & Gompper, G. 2015 Physics of microswimmer – single particle and collective behavior: a review. Rep. Prog. Phys. 78, 056601.Google Scholar
Feng, J. & Joseph, D. D. 1995 The unsteady motion of solid bodies in creeping flows. J. Fluid Mech. 303, 83102.Google Scholar
deGennes, P. G. 1976 Dynamics of entangled polymer solutions. I. The Rouse model. Macromolecules 9, 587598.Google Scholar
Ghose, S. & Adhikari, R. 2014 Irreducible representations of oscillatory and swirling flows in active soft matter. Phy. Rev. Lett. 112, 118102.Google Scholar
Goldstein, R. E. 2016 Fluid dynamics at the scale of the cell. J. Fluid Mech. 807, 139.Google Scholar
Golestanian, R., Liverpool, T. B. & Adjari, A. 2005 Propulsion of a molecular machine by asymmetric distribution of reaction products. Phys. Rev. Lett. 94, 220801.Google Scholar
Golestanian, R., Liverpool, T. B. & Adjari, A. 2007 Designing phoretic micro- and nanoswimmers. New J. Phys. 9, 126.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Springer.Google Scholar
Herminghaus, S., Maas, C. C., Krüger, C., Thutupalli, S., Goering, L. & Bahr, C. 2013 Interfacial mechanism in active emulsions. Soft Matt. 10, 7008.Google Scholar
Joanny, J. F., Jülicher, F., Kruse, K. & Prost, J. 2007 Hydrodynamic theory for active polar gels. New J. Phys. 9, 422.CrossRefGoogle Scholar
Kanevsky, A., Shelley, M. J. & Tornberg, A.-K. 2010 Modeling simple locomotors in Stokes flow. J. Comput. Phys. 229, 958977.Google Scholar
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications. Dover.Google Scholar
Köhler, S., Schaller, V. & Bausch, A. R. 2011 Structure formation in active networks. Nature Mat. 10, 462468.CrossRefGoogle ScholarPubMed
Kruse, K., Joanny, J. F., Jülicher, F., Prost, J. & Sekimoto, K. 2004 Aster, votices, and rotating spirals in active gels of polar filaments. Phys. Rev. Lett. 92, 078101.Google Scholar
Kruse, K., Joanny, J. F., Jülicher, F., Prost, J. & Sekimoto, K. 2005 Generic theory of active polar gels: a paradigm for cytoskeletal dynamics. Eur. Phys. J. E 16, 516.Google Scholar
Kumar, A., Maitra, A., Sumit, M., Ramaswamy, S. & Shivashankar, G. V. 2014 Actomyosin contractility rotates the cell nucleus. Sci. Rep. 4, 3781.Google Scholar
Lamb, H. 1879 Hydrodynamics. Cambridge University Press.Google Scholar
Landau, D. & Lifshitz, E. M. 1976 Mechanics, 3rd edn. Course of Theoretical Physics, vol. 1. Elsevier.Google Scholar
Lauga, E. 2016 Bacterial hydrodynamics. Annu. Rev. Fluid Mech. 48 (1), 105130.Google Scholar
Lauga, E. & Powers, T. R. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.Google Scholar
Lighthill, M. J. 1952 On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers. Commun. Pure Appl. Maths 9, 109.Google Scholar
Lighthill, M. J. 1976 Flagellar hydrodynamics. SIAM Rev. 18, 161.Google Scholar
Loisel, T. P., Boujemaa, R., Pantaloni, D. & Carlier, M.-F. 1999 Reconstitution of actin-based motility of listeria and shigella using pure proteins. Nature 401, 613.Google Scholar
Maass, C. C., Krüger, C., Herminghaus, S. & Bahr, C. 2016 Swimming droplets. Annu. Rev. Condens. Mat. Phys. 7, 171193.Google Scholar
Michaelidis, E. E. & Feng, Z.-G. 1995 The equation of motion of a small viscous sphere in an unsteady flow with interface slip. Intl J. Multiphase Flow 21, 315321.Google Scholar
