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

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Mixing by squirmers in stratified fluids

Vaseem A. Shaik1,2,* and Gwynn J. Elfring1,†

  • *Contact author: vshaik@northwestern.edu
  • †Contact author: gelfring@mech.ubc.ca
PDF

Phys. Rev. Fluids 10, 024102 – Published 24 February, 2025

DOI: https://doi.org/10.1103/PhysRevFluids.10.024102

Abstract

We find the mixing induced by a small swimming organism in a density stratified fluid. We model the swimmer as a spherical squirmer and quantify mixing through a mixing efficiency that is the ratio of the rate of change of the potential energy of the fluid to the total work done by the swimmer. Assessing mixing in the near field and far field of the swimmer separately, we find that the near-field mixing aligns with past work, but the overall mixing is much larger than that caused by a point-sized swimmer (like force dipole), although still small in weak stratification. Equivalent results are also obtained for a homogeneous dilute suspension of noninteracting swimmers. Our study highlights the impact of swimmer size on mixing, revealing that small-sized swimmers cannot induce substantial mixing. However, we propose a few (unexplored) pathways through which small swimmers could induce significant mixing, ultimately contributing toward oceanic mixing.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (85)

  1. A. P. Berke, L. Turner, H. C. Berg, and E. Lauga, Hydrodynamic attraction of swimming microorganisms by surfaces, Phys. Rev. Lett. 101, 038102 (2008).
  2. B. Ezhilan and D. Saintillan, Transport of a dilute active suspension in pressure-driven channel flow, J. Fluid Mech. 777, 482 (2015).
  3. J. Hill, O. Kalkanci, J. L. McMurry, and H. Koser, Hydrodynamic surface interactions enable Escherichia coli to seek efficient routes to swim upstream, Phys. Rev. Lett. 98, 068101 (2007).
  4. T. Kaya and H. Koser, Characterization of hydrodynamic surface interactions of Escherichia coli cell bodies in shear flow, Phys. Rev. Lett. 103, 138103 (2009).
  5. T. Kaya and H. Koser, Direct Upstream Motility in Escherichia coli, Biophys. J. 102, 1514 (2012).
  6. V. Kantsler, J. Dunkel, M. Blayney, and R. E. Goldstein, Rheotaxis facilitates upstream navigation of mammalian sperm cells, eLife 3, e02403 (2014).
  7. Z. Peng and J. F. Brady, Upstream swimming and Taylor dispersion of active Brownian particles, Phys. Rev. Fluids 5, 073102 (2020).
  8. M. E. Cates and J. Tailleur, Motility-induced phase separation, Annu. Rev. Condens. Matter Phys. 6, 219 (2015).
  9. R. Alert, J. Casademunt, and J.-F. Joanny, Active turbulence, Annu. Rev. Condens. Matter Phys. 13, 143 (2022).
  10. C. Brennen and H. Winet, Fluid mechanics of propulsion by cilia and flagella, Annu. Rev. Fluid Mech. 9, 339 (1977).
  11. H. C. Berg, E. coli in Motion (Springer, New York, 2004).
  12. L. J. Fauci and R. Dillon, Biofluidmechanics of reproduction, Annu. Rev. Fluid Mech. 38, 371 (2006).
  13. E. Lauga and T. R. Powers, The hydrodynamics of swimming microorganisms, Rep. Prog. Phys. 72, 096601 (2009).
  14. D. L. Koch and G. Subramanian, Collective hydrodynamics of swimming microorganisms: Living fluids, Annu. Rev. Fluid Mech. 43, 637 (2011).
