Hostname: page-component-76d6cb85b7-7262s Total loading time: 0 Render date: 2026-07-22T06:16:28.565Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-dimensional study on the motion and interactions of squirmers under gravity in a vertical channel

Published online by Cambridge University Press:  11 April 2023

Deming Nie Open the ORCID record for Deming Nie [Opens in a new window] ,
Yuxiang Ying ,
Geng Guan ,
Jianzhong Lin Open the ORCID record for Jianzhong Lin [Opens in a new window]  and
Zhenyu Ouyang Open the ORCID record for Zhenyu Ouyang [Opens in a new window]
Deming Nie
Affiliation:
Institute of Fluid Mechanics, China Jiliang University, Hangzhou 310018, PR China
Yuxiang Ying
Affiliation:
Institute of Fluid Mechanics, China Jiliang University, Hangzhou 310018, PR China
Geng Guan
Affiliation:
Institute of Fluid Mechanics, China Jiliang University, Hangzhou 310018, PR China
Jianzhong Lin*
Affiliation:
Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education, Ningbo 315201, PR China Institute of Fluid Engineering, Zhejiang University, Hangzhou 310027, PR China
Zhenyu Ouyang
Affiliation:
Key Laboratory of Impact and Safety Engineering (Ningbo University), Ministry of Education, Ningbo 315201, PR China
*
† Email address for correspondence: mecjzlin@public.zju.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We simulated the motions and interactions of circular squirmers under gravity in a two-dimensional channel at finite fluid inertia, aiming to provide a comprehensive analysis of the dynamic features of swimming microorganisms or engineered microswimmers. In addition to a squirmer-type factor (β), another control parameter (α) was introduced, representing ratio of the self-propelling strength to the sedimentation strength of squirmers. Simulations were performed at 0.4 ≤ α ≤ 1.2 and −5 ≤ β ≤ 5. We first considered the sedimentation of a single squirmer. Five patterns were revealed, depending on both α and β: steady downward falling, steady inclined falling or rising and small-scale or large-scale oscillating. Compared with a pusher (β < 0, gaining thrust from rear), a puller (β > 0, gaining thrust from front) is more likely to break down its symmetrical structure and subsequently lose stability, owing to the high-pressure regions on its lateral sides. Typically, a pusher settles faster than a puller, whereas a neutral squirmer (β = 0) settles in between. This is related to the ‘trailing negative flow’ behind a pusher and ‘leading negative flow’ before a puller. We then placed two squirmers in line with the gravity direction to study their interactions. Results show pullers attract each other and come into contact as a result of the low-pressure regions between them, whereas the opposite is observed for pushers. The interactions between two pullers are illustrated by their respective patterns. In contrast, pushers never come into contact and maintain distance from each other with increasing separation. We finally examined how a puller interacts with a pusher.

JFM classification

Biological Fluid Dynamics: Micro-organism dynamics Complex Fluids: Suspensions

Information

Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 960 , 10 April 2023 , A31
Check for updates
Copyright
© The Author(s), 2023. 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.)

