Secondary flows of viscoelastic fluids in serpentine microchannels

  • Lucie Ducloué
  • Laura Casanellas
  • Simon J. Haward
  • Robert J. Poole
  • Manuel A. Alves
  • Sandra Lerouge
  • Amy Q. Shen
  • Anke LindnerEmail author
Research Paper


Secondary flows are ubiquitous in channel flows, where small velocity components perpendicular to the main velocity appear due to the complexity of the channel geometry and/or that of the flow itself such as from inertial or non-Newtonian effects. We investigate here the inertialess secondary flow of viscoelastic fluids in curved microchannels of rectangular cross-section and constant but alternating curvature: the so-called “serpentine channel” geometry. Numerical calculations (Poole et al. J Non-Newton Fluid Mech 201:10–16, 2013) have shown that in this geometry, in the absence of elastic instabilities, a steady secondary flow develops that takes the shape of two counter-rotating vortices in the plane of the channel cross-section. We present the first experimental visualization evidence and characterisation of these steady secondary flows, using the complementary techniques of quantitative microparticle image velocimetry in the centreplane of the channel, and confocal visualisation of dye-stream transport in the cross-sectional plane. We show that the measured streamlines and the relative velocity magnitude of the secondary flows are in qualitative agreement with the numerical results. In addition to our techniques being broadly applicable to the characterisation of three-dimensional flow structures in microchannels, our results are important for understanding the onset of instability in serpentine viscoelastic flows.


Polymer solutions Non-Newtonian fluids Vortices Confocal microscopy Particle image velocimetry 



AL and LD acknowledge funding from the ERC Consolidator Grant PaDyFlow (Grant Agreement no. 682367). RJP acknowledges funding for a “Fellowship” in Complex Fluids and Rheology from the Engineering and Physical Sciences Research Council (EPSRC, UK) under grant number EP/M025187/1, and support from Chaire Total. SJH, AQS and LD gratefully acknowledge the support of the Okinawa Institute of Science and Technology Graduate University (OIST) with subsidy funding from the Cabinet Office, Government of Japan, and funding from the Japan Society for the Promotion of Science (Grant nos. 17K06173, 18H01135 and 18K03958). SL acknowledges funding from the Institut Universitaire de France. We would also like to acknowledge discussions on the nature of the secondary flow with Philipp Bohr and Christian Wagner. This work has received the support of Institut Pierre-Gilles de Gennes (Équipement d’Excellence, “Investissements d’avenir”, program ANR-10-EQPX-34).