Milner, S. T. 1993 Dynamical theory of concentration fluctuations in polymer solutions under shear. Phys. Rev. E 48, 36743691.Google Scholar
Moeendarbary, E., Valon, L., Fritzsche, M., Harris, A. R., Moulding, D. A., Thrasher, A. J., Stride, E., Mahadevan, L. & Charras, G. T. 2013 The cytoplasm of living cells behaves as a poroelastic material. Nat. Mater. 12, 253261.CrossRefGoogle ScholarPubMed
Morse, P. M. & Feshbach, H. 1953 Methods of Theoretical Physics, vol. 3. McGraw-Hill.Google Scholar
Moses, H. E. 1974 The use of vector spherical harmonics in global meteorology. J. Atmos. Sci. 31, 14901499.Google Scholar
Mustacich, R. V. & Ware, B. R. 1976 A study of protoplasmic streaming in nitella by laser Doppler spectroscopy. Biophys. J. 16, 373388.Google Scholar
Nan, B., Chen, J., Neu, J. C., Berry, R. M., Oster, G. & Zusman, D. R. 2001 Myxobacteria gliding motility requires cytoskeleton rotation powered by proton motive force. Proc. Natl Acad. Sci. USA 108, 24982503.Google Scholar
Newman, E. T. & Penrose, R. 1966 Note on the Bondi–Metzner–Sachs group. J. Math. Phys. 7 (5), 863870.Google Scholar
Pak, O. S. & Lauga, E. 2014 Generalized squirming motion of a sphere. J. Engng Maths 88, 1.Google Scholar
Peddireddy, K., Kumar, P., Thutupalli, S., Herminghaus, S. & Bahr, C. 2012 Solubilization of thermotropic liquid crystal compounds in aequeous surfactant solution. Langmuir 28, 12426.Google Scholar
Pedley, T. J., Brumley, D. R. & Goldstein, R. E. 2016 Squirmers with swirl – a model for volvox swimming. J. Fluid Mech. 798, 165186.Google Scholar
Purcell, E. M. 1952 Life at low Reynolds number. Am. J. Phys. 45, 3.Google Scholar
Sanchez, T., Chen, D. T. N., DeCamp, S. J., Heymann, M. & Dogic, Z. 2012 Spontaneous motion in hierarchically assembled active matter. Nature 491, 431.Google Scholar
Schmitt, M. & Stark, H. 2013 Swimming active droplet: a theoretical analysis. Eur. Phys. Lett. 101, 44008.Google Scholar
Schmitt, M. & Stark, H. 2016 Marangoni flow at droplet interfaces: three dimensional solution and applications. Phys. Fluids 28, 012106.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of a pendulum. Trans. Camb. Phil. Soc. 9, 8106.Google Scholar
Strychalski, W. & Guy, R. D. 2016 Intracellular pressure dynamics in blebbing cells. Biophys. J. 110, 11681179.Google Scholar
Takabatake, F., Magome, N., Ichikawa, M. & Yoshikawa, K. 2011 Spontaneous mode-selection in the self-propelled motion of a solid/liquid composite driven by interfacial instability. J. Chem. Phys. 134, 114704.Google Scholar
Tarama, M., Menzel, A. M., ten Hagen, B., Wittkowski, R. & Ohta, T. 2013 Dynamics of deformable active particle under shear flow. J. Chem. Phys. 139, 104906.Google Scholar
Taylor, G. I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146, 501523.Google Scholar
Vlahoska, P. M., Blawzdziewicz, J. & Loewenberg, M. 2009 Small-deformation theory for a surfactant-covered drop in linear flows. J. Fluid Mech. 624, 293337.Google Scholar
Whitfield, C. A., Marenduzzo, D., Voituriez, R. & Hawkins, R. J. 2014 Active polar fluid flow in finite droplets. Eur. Phys. J. E 37, 8.Google Scholar
Zöttl, A. & Stark, H. 2013 Periodic and quasiperiodic motion of an elongated microswimmer in Poiseuille flow. Eur. Phys. J. E 36, 4.Google Scholar