Article purchase

Temporarily unavailable

References

Ahana, P. & Thampi, S.P. 2019 Confinement induced trajectory of a squirmer in a two dimensional channel. Fluid Dyn. Res. 51, 065504.CrossRefGoogle Scholar
Aidun, C.K., Lu, Y. & Ding, E. 1998 Direct analysis of particulate suspensions with inertia using the discrete Boltzmann equation. J. Fluid Mech. 373, 287–311.CrossRefGoogle Scholar
Berke, A.P., Turner, L., Berg, H.C. & Lauga, E. 2008 Hydrodynamic attraction of swimming microorganisms by surfaces. Phys. Rev. Lett. 101, 038102.CrossRefGoogle ScholarPubMed
Blake, J.R. 1971 a A spherical envelope approach to ciliary propulsion. J. Fluid Mech. 46, 199–208.CrossRefGoogle Scholar
Blake, J.R. 1971 b Self propulsion due to oscillations on the surface of a cylinder at low Reynolds number. Bull. Austral. Math. Soc. 5, 255–264.CrossRefGoogle Scholar
Chisholm, N., Legendre, D., Lauga, E. & Khair, A. 2016 A squirmer across Reynolds numbers. J. Fluid Mech. 796, 233–256.CrossRefGoogle Scholar
Di Leonardo, R., Angelani, L., Dell'Arciprete, D. & Di Fabrizio, E. 2010 Bacterial ratchet motors. Proc. Natl Acad. Sci. USA 107, 9541–9545.CrossRefGoogle ScholarPubMed
Drescher, K., Leptos, K.C., Tuval, I., Ishikawa, T., Pedley, T.J. & Goldstein, R.E. 2009 Dancing volvox: hydrodynamic bound states of swimming algae. Phys. Rev. Lett. 102, 168101.CrossRefGoogle ScholarPubMed
Elgeti, J., Winkler, R.G. & Gompper, G. 2015 Physics of microswimmers—single particle motion and collective behavior: a review. Rep. Prog. Phys. 78, 056601.CrossRefGoogle ScholarPubMed
Fadda, F., Molina, J.J. & Yamamoto, R. 2020 Dynamics of a chiral swimmer sedimenting on a flat plate. Phys. Rev. E 101, 052608.CrossRefGoogle ScholarPubMed
Fenchel, T. & Finlay, B.J. 1986 Photobehavior of the ciliated protozoon Loxodes: taxic, transient, and kinetic responses in the presence and absence of oxygen. J. Protozool. 33, 139–145.CrossRefGoogle Scholar
Feng, J., Hu, H.H. & Joseph, D.D. 1994 Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261, 95–134.CrossRefGoogle Scholar
Friedrich, B.M., Riedel-Kruse, I.H., Howard, J. & Jülicher, F. 2010 High-precision tracking of sperm swimming fine structure provides strong test of resistive force theory. J. Expl Biol. 213, 1226–1234.CrossRefGoogle ScholarPubMed
Geyer, V.F., Jülicher, F., Howard, J. & Friedrich, B.M. 2013 Cell-body rocking is a dominant mechanism for flagellar synchronization in a swimming alga. Proc. Natl Acad. Sci. USA 110, 18058–18063.CrossRefGoogle Scholar
Glowinski, R., Pan, T.-W., Hesla, T.I., Joseph, D.D. & Periaux, J. 2001 A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169, 363–426.CrossRefGoogle Scholar
Götze, I.O. & Gompper, G. 2010 Mesoscale simulations of hydrodynamic squirmer interactions. Phys. Rev. E 82, 041921.CrossRefGoogle ScholarPubMed
ten Hagen, B., Kümmel, F., Wittkowski, R., Takagi, D., Löwen, H. & Bechinger, C. 2014 Gravitaxis of asymmetric self-propelled colloidal particles. Nat. Commun. 5, 4829.CrossRefGoogle ScholarPubMed
Happel, J. & Brenner, H. 1965 Low Reynolds Number Hydrodynamics. Prentice-Hall.Google Scholar
Hill, N.A. & Pedley, T.J. 2005 Bioconvection. Fluid Dyn. Res. 37, 1–20.CrossRefGoogle Scholar
Hu, J., Wysock, A., Winkler, R.G. & Gompper, G. 2015 Physical sensing of surface properties by microswimmers – directing bacterial motion via wall slip. Sci. Rep. 5, 9586.CrossRefGoogle ScholarPubMed
Hwang, Y. & Pedley, T.J. 2014 Bioconvection under uniform shear: linear stability analysis. J. Fluid Mech. 738, 522–562.CrossRefGoogle Scholar
Ishikawa, T. & Hota, M. 2006 Interaction of two swimming Paramecia. J. Expl Biol. 209, 4452–4463.CrossRefGoogle ScholarPubMed
