Brazilian Journal of Physics

, Volume 49, Issue 1, pp 140–150 | Cite as

Self-assembly of Droplets in a Straight Microchannel

  • Erfan KadivarEmail author
  • Mojtaba Farrokhbin
  • Fatemeh Ghasemipour


In this work, self-assembly of periodic zigzag arrangement of monodisperse droplets through a flat microfluidic channel is numerically investigated. Our numerical technique is based on the boundary element method (BEM). It is found that droplets having zigzag arrangement tend to travel to channel centerline. We exhibit that non-deformable droplets do not drift normal to the channel centerline. While, as the capillary number increases, deformable droplets tend to approach more to the center of channel and their vertical velocity component increases. Our numerical results illustrate that droplets are dragged by a constant horizontal velocity component which is governed by the continuous phase flow rate. However, this situation is completely different for the vertical velocity component of droplets. We report how the vertical velocity component of droplet towards the channel centerline depends on control parameters such as droplet size, droplet distance, initial configuration, relative orientation of droplets, and capillary number. This dependency plays an important role in estimating necessary time to reach self-assembly.


Droplet migration Deformable droplets Solid particles Microfluidics Boundary element method 



Kadivar acknowledges the support of Shiraz University of Technology Research Council.

Author Contributions

EK designed the research and developed the simulation code. MF and FG ran the simulations. EK analyzed and interpreted the results and wrote the manuscript.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflicts of interest.