  1. Afik E, Steinberg V (2017) On the role of initial velocities in pair dispersion in a microfluidic chaotic flow. Nature Commun 8(1):468CrossRefGoogle Scholar
  2. Afonso AM, Oliveira PJ, Pinho FT, Alves MA (2009) The log-conformation tensor approach in the finite-volume method framework. J Non-Newton Fluid Mech 157(1–2):55–65CrossRefGoogle Scholar
  3. Afonso AM, Oliveira PJ, Pinho FT, Alves MA (2011) Dynamics of high-Deborah-number entry flows: a numerical study. J Fluid Mech 677:272–304MathSciNetCrossRefGoogle Scholar
  4. Alves MA, Oliveira PJ, Pinho FT (2003a) A convergent and universally bounded interpolation scheme for the treatment of advection. Int J Numer Meth Fluids 41:47–75CrossRefGoogle Scholar
  5. Alves MA, Oliveira PJ, Pinho FT (2003b) Benchmark solutions for the flow of Oldroyd-B and PTT fluids in planar contractions. J Non-Newton Fluid Mech 110:45–75CrossRefGoogle Scholar
  6. Amini H, Sollier E, Masaeli M, Xie Y, Ganapathysubramanian B, Stone HA, Di Carlo D (2013) Engineering fluid flow using sequenced microstructures. Nature Commun 4:1826CrossRefGoogle Scholar
  7. Amini H, Lee W, Di Carlo D (2014) Inertial microfluidic physics. Lab Chip 14(15):2739–2761CrossRefGoogle Scholar
  8. Arratia PE, Thomas C, Diorio J, Gollub JP (2006) Elastic instabilities of polymer solutions in cross-channel flow. Phys Rev Lett 96(14):144502CrossRefGoogle Scholar
  9. Bird RB, Armstrong RC, Hassager O (1987) Dynamics of polymeric liquids, 2nd edn. Wiley, HobokenGoogle Scholar
  10. Bohr P (2015) Experimental study of secondary vortex flows in viscoelastic fluids. Dissertation, Saarland UniversityGoogle Scholar
  11. Burshtein N, Zografos K, Shen AQ, Poole RJ, Haward SJ (2017) Inertioelastic flow instability at a stagnation point. Phys Rev X 7(4):1–18Google Scholar
  12. Casanellas L, Alves MA, Poole RJ, Lerouge S, Lindner A (2016) The stabilizing effect of shear thinning on the onset of purely elastic instabilities in serpentine microflows. Soft Matter 12(29):6167–6175CrossRefGoogle Scholar
  13. Dean W (1928) Lxxii, the stream-line motion of fluid in a curved pipe (second paper). Lond Edinb Dublin Philos Mag J Sci 5(30):673–695CrossRefGoogle Scholar
  14. Dean WR (1927) Xvi, note on the motion of fluid in a curved pipe. Lond Edinb Dublin Philos Mag J Sci 4(20):208–223CrossRefGoogle Scholar
  15. Debbaut B, Avalosse T, Dooley J, Hughes K (1997) On the development of secondary motions in straight channels induced by the second normal stress difference: experiments and simulations. J Non-Newton Fluid Mech 69(2–3):255–271CrossRefGoogle Scholar
  16. Del Giudice F, D’Avino G, Greco F, De Santo I, Netti PA, Maffettone PL (2015) Rheometry-on-a-chip: measuring the relaxation time of a viscoelastic liquid through particle migration in microchannel flows. Lab Chip 15:783–792CrossRefGoogle Scholar
  17. Di Carlo D (2009) Inertial microfluidics. Lab Chip 9(21):3038–3046CrossRefGoogle Scholar
  18. Di Carlo D, Irimia D, Tompkins RG, Toner M (2007) Continuous inertial focusing, ordering, and separation of particles in microchannels. Proc Natl Acad Sci 104(48):18892–18897CrossRefGoogle Scholar
  19. Fan Y, Tanner RI, Phan-Thien N (2001) Fully developed viscous and viscoelastic flows in curved pipes. J Fluid Mech 440:327–357CrossRefGoogle Scholar
  20. Fani A, Camarri S, Salvetti MV (2013) Investigation of the steady engulfment regime in a three-dimensional t-mixer. Phys Fluids 25(6):064102CrossRefGoogle Scholar
  21. Furukawa R, Arauz-Lara JL, Ware BR (1991) Self-diffusion and probe diffusion in dilute and semidilute aqueous solutions of dextran. Macromolecules 24(2):599–605CrossRefGoogle Scholar
  22. Gervang B, Larsen PS (1991) Secondary flows in straight ducts of rectangular cross section. J Non-Newton Fluid Mech 39(3):217–237CrossRefGoogle Scholar
  23. Groisman A, Steinberg V (2000) Elastic turbulence in a polymer solution flow. Nature 405:53–55CrossRefGoogle Scholar
  24. Guglielmini L, Rusconi R, Lecuyer S, Stone H (2011) Three-dimensional features in low-reynolds-number confined corner flows. J Fluid Mech 668:33–57CrossRefGoogle Scholar
  25. Hardt S, Drese KS, Hessel V, Schönfeld F (2005) Passive micromixers for applications in the microreactor and \(\mu\)TAS fields. Microfluid Nanofluid 1(2):108–118CrossRefGoogle Scholar