Ishikawa, T., Locsei, J.T. & Pedley, T.J. 2008 Development of coherent structures in concentrated suspensions of swimming model micro-organisms. J. Fluid Mech. 615, 401–431.CrossRefGoogle Scholar
Ishikawa, T., Sekiya, G., Imai, Y. & Yamaguchi, T. 2007 Hydrodynamic interactions between two swimming bacteria. Biophys. J. 93, 2217–2225.CrossRefGoogle ScholarPubMed
Ishikawa, T., Simmonds, M.P. & Pedley, T.J. 2006 Hydrodynamic interaction of two swimming model micro-organisms. J. Fluid Mech. 568, 119–160.CrossRefGoogle Scholar
Jalali, M.A., Khoshnood, A. & Alam, M.-R. 2015 Microswimmer-induced chaotic mixing. J. Fluid Mech. 779, 669–683.CrossRefGoogle Scholar
Khair, A.S. & Chisholm, N.G. 2014 Expansions at small Reynolds numbers for the locomotion of a spherical squirmer. Phys. Fluids 26, 011902.CrossRefGoogle Scholar
Klindt, G.S. & Friedrich, B.M. 2015 Flagellar swimmers oscillate between pusher- and puller-type swimming. Phys. Rev. E 92, 063019.CrossRefGoogle ScholarPubMed
Koch, D. & Subramanian, G. 2011 Collective hydrodynamics of swimming microorganisms: living fluids. Annu. Rev. Fluid Mech. 43, 637–659.CrossRefGoogle Scholar
Kyoya, K., Matsunaga, D., Imai, Y., Omori, T. & Ishikawa, T. 2015 Shape matters: near-field fluid mechanics dominate the collective motions of ellipsoidal squirmers. Phys. Rev. E 92, 063027.CrossRefGoogle ScholarPubMed
Lallemand, P. & Luo, L.S. 2003 Lattice Boltzmann method for moving boundaries. J. Comput. Phys. 184, 406–421.CrossRefGoogle Scholar
Lauga, E. & Powers, T. 2009 The hydrodynamics of swimming microorganisms. Rep. Prog. Phys. 72, 096601.CrossRefGoogle Scholar
Li, G.J. & Ardekani, A.M. 2014 Hydrodynamic interaction of micorswimmers near a wall. Phys. Rev. E 90, 013010.CrossRefGoogle ScholarPubMed
Li, G.J., Ostace, A. & Ardekani, A.M. 2016 Hydrodynamic interaction of swimming organisms in an inertial regime. Phys. Rev. E 94, 053104.CrossRefGoogle 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 5, 109–118.CrossRefGoogle Scholar
Lin, Z.W. & Gao, T. 2019 Direct-forcing fictitious domain method for simulating non-Brownian active particles. Phys. Rev. E 100, 013304.CrossRefGoogle ScholarPubMed
Lushi, E., Wioland, H. & Goldstein, R. 2014 Fluid flows generated by swimming bacteria drive self-organization in confined fluid suspensions. Proc. Natl Acad. Sci. USA 111, 9733–9738.CrossRefGoogle Scholar
Matas-Navarro, R., Golestanian, R., Liverpool, T.B. & Fielding, S.M. 2014 Hydrodynamic suppression of phase separation in active suspensions. Phys. Rev. E 90, 032304.CrossRefGoogle ScholarPubMed
Mei, R., Yu, D., Shyy, W. & Luo, L.-S. 2002 Force evaluation in the lattice Boltzmann method involving curved geometry. Phys. Rev. E 65, 041203.CrossRefGoogle ScholarPubMed
Mirzakhanloo, M., Jalali, M.A. & Alam, M.R. 2018 Hydrodynamic choreographies of microswimmers. Sci. Rep. 8, 3670.CrossRefGoogle ScholarPubMed
Mogami, Y., Ishii, J. & Baba, S.A. 2001 Theoretical and experimental dissection of gravity-dependent mechanical orientation in gravitactic microorganisms. Biol. Bull. 201, 26–33.CrossRefGoogle ScholarPubMed
More, R.V. & Ardekani, A.M. 2021 Hydrodynamic interactions between swimming microorganisms in a linearly density stratified fluid. Phys. Rev. E 103, 013109.CrossRefGoogle Scholar
Najafi, A., Raad, S.S.H. & Yousefi, R. 2013 Self-propulsion in a low-Reynolds-number fluid confined by two walls of a microchannel. Phys. Rev. E 88, 045001.CrossRefGoogle Scholar
Nie, D.M., Guan, G. & Lin, J.Z. 2021 Interaction between two unequal particles at intermediate Reynolds numbers: a pattern of horizontal oscillatory motion. Phys. Rev. E 103, 013105.CrossRefGoogle Scholar
Nie, D.M. & Lin, J.Z. 2019 Discontinuity in the sedimentation system with two particles having different densities in a vertical channel. Phys. Rev. E 99, 053112.CrossRefGoogle Scholar