  1. 1.
    N.T. Nguyen, S. Wereley. Fundamentals and Applications of Microfluidics (Artech House, Boston, 2002)zbMATHGoogle Scholar
  2. 2.
    N.T. Nguyen, Micro-magnetofluidics: interactions between magnetism and fluid flow on the microscale. Microfluid. Nanofluid. 12, 1 (2012)CrossRefGoogle Scholar
  3. 3.
    S. Hutzler, N. Peron, D. Weaire, W. drenckhan, The foam/emulsion analogy in structure and drainage. Eur. Phys. J. E. 14, 381 (2004)CrossRefGoogle Scholar
  4. 4.
    C. Priest, S. Herminghaus, R. Seemann, Controlled electrocoalescence in microfluidics: targeting a single lamella. Appl. Phys. Lett. 88, 024106 (2006)ADSCrossRefGoogle Scholar
  5. 5.
    J. Sivasamy, Y.C. Chim, T.N. Wong, N.T. Nguyen, L. Yobas, Reliable addition of reagents into microfluidic droplets. Microfluid. Nanofluid. 8, 409 (2010)CrossRefGoogle Scholar
  6. 6.
    L.G. Leal, Particle motions in a viscous fluid. Ann. Rev. Fluid Mech. 12, 435 (1980)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    P.C.H. Chan, L.G. Leal, The motion of a deformable drop in a second-order fluid. J. Fluid Mech. 92, 131 (1979)ADSCrossRefzbMATHGoogle Scholar
  8. 8.
    J.A. Stoos, S.M. Yang, L.G. Leal, Hydrodynamic interaction of a small fluid particle and a spherical drop in low-Reynolds number flow. Int. J. Multiphase Flow. 18, 1019 (1992)CrossRefzbMATHGoogle Scholar
  9. 9.
    J. Bibette, D. Morse, T. Witten, D.A. Weitz, Stability criteria for emulsions. Phys. Rev. Lett. 69, 2439 (1992)ADSCrossRefGoogle Scholar
  10. 10.
    J. Bibette, F.L. Calderon, P. Poulin, Emulsions: basic principles. Rep. Prog. Phys. 62, 696 (1999)CrossRefGoogle Scholar
  11. 11.
    J.C. Baret, Surfactants in droplet-based microfluidics. Lab Chip. 12, 422 (2012)CrossRefGoogle Scholar
  12. 12.
    J.C. Baret, F. Kleinschmidt, A.E. Harrak, A.D. Griffith, Kinetic aspects of emulsion stabilization by surfactants: a microfluidic analysis. Langmuir. 25, 6088 (2009)CrossRefGoogle Scholar
  13. 13.
    M.K. Lyon, L.G. Leal, An experimental study of the motion of concentrated suspensions in two-dimensional channel flow. Part 2. Bidisperse systems. J. Fluid Mech. 363, 57 (1998)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    S. Mortazavi, G. Tryggvason, A numerical study of the motion of drops in Poiseuille flow. Part 1. Lateral migration of one drop. J. Fluid Mech. 411, 325 (2000)ADSCrossRefzbMATHGoogle Scholar
  15. 15.
    E. Kadivar, M. Farrokhbin, . Phys. A. 479, 449 (2017)MathSciNetCrossRefGoogle Scholar
  16. 16.
    E. Kadivar, A. Alizadeh, . Eur. Phys. J. E. 40, 31 (2017)CrossRefGoogle Scholar
  17. 17.
    H.L. Goldsmith, S.G. Mason, The flow of suspensions through tubes. I. Single spheres, rods, and discs. J. Colloid Sci. 17, 448 (1962)CrossRefGoogle Scholar
  18. 18.
    P.C.H. Chan, L.G. Leal, An experimental study of drop migration in shear flow between concentric cylinders. Int. J. Multiphase Flow. 7, 83 (1981)CrossRefGoogle Scholar
  19. 19.
    J.R. Smart, D.T. Leighton, Measurement of the drift of a droplet due to the presence of a plane. Phys. Fluids A. 3, 21 (1991)ADSCrossRefGoogle Scholar
  20. 20.
    E. Kadivar, Droplet trajectories in a flat microfluidic network. Eur. J. Mech. B. Fluids. 57, 75 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    M. Seo, Z. Nie, S. Xu, P.C. Lewis, E. Kumacheva, Microfluidics: from dynamic lattices to periodic arrays of polymer disks. Langmuir. 21, 4773 (2005)CrossRefGoogle Scholar
  22. 22.
    J.P. Raven, P. Marmottant, Periodic microfluidic bubbling oscillator: insight into the stability of two-phase microflows. Phys. Rev. Lett. 97, 154501 (2006)ADSCrossRefGoogle Scholar
  23. 23.
    R. Mehrotra, N. Jing, J. Kameoka, Monodispersed polygonal water droplets in microchannel. Appl. Phys. Lett. 92, 213109 (2008)ADSCrossRefGoogle Scholar
  24. 24.
    P. Marmottant, J.P. Raven, Microfluidics with foams. Soft Matter. 5, 3385 (2009)ADSCrossRefGoogle Scholar
  25. 25.
    N. Kern, D. Weaire, A. Martin, S. Hutzler, S.J. Cox, Two-dimensional viscous froth model for foam dynamics. Phys. Rev. E. 70, 041411 (2004)ADSCrossRefGoogle Scholar
  26. 26.
    P. Garstecki, Whitesides GM Tessellation of a stripe. Phys. Rev. E. 73, 031603 (2006)ADSCrossRefGoogle Scholar
  27. 27.
    E. Kadivar, Quasistatic packings of droplets in flat microfluidic channels. Phys. A. 443, 486 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    T. Beatus, T. Tlusty, R.B. Ziv, Phonons in a one-dimensional microfluidic crystal. Nat. Phys. 2, 743 (2006)CrossRefGoogle Scholar
  29. 29.
    T. Beatus, T. Tlusty, R.B. Ziv, Burgers shock waves and sound in a 2d microfluidic droplets ensemble. Phys. Rev. Lett. 103, 114502 (2009)ADSCrossRefGoogle Scholar
  30. 30.
    M. Loewenberg, E.J. Hinch, Collision of two deformable drops in shear flow. J. Fluid Mech. 338, 299 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    K.G. Hollingsworth, M.L. Johns, Droplet migration in emulsion systems measured using MR methods. J. Colloid Interface Sci. 296, 700 (2006)ADSCrossRefGoogle Scholar
  32. 32.
    X Li, H Zhou, C Pozrikidis, A numerical study of the shearing motion of emulsions and foams. J. Fluid Mech. 286, 379404 (1995)CrossRefzbMATHGoogle Scholar
  33. 33.
    P.J.A. Janssen, M.D. Baron, P.D. Anderson, J. Blawzdziewicz, M. Loewenberg, E. Wajnryb, Collective dynamics of confined rigid spheres and deformable drops. Soft Matter. 8, 7495 (2012)ADSCrossRefGoogle Scholar
  34. 34.
    JB Fleury, UD Schiller, S Thutupalli, G Gompper, R Seemann, Mode coupling of phonons in a dense one-dimensional microfluidic crystal. New J. Phys. 16, 063029 (2014)ADSCrossRefGoogle Scholar
  35. 35.
    K.V. McCloud, J.V. Maher, Experimental perturbations to Saffman-Taylor flow. Phys. Rep. 260, 139 (1995)ADSCrossRefGoogle Scholar
  36. 36.
    C. Liu, Z. Li, On the validity of the Navier-Stokes equations for nanoscale liquid flows: The role of channel size. AIP Adv. 1, 032108 (2011)ADSCrossRefGoogle Scholar
  37. 37.
    C.W. Park, M. Homsy, Two-phase displacement in hele shaw cells: theory. J. Fluid Mech. 139, 291 (1984)ADSCrossRefzbMATHGoogle Scholar
  38. 38.
    C. Pozrikids. A Practical Guide to Boundary Element Methods (CRC Press, Florida, 2002)Google Scholar
  39. 39.
    T. Dessup, T. Maimbourg, C. Coste, M.S. Jean, Linear instability of a zigzag pattern. Phys. Rev. E. 91, 022908 (2015)ADSMathSciNetCrossRefGoogle Scholar
  40. 40.
    E Kadivar, S Herminghaus, M Brinkmann, Droplet sorting in a loop of flat microfluidic channels. J. Phys. Condens. Matter. 25, 285102 (2013)CrossRefGoogle Scholar
  41. 41.
    R. Mehrotra, Monodispersed polygonal water droplets in microchannel, Texas A&M University (2008)Google Scholar
  42. 42.
    A. Karins, S. Mason, Particle motions in sheared suspensions: XXIII. Wall migration of fluid drops. J. Colloid Interface Sci. 24, 164 (1967)ADSCrossRefGoogle Scholar
  43. 43.
    B. Kaoui, T. Krüger, J. Harting, How does confinement affect the dynamics of viscous vesicles and red blood cells? Soft Matter. 8, 9246 (2012)ADSCrossRefGoogle Scholar

Copyright information

© Sociedade Brasileira de Física 2018

Authors and Affiliations

  • Erfan Kadivar
    • 1
    Email author
  • Mojtaba Farrokhbin
    • 2
  • Fatemeh Ghasemipour
    • 3
  1. 1.Department of PhysicsShiraz University of TechnologyShirazIran
  2. 2.Department of Physics, Faculty of SciencesYazd UniversityYazdIran
  3. 3.Department of PhysicsShahid Beheshti UniversityTehranIran

Personalised recommendations