Journal of Mathematical Biology

, Volume 62, Issue 5, pp 707–740 | Cite as

Effective shear viscosity and dynamics of suspensions of micro-swimmers from small to moderate concentrations

  • V. Gyrya
  • K. Lipnikov
  • I. S. Aranson
  • L. Berlyand
Article
First Online:
Received:
Revised:

Abstract

Recently, there has been a number of experimental studies convincingly demonstrating that a suspension of self-propelled bacteria (microswimmers in general) may have an effective viscosity significantly smaller than the viscosity of the ambient fluid. This is in sharp contrast with suspensions of hard passive inclusions, whose presence always increases the viscosity. Here we present a 2D model for a suspension of microswimmers in a fluid and analyze it analytically in the dilute regime (no swimmer–swimmer interactions) and numerically using a Mimetic Finite Difference discretization. Our analysis shows that in the dilute regime (in the absence of rotational diffusion) the effective shear viscosity is not affected by self-propulsion. But at the moderate concentrations (due to swimmer–swimmer interactions) the effective viscosity decreases linearly as a function of the propulsion strength of the swimmers. These findings prove that (i) a physically observable decrease of viscosity for a suspension of self-propelled microswimmers can be explained purely by hydrodynamic interactions and (ii) self-propulsion and interaction of swimmers are both essential to the reduction of the effective shear viscosity. We also performed a number of numerical experiments analyzing the dynamics of swimmers resulting from pairwise interactions. The numerical results agree with the physically observed phenomena (e.g., attraction of swimmer to swimmer and swimmer to the wall). This is viewed as an additional validation of the model and the numerical scheme.

Mathematics Subject Classification (2000)

