Zeitschrift für angewandte Mathematik und Physik

, Volume 66, Issue 1, pp 149–169 | Cite as

Stokes flow in a two-dimensional micro-device combined by a cross-slot and a microfluidic four-roll mill

  • Jing Guan
  • Jinxia Liu
  • Xiaoduan Li
  • Jun Tao
  • Jingtao WangEmail author


The flow structures in a novel microfluidics device (CS-MFRM) combining a cross-slot (CS) and a microfluidics four-roll mill (MFRM) have been investigated through a two-dimensional boundary element method. By changing the volume flow rates at various inlets of a CS-MFRM, diverse flow structures can be generated. Some of them are proposed to be employed to achieve some functions in the fabrication process of anisotropic particles. The stagnant points and eddies in those flows are particularly discussed since they are critical to trap and/or rotate droplets. Energy consumption of eddies generated in branches in some flow structures is also investigated in this paper.

Mathematics Subject Classification (2000)

76D07 76T25 

List of symbols


The viscosity of continuous phase


The viscosity ratio of the droplet to the continuous phase


Shear rate




Velocity vector


Surface stress


Fundamental solution of the two-dimensional Stokes equations


Associated stress kernel of the fundamental solution


Volume flow rate in a channel


Half width of a channel


Radius of central circular cavity


Boundary element method Microchannels Stagnation points Flow branches Stokes eddies 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Helen S., Delai L.C., Rustem F.I.: Reactions in droplets in microfluidic channels. Angew. Chem. Int. Ed. 45, 7336–7356 (2006)CrossRefGoogle Scholar
  2. 2.
    Whitesides G.M.: The origins and the future of microfluidics. Nature 442, 368–373 (2006)CrossRefGoogle Scholar
  3. 3.
    Teh S.Y., Lin R., Hung L.H., Lee A.P.: Droplet microfluidics. Lab Chip 8, 198–220 (2008)CrossRefGoogle Scholar
  4. 4.
    Wang J.T., Wang J., Han J.J.: Fabrication of advanced particles and particle-based materials assisted by droplet-based microfluidics. Small 7, 1728–1754 (2011)CrossRefGoogle Scholar
  5. 5.
    Lingxin C., Sangyeop L., Jaebum K.L., Eun C.: Continuous dynamic flow micropumps for microfluid manipulation. J. Micromech. Microeng. 18, 013001 (2008)CrossRefGoogle Scholar
  6. 6.
    Um E., Park J.K.: A microfluidic abacus channel for controlling the addition of droplets. Lab Chip 9, 207–212 (2009)CrossRefGoogle Scholar
  7. 7.
    Dendukuri D., Doyle P.S.: The synthesis and assembly of polymeric microparticles using microfluidics. Adv. Mater. 21, 1–16 (2009)CrossRefGoogle Scholar
  8. 8.
    Aubin J., Ferrando M., Jiricny V.: Current methods for characterising mixing and flow in microchannels. Chem. Eng. Sci. 65, 2065–2093 (2010)CrossRefGoogle Scholar
  9. 9.
    Hao G., Michel H.G.D., Frieder M.: Droplets formation and merging in two-phase flow microfluidics. Int. J. Mol. Sci. 12, 2572–2597 (2011)CrossRefGoogle Scholar
  10. 10.
    Seemann R., Brinkmann M., Pfohl T., Herminghaus S.: Droplet based microfluidics. Rep. Prog. Phys. 75, 016601 (2012)CrossRefGoogle Scholar
  11. 11.
    Tanyeri M., Johnson-Chavarria E.M., Schroeder C.M.: Hydrodynamic trap for single particles and cells. Appl. Phys. Lett. 96, 224101 (2010)CrossRefGoogle Scholar
  12. 12.
    Janssen J.J.M., Boon A., Agterof W.G.M.: Influence of dynamic interfacial properties on droplet breakup in plane hyperbolic flow. AICHE J. 43, 1436–1447 (1997)CrossRefGoogle Scholar
  13. 13.
    Perkins T.T., Smith D.E., Chu S.: Single polymer dynamics in an elongational flow. Science 276, 2016–2021 (1997)CrossRefGoogle Scholar
  14. 14.
    Schroeder C.M., Babcock H.P.E., Shaqfeh S.G., Chu S.: Observation of polymer conformation hysteresis in extensional flow. Science 301, 1515–1519 (2003)CrossRefGoogle Scholar
  15. 15.
    Hellou M., Bach T.D.P.: Stokes flow in a junction of two-dimensional orthogonal channels. Z. Angew. Math. Phys. 62, 135–147 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Lee J.S., Dylla-Spears R., Teclemariam N.P., Muller S.J.: Microfluidic four-roll mill for all flow types. Appl. Phys. Lett. 90, 074103 (2007)CrossRefGoogle Scholar