  15. J. S. Guasto, R. Rusconi, and R. Stocker, Fluid mechanics of planktonic microorganisms, Annu. Rev. Fluid Mech. 44, 373 (2012).
  16. 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).
  17. E. Lauga, Bacterial hydrodynamics, Annu. Rev. Fluid Mech. 48, 105 (2016).
  18. E. Lauga, The Fluid Dynamics of Cell Motility (Cambridge University Press, Cambridge, 2020).
  19. W. Munk and C. Wunsch, Abyssal recipes ii: Energetics of tidal and wind mixing, Deep Sea Res. Part I: Oceanogr. Res. Papers 45, 1977 (1998).
  20. W. K. Dewar, R. J. Bingham, R. L. Iverson, D. P. Nowacek, L. C. St. Laurent, and P. H. Wiebe, Does the marine biosphere mix the ocean? J. Mar. Res. 64, 541 (2006).
  21. M. Huntley and M. Zhou, Influence of animals on turbulence in the sea, Marine Ecol. Prog. Ser. 273, 65 (2004).
  22. E. Kunze, J. F. Dower, I. Beveridge, R. Dewey, and K. P. Bartlett, Observations of biologically generated turbulence in a coastal inlet, Science 313, 1768 (2006).
  23. A. W. Visser, Biomixing of the oceans? Science 316, 838 (2007).
  24. M. C. Gregg and J. K. Horne, Turbulence, acoustic backscatter, and pelagic nekton in monterey bay, J. Phys. Oceanogr. 39, 1097 (2009).
  25. K. Katija and J. O. Dabiri, A viscosity-enhanced mechanism for biogenic ocean mixing, Nature (London) 460, 624 (2009).
  26. G. Subramanian, Viscosity-enhanced bio-mixing of the oceans, Curr. Sci. 98, 1103 (2010).
  27. A. M. Leshansky and L. M. Pismen, Do small swimmers mix the ocean? Phys. Rev. E 82, 025301(R) (2010).
  28. X.-L. Wu and A. Libchaber, Particle diffusion in a quasi-two-dimensional bacterial bath, Phys. Rev. Lett. 84, 3017 (2000).
  29. Z. Lin, J.-L. Thiffeault, and S. Childress, Stirring by squirmers, J. Fluid Mech. 669, 167 (2011).
  30. E. Kunze, Fluid mixing by swimming organisms in the low-Reynolds-number limit, J. Mar. Res. 69, 591 (2011).
  31. K. Katija, Biogenic inputs to ocean mixing, J. Exp. Biol. 215, 1040 (2012).
  32. T. Jephson and P. Carlsson, Species- and stratification-dependent diel vertical migration behaviour of three dinoflagellate species in a laboratory study, J. Plankton Res. 31, 1353 (2009).
  33. B. Bergström and J.-O. Strömberg, Behavioural differences in relation to pycnoclines during vertical migration of the euphausiids Meganyctiphanes norvegica (M. Sars) and Thysanoessa raschii (M. Sars), J. Plankton Res. 19, 255 (1997).
  34. B. S. Sherman, I. T. Webster, G. J. Jones, and R. L. Oliver, Transitions between Auhcoseira and Anabaena dominance in a turbid river weir pool, Limnol. Oceanogr. 43, 1902 (1998).
  35. V. A. Shaik and A. M. Ardekani, Drag, deformation, and drift volume associated with a drop rising in a density stratified fluid, Phys. Rev. Fluids 5, 013604 (2020).
  36. V. A. Shaik and A. M. Ardekani, Far-field flow and drift due to particles and organisms in density-stratified fluids, Phys. Rev. E 102, 063106 (2020).
  37. E. J. List, Laminar momentum jets in a stratified fluid, J. Fluid Mech. 45, 561 (1971).
  38. A. M. Ardekani and R. Stocker, Stratlets: Low reynolds number point-force solutions in a stratified fluid, Phys. Rev. Lett. 105, 084502 (2010).