Nie, D.M. & Lin, J.Z. 2020 Simulation of sedimentation of two spheres with different densities in a square tube. J. Fluid Mech. 896, A12.CrossRefGoogle Scholar
Ntefidou, M., Iseki, M., Watanabe, M., Lebert, M. & Häder, D.-P. 2003 Photoactivated adenylyl cyclase controls phototaxis in the flagellate Euglena gracilis. Plant Physiol. 133, 1517–1521.CrossRefGoogle ScholarPubMed
Ouyang, Z.Y. & Lin, J.Z. 2022 Behaviors of a settling microswimmer in a narrow vertical channel. Powder Technol. 398, 117042.CrossRefGoogle Scholar
Ouyang, Z.Y., Lin, J.Z. & Ku, X.K. 2018 The hydrodynamic behavior of a squirmer swimming in power-law fluid. Phys. Fluids 30, 083301.CrossRefGoogle Scholar
Papavassiliou, D. & Alexander, G.P. 2017 Exact solutions for hydrodynamic interactions of two squirming spheres. J. Fluid Mech. 813, 618–646.CrossRefGoogle Scholar
Pedley, T.J. & Kessler, J.O. 1992 Hydrodynamic phenomena in suspensions of swimming microorganisms. Annu. Rev. Fluid Mech. 24, 313–358.CrossRefGoogle Scholar
Polin, M., Tuval, I., Drescher, K., Gollub, J.P. & Goldstein, R.E. 2009 Chlamydomonas swims with two “gears” in a eukaryotic version of run-and-tumble locomotion. Science 325, 487–490.CrossRefGoogle Scholar
Pooley, C.M., Alexander, G.P. & Yeomans, J.M. 2012 Hydrodynamic interaction between two swimmers at low Reynolds number. Phys. Rev. Lett. 99, 228103.CrossRefGoogle Scholar
Qi, K., Westphal, E., Gompper, G. & Winkler, R.G. 2022 Emergence of active turbulence in microswimmer suspensions due to active hydrodynamic stress and volume exclusion. Commun. Phys. 5, 49.CrossRefGoogle Scholar
Qian, Y.H. 1992 Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17, 479–484.CrossRefGoogle Scholar
Roberts, A.M. 2010 The mechanics of gravitaxis in Paramecium. J. Expl Biol. 213, 4158–4162.CrossRefGoogle ScholarPubMed
Rühle, F., Blaschke, J., Kuhr, J.T. & Stark, H. 2018 Gravity-induced dynamics of a squirmer microswimmer in wall proximity. New J. Phys. 20, 025003.CrossRefGoogle Scholar
Saintillan, D. 2018 Rheology of active fluids. Annu. Rev. Fluid Mech. 50, 563–592.CrossRefGoogle Scholar
Sengupta, A., Carrara, F. & Stocker, R. 2017 Phytoplankton can actively diversify their migration strategy in response to turbulent cues. Nature 543, 555–558.CrossRefGoogle ScholarPubMed
Shen, Z., Würger, A. & Lintuvuori, J.S. 2018 Hydrodynamic interaction of a self-propelling particle with a wall. Eur. Phys. J. E 41, 39.CrossRefGoogle ScholarPubMed
Sokolov, A., Apodaca, M.M., Grzybowski, B.A. & Aranson, I.S. 2009 Swimming bacteria power microscopic gears. Proc. Natl Acad. Sci. USA 107, 969–974.CrossRefGoogle ScholarPubMed
Sokolov, A. & Aranson, I.S. 2012 Physical properties of collective motion in suspensions of bacteria. Phys. Rev. Lett. 109, 248109.CrossRefGoogle ScholarPubMed
Theers, M., Westphal, E., Gompper, G. & Winkler, R.G. 2016 Modeling a spheroidal microswimmer and cooperative swimming in a narrow slit. Soft Matt. 35, 7372–7385.CrossRefGoogle Scholar
Theers, M., Westphal, E., Qi, K., Winkler, R.G. & Gompper, G. 2018 Clustering of microswimmers: interplay of shape and hydrodynamics. Soft Matt. 14, 8590–8603.CrossRefGoogle ScholarPubMed
Wang, S. & Ardekani, A. 2012 Inertial squirmer. Phys. Fluids 24, 101902.CrossRefGoogle Scholar
Wang, S. & Ardekani, A. 2015 Biogenic mixing induced by intermediate Reynolds number swimming in stratified fluids. Sci. Rep. 5, 17448.CrossRefGoogle ScholarPubMed
Wioland, H., Lushi, E. & Goldstein, R. 2016 Directed collective motion of bacterial under channel confinement. New J. Phys. 18, 075002.CrossRefGoogle Scholar
Wolff, K., Hahn, A.M. & Stark, H. 2013 Sedimentation and polar order of active bottom-heavy particles. Eur. Phys. J. E 36, 43.CrossRefGoogle ScholarPubMed