  26. Holyst R, Bielejewska A, Szymański J, Wilk A, Patkowski A, Gapiński J, Zywociński A, Kalwarczyk T, Kalwarczyk E, Tabaka M (2009) Scaling form of viscosity at all length-scales in poly (ethylene glycol) solutions studied by fluorescence correlation spectroscopy and capillary electrophoresis. Phys Chem Chem Phys 11(40):9025–9032CrossRefGoogle Scholar
  27. Ismagilov RF, Stroock AD, Kenis PJ, Whitesides G, Stone HA (2000) Experimental and theoretical scaling laws for transverse diffusive broadening in two-phase laminar flows in microchannels. Appl Phys Lett 76(17):2376–2378CrossRefGoogle Scholar
  28. Kockmann N, Föll C, Woias P (2003) Flow regimes and mass transfer characteristics in static micromixers. Microfluid BioMEMS Med Microsyst Int Soc Opt Photon 4982:319–330CrossRefGoogle Scholar
  29. Lauga E, Stroock AD, Stone HA (2004) Three-dimensional flows in slowly varying planar geometries. Phys Fluids 16(8):3051–3062 0306572MathSciNetCrossRefGoogle Scholar
  30. Lee CY, Chang CL, Wang YN, Fu LM (2011) Microfluidic mixing: a review. Int J Mol Sci 12(5):3263–3287CrossRefGoogle Scholar
  31. Li XB, Oishi M, Oshima M, Li FC, Li SJ (2016) Measuring elasticity-induced unstable flow structures in a curved microchannel using confocal micro particle image velocimetry. Exp Therm Fluid Sci 75:118–128CrossRefGoogle Scholar
  32. Meinhart CD, Wereley ST, Gray MHB (2000) Volume illumination for two-dimensional particle image velocimetry. Meas Sci Technol 11:809–814CrossRefGoogle Scholar
  33. Mitchell P (2001) Microfluidics-downsizing large-scale biology. Nature Biotechnol 19(8):717CrossRefGoogle Scholar
  34. Mustafa MB, Tipton DL, Barkley MD, Russo PS, Blum FD (1993) Dye diffusion in isotropic and liquid-crystalline aqueous (hydroxypropyl) cellulose. Macromolecules 26(2):370–378CrossRefGoogle Scholar
  35. Oldroyd J (1950) On the formulation of rheological equations of state. Proc R Soc London A 200:523–541MathSciNetCrossRefGoogle Scholar
  36. Ottino JM (1989) The kinematics of mixing: stretching, chaos, and transport, vol 3. Cambridge University Press, CambridgezbMATHGoogle Scholar
  37. Poole RJ, Lindner A, Alves MA (2013) Viscoelastic secondary flows in serpentine channels. J Non-Newton Fluid Mech 201:10–16CrossRefGoogle Scholar
  38. Robertson AM, Muller SJ (1996) Flow of Oldroyd-B fluids in curved pipes of circular and annular cross-section. Int J Non-linear Mech 31(1):1–20CrossRefGoogle Scholar
  39. Salipante P, Hudson SD, Schmidt JW, Wright JD (2017) Microparticle tracking velocimetry as a tool for microfluidic flow measurements. Exp Fluids 58(7):1–10CrossRefGoogle Scholar
  40. Souliès A, Aubril J, Castelain C, Burghelea T (2017) Characterisation of elastic turbulence in a serpentine micro-channel. Phys Fluids 29:083102CrossRefGoogle Scholar
  41. Stroock AD, Dertinger SKW, Ajdari A, Mezić I, Stone HA, Whitesides GM (2002) Chaotic mixer for microchannels. Science 295(5555):647–651CrossRefGoogle Scholar
  42. Sznitman J, Guglielmini L, Clifton D, Scobee D, Stone HA, Smits AJ (2012) Experimental characterization of three-dimensional corner flows at low Reynolds numbers. J Fluid Mech 707:37–52CrossRefGoogle Scholar
  43. Tabeling P (2005) Introduction to microfluidics. Oxford University Press on Demand, OxfordGoogle Scholar
  44. Wereley ST, Meinhart CD (2005) Micron-resolution particle image velocimetry. In: Breuer KS (ed) Microscale diagnostic techniques. Springer, Heidelberg, pp 51–112CrossRefGoogle Scholar
  45. Xue SC, Phan-Thien N, Tanner RI (1995) Numerical study of secondary flows of viscoelastic fluid in straight pipes by an implicit finite volume method. J Non-Newton Fluid Mech 59(2–3):191–213CrossRefGoogle Scholar
  46. Zilz J, Poole RJ, Alves M, Bartolo D, Lindner BLA (2012) Geometric scaling of a purely elastic flow instability in serpentine channels. J Fluid Mech 712:203–218MathSciNetCrossRefGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR 7636, CNRS, ESPCI ParisPSL Research University, Université Paris Diderot, Sorbonne UniversitéParisFrance
  2. 2.Laboratoire Charles Coulomb UMR 5221 CNRS-UMUniversité de MontpellierMontpellier Cedex 5France
  3. 3.Okinawa Institute of Science and Technology Graduate UniversityOnnaJapan
  4. 4.School of EngineeringUniversity of LiverpoolLiverpoolUK
  5. 5.Departamento de Engenharia Química, Centro de Estudos de Fenómenos de TransporteFaculdade de Engenharia da Universidade do PortoPortoPortugal
  6. 6.Laboratoire Matière et Systèmes ComplexesCNRS UMR 75057-Université Paris DiderotParis CedexFrance

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