  39. A. K. Varanasi and G. Subramanian, Motion of a sphere in a viscous density stratified fluid, J. Fluid Mech. 949, A29 (2022).
  40. A. Doostmohammadi, R. Stocker, and A. M. Ardekani, Low-Reynolds-number swimming at pycnoclines, Proc. Natl. Acad. Sci. USA 109, 3856 (2012).
  41. R. Dandekar, V. A. Shaik, and A. M. Ardekani, Swimming sheet in a density-stratified fluid, J. Fluid Mech. 874, 210 (2019).
  42. R. V. More and A. M. Ardekani, Motion of an inertial squirmer in a density stratified fluid, J. Fluid Mech. 905, A9 (2020).
  43. V. A. Shaik and A. M. Ardekani, Squirming in density-stratified fluids, Phys. Fluids 33, 101903 (2021).
  44. V. A. Shaik and G. J. Elfring, Densitaxis: Active particle motion in density gradients, Proc. Natl. Acad. Sci. USA 121, e2405466121 (2024).
  45. G. L. Wagner, W. R. Young, and E. Lauga, Mixing by microorganisms in stratified fluids, J. Mar. Res. 72, 47 (2014).
  46. S. Wang and A. M. Ardekani, Biogenic mixing induced by intermediate Reynolds number swimming in stratified fluids, Sci. Rep. 5, 17448 (2015).
  47. R. Tait and M. Howe, Some observations of thermo-haline stratification in the deep ocean, Deep Sea Res. Oceanogr. Abstr. 15, 275 (1968).
  48. V. T. Neal, S. Neshyba, and W. Denner, Thermal stratification in the arctic ocean, Science 166, 373 (1969).
  49. M. Howe and R. Tait, Further observations of thermo-haline stratification in the deep ocean, Deep Sea Res. Oceanogr. Abstr. 17, 963 (1970).
  50. R. I. Tait and M. R. Howe, Thermohaline staircase, Nature (London) 231, 178 (1971).
  51. S. Neshyba, V. T. Neal, and W. Denner, Temperature and conductivity measurements under ice island t-3, J. Geophys. Res. 76, 8107 (1971).
  52. O. Phillips, Turbulence in a strongly stratified fluid—is it unstable? Deep Sea Res. Oceanogr. Abstr. 19, 79 (1972).
  53. E. S. Posmentier, The generation of salinity finestructure by vertical diffusion, J. Phys. Oceanogr. 7, 298 (1977).
  54. L. Padman and T. M. Dillon, Vertical heat fluxes through the beaufort sea thermohaline staircase, J. Geophys. Res.: Oceans 92, 10799 (1987).
  55. W. J. Merryfield, Origin of thermohaline staircases, J. Phys. Oceanogr. 30, 1046 (2000).
  56. M. Timmermans, J. Toole, R. Krishfield, and P. Winsor, Ice-tethered profiler observations of the double-diffusive staircase in the canada basin thermocline, J. Geophys. Res.: Oceans 113, C00A02 (2008).
  57. N. Petropoulos, A. Mashayek, and C.-c. P. Caulfield, Turbulent disruption of density staircases in stratified shear flows, J. Fluid Mech. 961, A30 (2023).
  58. D. D. Gray and A. Giorgini, The validity of the boussinesq approximation for liquids and gases, Int. J. Heat Mass Transf. 19, 545 (1976).
  59. L. G. Leal, Advanced Transport Phenomena (Cambridge University Press, Cambridge, 2007).
  60. F. Candelier, R. Mehaddi, and O. Vauquelin, The history force on a small particle in a linearly stratified fluid, J. Fluid Mech. 749, 184 (2014).
  61. 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).
  62. J. R. Blake, A spherical envelope approach to ciliary propulsion, J. Fluid Mech. 46, 199 (1971).
  63. T. Ishikawa, M. P. Simmonds, and T. J. Pedley, Hydrodynamic interaction of two swimming model micro-organisms, J. Fluid Mech. 568, 119 (2006).