35 76 92 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aranson IS, Tsimring LS (2006) Patterns and collective behavior in granular media: theoretical concepts. Rev Mod Phys 78: 641–692CrossRefGoogle Scholar
  2. Batchelor GK (1970) The stress system in a suspension of force-free particles. J Fluid Mech 41: 545–570CrossRefMATHMathSciNetGoogle Scholar
  3. Batchelor GK, Green JT (1972) The determination of the bulk stress in a suspension of spherical particles to order c 2. J Fluid Mech 56: 401–427CrossRefMATHGoogle Scholar
  4. Beirão da Veiga L, Gyrya V, Lipnikov K, Manzini G (2009) Mimetic finite difference method for the Stokes problem on polygonal meshes. J Comput Phys 228(19): 7215–7232CrossRefMATHMathSciNetGoogle Scholar
  5. Berlyand L, Borcea L, Panchenko A (2005) Network approximation for effective viscosity of concentrated suspensions with complex geometry. SIAM J Math Anal 36(5): 1580–1628 (electronic)CrossRefMATHMathSciNetGoogle Scholar
  6. Berlyand L, Panchenko A (2007) Strong and weak blow up of the viscous dissipation rates for concentrated suspensions. J Fluid Mech 578: 1–34CrossRefMATHMathSciNetGoogle Scholar
  7. Berlyand L, Gorb Y, Novikov A (2009) Fictitious fluid approach and anomalous blow-up of the dissipation rate in a 2D model of concentrated suspensions. Arch Ration Mech Anal 193(3): 585–622CrossRefMATHMathSciNetGoogle Scholar
  8. Berke AP, Turner L, Berg HC, Lauga E (2008) Hydrodynamic attraction of swimming microorganisms by surfaces. Phys Rev Lett 101: 038102CrossRefGoogle Scholar
  9. Brezzi F, Lipnikov K, Shashkov M, Simoncini V (2007) A new discretization methodology for diffusion problems on generalized polyhedral meshes. Comput Methods Appl Mech Eng 196: 3682–3692CrossRefMATHMathSciNetGoogle Scholar
  10. Cates ME, Fielding SM, Marenduzzo D, Orlandini E, Yeomans JM (2008) Shearing active gels close to the isotropic-nematic transition. Phys Rev Lett 2008:068102:1–4Google Scholar
  11. Dombrowski C, Cisneros L, Chatkaew S, Goldstein RE, Kessler JO (2004) Self-concentration and large-scale coherence in bacterial dynamics. Phys Rev Lett 93(9):098103:1–4Google Scholar
  12. Dreyfus R, Baudry J, Roper ML, Fermigier M, Stone HA, Bibette J (2007) Microscopic artificial swimmers. Nature 437(7060): 862–865CrossRefGoogle Scholar
  13. Duvaut G, Lions JL (1976) Inequalities in mechanics and physics. Springer, BerlinMATHGoogle Scholar
  14. Drescher K, Leptos KC, Tuval I, Ishikawa T, Pedley TJ, Goldstein RE (2009) Dancing volvox: hydrodynamic bound states of swimming algae. Phys Rev Lett 102(16): 168101CrossRefGoogle Scholar
  15. Einstein A (1906) A new determination of the molecular dimensions. Ann Phys 19(2): 289–306CrossRefGoogle Scholar
  16. Galdi GP (1994) An introduction to the mathematical theory of the Navier–Stokes equations, vol I. Springer Tracts in Natural Philosophy, vol 38. Linearized steady problems. Springer, New YorkGoogle Scholar
  17. Golestanian R, Ajdari A (2008) Mechanical response of a small swimmer driven by conformational transitions. Phys Rev Lett 100: 038101CrossRefGoogle Scholar
  18. Gyrya V, Berlyand L, Aranson I, Karpeev D (2009) A model of hydrodynamic interaction between swimming bacteria. Bull Math Biol 72: 148–183CrossRefMathSciNetGoogle Scholar
  19. Haines B, Aranson I, Berlyand L, Karpeev D (2008) Effective viscosity of dilute bacterial suspensions: a two-dimensional model. Phys Biol 5:046003:1–9Google Scholar
  20. Haines BM, Sokolov A, Aranson IS, Berlyand L, Karpeev DA (2009) Three-dimensional model for the effective viscosity of bacterial suspensions. Phys Rev E 80: 041922CrossRefGoogle Scholar
  21. Hatwalne Y, Ramaswamy S, Rao M, Simha RA (2004) Rheology of active-particle suspensions. Phys Rev Lett 92: 118101CrossRefGoogle Scholar
  22. Hernandez-Ortiz J, Stoltz C, Graham M (2005) Transport and collective dynamics in suspensions of confined swimming particles. Phys Rev Lett 95:204501:1–4Google Scholar
  23. Hinch EJ, Leal LG (1972) The effect of brownian motion on the rheological properties of a suspension of non-spherical particles. J Fluid Mech 52(4): 683–712CrossRefMATHGoogle Scholar
  24. Ishikawa T, Pedley TJ (2007) The rheology of a semi-dilute suspension of swimming model micro-organisms. J Fluid Mech 588: 399–435MATHMathSciNetGoogle Scholar
  25. Ishikawa T, Pedley TJ (2007) Diffusion of swimming model micro-organisms in a semi-dilute suspension. J Fluid Mech 588: 437–462MATHMathSciNetGoogle Scholar
  26. Jeffery GB (1922) The motion of ellipsoidal particles immersed in a viscous fluid. Proc R Soc Lond Ser A 102: 161–179CrossRefGoogle Scholar
  27. Leal LG, Hinch EJ (1971) The effect of weak brownian rotations on particles in shear flow. J Fluid Mech 46(4): 685–703CrossRefMATHGoogle Scholar
  28. Leptos KC, Guasto JS, Gollub JP, Pesci AI, Goldstein RE (2009) Dynamics of enhanced tracer diffusion in suspensions of swimming eukaryotic microorganisms. Phys Rev Lett 103: 198103CrossRefGoogle Scholar
  29. Levy T, Sanchez-Palencia E (1983) Suspension of solid particles in a newtonian fluid. J Fluid Mech 56: 401–427Google Scholar
  30. Najafi A, Golestanian R (2004) Simple swimmer and low reynolds number: three linked spheres. Phys Rev E 69: 062901CrossRefGoogle Scholar
  31. Nasseri S, Phan-Thien N (1997) Hydrodynamic interaction between two nearby swimming micromachines. Comput Mech 20: 551–559CrossRefMATHGoogle Scholar
  32. Nunan KC, Keller JB (1984) Effective viscosity of periodic suspensions. J Fluid Mech 142: 269–287CrossRefMATHGoogle Scholar
  33. Purcell E (1977) Life at low Reynolds number. Am J Phys 309(45): 3–11CrossRefGoogle Scholar
  34. Saintillan S, Shelley M (2007) Orientational order and instabilities in suspensions of self-locomoting rods. Phys Rev Lett 99:058102:1–4Google Scholar
  35. Short M, Solari C, Ganguly S, Powers T, Kessler J, Goldstein R (2006) Flows driven by flagella of multicellular organisms enhance long-range molecular transport. Proc Natl Acad Sci USA 103: 8315–8319CrossRefGoogle Scholar
  36. Sokolov A, Aranson IS (2009) Reduction of viscosity in suspension of swimming bacteria. Phys Rev Lett 103: 148101CrossRefGoogle Scholar
  37. Sokolov A, Goldstein RE, Feldchtein FI, Aranson IS (2009) Enhanced mixing and spatial instability in concentrated bacterial suspensions. Phys Rev E (Statistical, Nonlinear, and Soft Matter Physics) 80(3): 031903CrossRefGoogle Scholar
  38. Sokolov A, Aranson I, Kessler J, Goldstein R (2007) Concentration dependence of the collective dynamics of swimming bacteria. Phys Rev Lett 98:158102:1–4Google Scholar
  39. Sokolov A, Apodaca MM, Grzybowski BA, Aranson IS (2010) Swimming bacteria power microscopic gears. Proc Natl Acad Sci USA 107: 969–974CrossRefGoogle Scholar
  40. Taylor G (1951) Analysis of the swimming of microscopic organisms. Proc R Soc Lond Ser A 209: 447–461CrossRefMATHGoogle Scholar
  41. Tuval I, Cisneros L, Dombrowski C, Wolgemuth CW, Kessler JO, Goldstein RE (2005) Bacterial swimming and oxygen transport near contact lines. Proc Natl Acad Sci 102: 2277–2282CrossRefGoogle Scholar
  42. Underhill PT, Hernandez-Ortiz JP, Graham MD (2008) Diffusion and spatial correlations in suspensions of swimming particles. Phys Rev Lett 100:248101–1–4Google Scholar
  43. Wu X-L, Libchaber A (2000) Particle diffusion in a quasi-two-dimensional bacterial bath. Phys Rev Lett 84: 3017–3020CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • V. Gyrya
    • 1
  • K. Lipnikov
    • 2
  • I. S. Aranson
    • 3
  • L. Berlyand
    • 1
  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Los Alamos National Laboratory, MS B284Los AlamosUSA
  3. 3.Division of Material Science, Argonne National LaboratoryArgonneUSA

Personalised recommendations