  17. 17.
    Wang J.T., Han J.J., Yu D.M.: Numerical studies of geometry effects of a two-dimensional microfluidic four-roll mill on droplet elongation and rotation. Eng. Anal. Bound. Elem. 36, 1453–1464 (2012)CrossRefGoogle Scholar
  18. 18.
    Lee J.S., Shaqfeh E.S.G., Muller S.J.: Dynamics of DNA tumbling in shear to rotational mixed flows: pathways and periods. Phys. Rev. E 75, 040802 (2007)CrossRefGoogle Scholar
  19. 19.
    Deschamps J., Kantsler V., Segre E., Steinberg V.: Dynamics of a vesicle in general flow. PNAS 106, 11444–11447 (2009)CrossRefzbMATHGoogle Scholar
  20. 20.
    Young Y.N., Blawzdziewicz J., Cristini V., Goodman R.H.: Hysteretic and chaotic dynamics of viscous drops in creeping flows with rotation. J Fluid Mech. 607, 209–234 (2008)zbMATHMathSciNetGoogle Scholar
  21. 21.
    Wang, J.T., Tao, J., Han, J.J.: Hydrodynamic Control of Droplets Coalescence in Microfluidic Devices to Fabricate Anisotropic Particles Through Boundary Element Method (unpublished)Google Scholar
  22. 22.
    Shankar P.N.: The eddy structure in Stokes flow in a cavity. J. Fluid Mech. 250, 371–383 (1993)CrossRefzbMATHGoogle Scholar
  23. 23.
    Georgiadou M., Mohr R., Alkire R.C.: Local mass transport in two-dimensional cavities in laminar shear flow. J. Electrochem. Soc. 147, 3021–3028 (2000)CrossRefGoogle Scholar
  24. 24.
    Lutz B.R., Chen J., Schwartz D.T.: Hydrodynamic Tweezers: 1. Non-contact cell trapping in a laminar oscillating flow. Anal. Chem. 78, 5429–5435 (2006)CrossRefGoogle Scholar
  25. 25.
    Lin C.M., Lai Y.S., Liu H.P., Chen C.Y., Wo A.M.: Trapping of bioparticles via microvortices in a microfluidic device for bioassay applications. Anal. Chem. 80, 8937–8945 (2008)CrossRefGoogle Scholar
  26. 26.
    Pozrikidis C.: Boundary Integral and Singularity Methods for Linearized Viscous Flow. Cambridge University Press, Cambridge (1992)CrossRefzbMATHGoogle Scholar
  27. 27.
    Youngren G.K., Acrivos A.: On the shape of a gas bubble in a viscous extensional flow. J Fluid Mech. 76, 433–42 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Muldowney G.P., Higdon J.J.L.: A spectral boundary element approach to three-dimensional Stokes flow. J. Fluid Mech. 298, 167–192 (1995)CrossRefzbMATHGoogle Scholar
  29. 29.
    Liang J., Subramaniam S.: Computation of molecular electrostatics with boundary element methods. Biophys. J. 73, 1830–1841 (1997)CrossRefGoogle Scholar
  30. 30.
    Pozrikidis C.: Interfacial dynamics for Stokes flow. J Comput. Phys. 169, 250–301 (2001)CrossRefzbMATHGoogle Scholar
  31. 31.
    Dimitrakopoulos P., Wang J.T.: A spectral boundary element algorithm for interfacial dynamics in two-dimensional Stokes flow based on Hermitian interfacial smoothing. Eng. Anal. Bound. Elem. 31, 646–656 (2007)CrossRefzbMATHGoogle Scholar
  32. 32.
    Wang J.T., Liu J.X., Han J.J., Guan J.: Effects of complex internal structures on rheology of multiple emulsions particles in 2D from a boundary integral method. Phys. Rev. Lett. 110, 066001 (2013)CrossRefGoogle Scholar
  33. 33.
    Wang J.T, Liu J.X., Han J.J., Guan J.: Rheology investigation of the globule of multiple emulsions with complex internal structures through a boundary element method. Chem. Eng. Sci. 96, 87–97 (2013)CrossRefGoogle Scholar
  34. 34.
    Baroud C.N., Gallaire F., Dangla R.: Dynamics of microfluidic droplets. Lab Chip 10, 2032–2045 (2010)CrossRefGoogle Scholar
  35. 35.
    Stone H.A., Leal L.G.: Breakup of concentric double emulsion droplets in linear flows. J Fluid Mech. 211, 123–156 (1990)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Jing Guan
    • 1
  • Jinxia Liu
    • 2
  • Xiaoduan Li
    • 2
  • Jun Tao
    • 2
  • Jingtao Wang
    • 2
    • 3
    Email author
  1. 1.School of ScienceTianjin UniversityTianjinPeople’s Republic of China
  2. 2.School of Chemical Engineering and TechnologyTianjin UniversityTianjinPeople’s Republic of China
  3. 3.State Key Laboratory of Chemical EngineeringTianjin UniversityTianjinPeople’s Republic of China

Personalised recommendations