  64. G.-J. Li and A. M. Ardekani, Hydrodynamic interaction of microswimmers near a wall, Phys. Rev. E 90, 013010 (2014).
  65. V. A. Shaik and A. M. Ardekani, Motion of a model swimmer near a weakly deforming interface, J. Fluid Mech. 824, 42 (2017).
  66. L. Zhu, M. Do-Quang, E. Lauga, and L. Brandt, Locomotion by tangential deformation in a polymeric fluid, Phys. Rev. E 83, 011901 (2011).
  67. L. Zhu, E. Lauga, and L. Brandt, Self-propulsion in viscoelastic fluids: Pushers vs. pullers, Phys. Fluids 24, 051902 (2012).
  68. G. J. Li, A. Karimi, and A. M. Ardekani, Effect of solid boundaries on swimming dynamics of microorganisms in a viscoelastic fluid, Rheol. Acta 53, 911 (2014).
  69. S. Yazdi, A. M. Ardekani, and A. Borhan, Locomotion of microorganisms near a no-slip boundary in a viscoelastic fluid, Phys. Rev. E 90, 043002 (2014).
  70. S. Yazdi, A. M. Ardekani, and A. Borhan, Swimming dynamics near a wall in a weakly elastic fluid, J. Nonlinear Sci. 25, 1153 (2015).
  71. S. Yazdi and A. Borhan, Effect of a planar interface on time-averaged locomotion of a spherical squirmer in a viscoelastic fluid, Phys. Fluids 29, 093104 (2017).
  72. S. Wang and A. Ardekani, Inertial squirmer, Phys. Fluids 24, 101902 (2012).
  73. A. S. Khair and N. G. Chisholm, Expansions at small Reynolds numbers for the locomotion of a spherical squirmer, Phys. Fluids 26, 011902 (2014).
  74. N. G. Chisholm, D. Legendre, E. Lauga, and A. S. Khair, A squirmer across Reynolds numbers, J. Fluid Mech. 796, 233 (2016).
  75. H. Lee, I. Fouxon, and C. Lee, Sedimentation of a small sphere in stratified fluid, Phys. Rev. Fluids 4, 104101 (2019).
  76. A. K. Varanasi, N. K. Marath, and G. Subramanian, The rotation of a sedimenting spheroidal particle in a linearly stratified fluid, J. Fluid Mech. 933, A17 (2022).
  77. T. Villareal and E. Carpenter, Buoyancy regulation and the potential for vertical migration in the oceanic cyanobacterium Trichodesmium, Microb. Ecol. 45, 1 (2003).
  78. C. Boyd and D. Gradmann, Impact of osmolytes on buoyancy of marine phytoplankton, Mar. Biol. 141, 605 (2002).
  79. F. J. Sartoris, D. N. Thomas, A. Cornils, and S. B. S. Schiela, Buoyancy and diapause in Antarctic copepods: The role of ammonium accumulation, Limnol. Oceanogr. 55, 1860 (2010).
  80. A. C. King, J. Billingham, and S. R. Otto, Differential Equations. Linear, Nonlinear, Ordinary, Partial (Cambridge University Press, Cambridge, 2003).
  81. E. Villermaux, Mixing versus stirring, Annu. Rev. Fluid Mech. 51, 245 (2019).
  82. W. R. Peltier and C. P. Caulfield, Mixing efficiency in stratified shear flows, Annu. Rev. Fluid Mech. 35, 135 (2003).
  83. C. Caulfield, Layering, instabilities, and mixing in turbulent stratified flows, Annu. Rev. Fluid Mech. 53, 113 (2021).
  84. M. Gregg, E. D'Asaro, J. Riley, and E. Kunze, Mixing efficiency in the ocean, Annu. Rev. Mar. Sci. 10, 443 (2018).
  85. J. Gong, V. A. Shaik, and G. J. Elfring, Active particles crossing sharp viscosity gradients, Sci. Rep. 13, 596 (2023).

Outline

Information

Phys. Rev. Fluids 10, 024102– Published 24 February, 2025
  • Received 29 June 2024
  • Accepted 29 January 2025
Crossmark - Check for updates button

©2025 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation