Rheologica Acta

, Volume 58, Issue 11–12, pp 771–779 | Cite as

Rheology of a nematic active suspension undergoing oscillatory shear and step strain flows

  • Sara Malvar
  • Bruno S. Carmo
  • Francisco R. CunhaEmail author
Original Contribution


We investigate experimentally the rheology of an active suspension of nematodes C. elegans under oscillatory shear. There are few experimental investigations and theoretical work on the oscillatory rheological properties including elastic and viscous moduli of micro-swimmer suspensions. The viscous and elastic moduli are evaluated by our experiments in the regime of linear viscoelasticity. These viscoelastic quantities are explored at low frequency given the active suspension viscosity and the shear elastic modulus. The experiments have revealed an anomalous behavior of the viscosity and the shear elastic modulus with the variation of the suspension volume fraction. The suspension relative viscosity decreased with the increase of active particles within a certain range of volume fraction. However, above a critical particle volume fraction, the relative viscosity increases. This observed increase of the viscosity for larger concentration is a direct consequence of formation of large and coherent oriented structures and active particle interactions. This collective behavior also increases the first normal stress difference N1, obtained through Cox–Merz rule. Three different regions are obtained regarding different involved mechanisms and a physical interpretation is provided based on the particles dipole stresslet. Step strain tests are carried out and the active relaxation time as a function of volume fractions are obtained. An intrinsic oscillatory behavior is observed regardless the volume fraction, showing the non-equilibrium condition of the active suspension.


Collective behavior Active suspension viscosity Oscillatory shear Shear elastic modulus C. elegans 



We wish to acknowledge the support of Professor Carlos Winter from the Nematode’s Molecular Biology Laboratory (ICB-USP) and Professor Vicente de Paulo Martins from Pathogens Molecular Analysis Laboratory (IB-UnB) in obtainment and cultivation of C. elegans. Also we would like to acknowledge Professor Takuji Ishikawa for the comments and discussions during the preparation of this manuscript.

Funding information

The work was supported in part by the Brazilian funding agencies FAPESP—São Paulo State Research Support Foundation (Grant No. 2016/14337-5) and CNPq (Grant Nos. 552221/2009-0/142303/2015-1).


  1. Acierno D, La Mantia F, Marrucci G (1977) A non-linear viscoelastic model with structure-dependent relaxation times: III. Comparison with LD polyethylene creep and recoil data. J Non-Newtonian Fluid Mech 209(3):271–280. CrossRefGoogle Scholar
  2. Al-Hadithi TSR, Barnes HA, Walters K (1992) The relationship between the linear (oscillatory) and nonlinear (steady-state) flow properties of a series of polymer and colloidal systems. Colloid Polym Sci 270:40–46CrossRefGoogle Scholar
  3. Batchelor G, Green J (1972) The determination of the bulk stress in a suspension of spherical particles to order c 2. J Fluid Mech 56:401–427CrossRefGoogle Scholar
  4. Batchelor GK (1977) The effect of Brownian motion on the bulk stress in a suspension of spherical particles. J Fluid Mech 83(1):97–117. CrossRefGoogle Scholar
  5. Baugh LR (2013) To grow or not to grow: nutritional control of development during Caenorhabditis elegans L1 arrest. Genetics 194:539–555CrossRefGoogle Scholar
  6. Bechtel T, Khair A (2017) Linear viscoelasticity of a dilute active suspension. Rheol Acta 56(2):149–160. CrossRefGoogle Scholar
  7. Berri S, et al. (2010) Forward locomotion of the nematode C. elegans is achieved through modulation of a single gait. HFSP J 3(3). CrossRefGoogle Scholar
  8. Bratanov V, Jenko F, Frey E (2015) New class of turbulence in active fluids. PNAS 8(112(49)):15,048–53. CrossRefGoogle Scholar
  9. Cox WP, Merz EH (1958) Correlation of dynamic and steady flow properties. J Polym Sci 28:619CrossRefGoogle Scholar
  10. Doi M, Edwards SF (1986) The theory of polymer dynamics. Clarendon Press, Oxford University PressGoogle Scholar
  11. Dunkel J, Heidenreich S, Drescher K, Wensink HH, Bar M, Goldstein RE (2013) Fluid dynamics of bacterial turbulence. Phys Rev Lett 110(22):228,102. CrossRefGoogle Scholar
  12. Einstein A (1905) Eine neue Bestimmung der Molekuledimensionen. Annalen der physik 19:289–306Google Scholar
  13. Fang-Yen C, Wyart M, Xie J, Kawai R, Kodger T, Chen S, Wen Q, Samuel ADT (2010) Biomechanical analysis of gait adaptation in the nematode Caenorhabditis elegans. PNAS 107(47). CrossRefGoogle Scholar
  14. Gupta RK (2010) Polymer and Composite Rheology. CRC PressGoogle Scholar
  15. Haines BM, Aranson IS, Berlyand L, Karpeev DA (2008) Effective viscosity of dilute bacterial suspensions: a two-dimensional model. Phys Bio 5(4):046,003. CrossRefGoogle Scholar
  16. Hatwalne Y, Ramaswamy S, Rao M, Simha RA (2004) Rheology of active-particle suspensions. Phys Rev Lett 92:118,101. CrossRefGoogle Scholar
  17. Hess S (1976) Fokker-Planck-equation approach to flow alignment in liquid crystals. Zeitschrift fur Naturforschung A 31(9):1034–1037. CrossRefGoogle Scholar
  18. Ishikawa T, Pedley TJ (2014) Dispersion of model microorganisms swimming in a nonuniform suspension. Phys Rev Lett 90(3):033,008. CrossRefGoogle Scholar
  19. Koch DL, Subramanian G (2011) Collective hydrodynamics of swimming microorganisms: living fluids. Ann Rev Fluid Mech 43:637–659CrossRefGoogle Scholar
  20. Krieger I, Dougherty T (1959) A mechanism for non-Newtonian flow in suspensions of rigid spheres. Transactions of the Society of Rheology 3(1):137–152. CrossRefGoogle Scholar
  21. LH M (1986) Prediction of elastic strains of polymer melts in shear and elongation. J Rheol 30(3):459CrossRefGoogle Scholar
  22. Malvar S, Gontijo RG, Carmo BS, Cunha FR (2017) On the kinematics-wave motion of living particles in suspension. Biomicrofluidics 11(4):044,112. CrossRefGoogle Scholar
  23. Maron S, Pierce P (1956) Application of Ree-Eyring generalized flow theory to suspensions of spherical particles. J Colloid Sci 11(1):80–95. CrossRefGoogle Scholar
  24. Mooney M (1951) The viscosity of a concentrated suspension of spherical particles. J Colloid Sci 6(2):162–170. CrossRefGoogle Scholar
  25. Novikov VV (2006) Fractals, diffusion, and relaxation in disorderedcomplex systems, Part 2, chapter 7. Wiley-InterscienceGoogle Scholar
  26. Oliveira TF, Cunha FR (2015) Emulsion rheology for steady and oscillatory shear flows at moderate and high viscosity ratio. Rheol Acta 54:951–971CrossRefGoogle Scholar
  27. Rafai S, Jibuti L, Peyla P (2010) Effective viscosity of microswimmer suspensions. Phys Rev Lett 104 (9):098,102. CrossRefGoogle Scholar
  28. Saintillan D (2010) Extensional rheology of active suspensions. Phys Rev E 81(5):056,307. CrossRefGoogle Scholar
  29. Saintillan D, Shelley MJ (2011) Emergence of coherent structures and large-scale flows in motile suspensions. J R Soc Interface 9(68):571–585. CrossRefGoogle Scholar
  30. Sokolov A, Aranson IS (2009) Reduction of viscosity in suspension of swimming bacteria. Phys Rev Lett 103(14):148,101. CrossRefGoogle Scholar
  31. Stiernagle T (2006) Maintenance of C. elegans. WormBook: The Online Review of C. elegans BiologyGoogle Scholar
  32. Takatori S, Brady J (2014) Swim pressure: stress generation in active matter. Soft Matter 100(47):9433–9445. CrossRefGoogle Scholar
  33. Wensink H, Dunkel J, Heidenreich S, Drescher K, Goldstein RE, Lowen H, Yeomans JM (2012) Meso-scale turbulence in living fluids. PNAS 109(36):14,308–14,313. CrossRefGoogle Scholar
  34. Yasuda K, Armstrong RC, Cohen RE (1981) Shear flow properties of concentrated solutions of linear and star branched polystyrenes. Rheol Acta 20(2):163–178CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Sara Malvar
    • 1
  • Bruno S. Carmo
    • 1
  • Francisco R. Cunha
    • 2
    Email author
  1. 1.Fluid and Dynamics Research Group, Department of Mechanical EngineeringEscola Politécnica of the University of São Paulo - USPSão PauloBrazil
  2. 2.Microhydrodynamics and Rheology LAB - VORTEX Research Group, Department of Mechanical Engineering, Technology FacultyUniversity of Brasília - UnBBrasíliaBrazil

Personalised recommendations