Transient electroosmotic flow of general Maxwell fluids through a slit microchannel

  • Yongjun Jian
  • Jie Su
  • Long Chang
  • Quansheng Liu
  • Guowei He


Using Laplace transform method, semi-analytical solutions are presented for transient electroosmotic flow of Maxwell fluids between micro-parallel plates. The solution involves solving the linearized Poisson–Boltzmann equation, together with the Cauchy momentum equation and the Maxwell constitutive equation considering the depletion effect produced by the interaction between macro-molecules of the Maxwell fluids and the channel surface. The overall flow is divided into depletion layer and bulk flow outside of depletion layer. In addition, the Maxwell stress is incorporated to describe the boundary condition at the interface. The velocity expressions of these two layers were obtained respectively. By numerical computations of inverse Laplace transform, the influences of viscosity ratio μ, density ratio ρ, dielectric constant ratio \({\varepsilon}\) of layer II to layer I, relaxation time \({{\bar{\lambda}}_1}\), interface charge density jump Q, and interface zeta potential difference \({\Delta {\bar{\psi}}}\) on transient velocity amplitude are presented.

Mathematics Subject Classification (2000)

76A05 76D05 76W05 


Electric double layer (EDL) Unsteady electroosmotic flow (EOF) Maxwell fluids Micro-parallel plates Laplace transform 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Stone H.A., Stroock A.D., Ajdari A.: Engineering flows in small devices: microfluidics toward a Lab-on-a-chip. Ann. Rev. Fluid Mech. 36, 381–411 (2004)CrossRefGoogle Scholar
  2. 2.
    Bayraktar T., Pidugu S.B.: Characterization of liquid flows in microfluidic systems. Int. J. Heat Mass Trans. 49, 815–824 (2006)CrossRefMATHGoogle Scholar
  3. 3.
    Li D.: Electrokinetics in Microfluidics. Elsevier, Amsterdam (2004)Google Scholar
  4. 4.
    Karniadakis G., Beskok A., Aluru N.: Micorflows and Nanoflows: Fundamentals and Simulation. Springer, New York (2005)Google Scholar
  5. 5.
    Levine S., Marriott J.R., Neale G., Epstein N.: Theory of electrokinetic flow in fine cylindrical capillaries at high zeta potentials. J. Colloid Interface Sci. 52, 136–149 (1975)CrossRefGoogle Scholar
  6. 6.
    Tsao H.K.: Electroosmotic flow through an annulus. J. Colloid Interface Sci. 225, 247–250 (2000)CrossRefGoogle Scholar
  7. 7.
    Hsu J.P., Kao C.Y., Tseng S.J., Chen C.J.: Electrokinetic flow through an elliptical microchannel: effects of aspect ratio and electrical boundary conditions. J. Colloid Interface Sci. 248, 176–184 (2002)CrossRefGoogle Scholar
  8. 8.
    Yang C., Li D., Masliyah J.H.: Modeling forced liquid convection in rectangular microchannels with electrokinetic effects. Int. J. Heat Mass Transf. 41, 4229–4249 (1998)CrossRefMATHGoogle Scholar
  9. 9.
    Bianchi F., Ferrigno R., Girault H.H.: Finite element simulation of an electroosmotic driven flow division at a t-junction of microscale dimensions. Anal. Chem. 72, 1987–1993 (2000)CrossRefGoogle Scholar
  10. 10.
    Wang C.Y., Liu Y.H., Chang C.C.: Analytical solution of electro-osmotic flow in a semicircular microchannel. Phys. Fluids 20, 063105 (2008)CrossRefGoogle Scholar
  11. 11.
    Dutta P., Beskok A.: Analytical solution of time periodic electroosmotic flows: analogies to Stokes’ second problem. Anal. Chem. 73, 5097–5102 (2001)CrossRefGoogle Scholar
  12. 12.
    Keh H.J., Tseng H.C.: Transient electrokinetic flow in fine capillaries. J. Colloid Interface Sci. 242, 450–459 (2001)CrossRefGoogle Scholar
  13. 13.
    Kang Y.J., Yang C., Huang X.Y.: Dynamic aspects of electroosmotic flow in a cylindrical microcapillary. Int. J. Eng. Sci. 40, 2203–2221 (2002)CrossRefGoogle Scholar
  14. 14.
    Wang X.M., Chen B., Wu J.K.: a semianalytical solution of periodical electro-osmosis in a rectangular microchannel. Phys. Fluids 19, 127101 (2007)CrossRefGoogle Scholar
  15. 15.
    Chakraborty S., Ray S.: mass flow-rate control through time periodic electro-osmotic flows in circular microchannels. Phys. Fluids 20, 083602 (2008)CrossRefGoogle Scholar
  16. 16.
    Jian Y.J., Yang L.G., Liu Q.S.: Time periodic electro-osmotic flow through a microannulus. Phys. Fluids 22, 042001 (2010)CrossRefGoogle Scholar
  17. 17.
    Deng S.Y., Jian Y.J., Bi Y.H., Chang L., Wang H.J., Liu Q. S.: Unsteady electroosmotic flow of power-law fluid in a rectangular microchannel. Mech. Res. Commun. 39, 9–14 (2010)CrossRefGoogle Scholar
  18. 18.
    Das S., Chakraborty S.: Analytical solutions for velocity, temperature and concentration distribution in electroosmotic microchannel flows in a non-Newtonian bio-fluid. Anal. Chim. Acta 559, 15–24 (2006)CrossRefGoogle Scholar
  19. 19.
    Chakraborty S.: Electroosmotically driven capillary transport of typical non-Newtonian biofluids in rectangular microchannels. Anal. Chim. Acta 605, 175–184 (2007)CrossRefGoogle Scholar
  20. 20.
    Zhao C., Zholkovskij E., Masliyah J.H., Yang C.: Analysis of electroosmotic flow of power-law fluids in a slit microchannel. J. Colloid Interface Sci. 326, 503–510 (2008)CrossRefGoogle Scholar
  21. 21.
    Zhao C., Yang C.: Nonlinear Smoluchowski velocity for electroosmosis of power-law fluids over a surface with arbitrary zeta potentials. Electrophoresis 31, 973–979 (2010)CrossRefGoogle Scholar
  22. 22.
    Tang G.H., Li X.F., He Y.L., Tao W.Q.: Electroosmotic flow of non-newtonian fluid in microchannels. J. Non-Newton. Fluid Mech. 157, 133–137 (2009)CrossRefMATHGoogle Scholar
  23. 23.
    Park H.M., Lee W.M.: Helmholtz–Smoluchowski velocity for viscoelastic electroosmotic flows. J. Colloid Interface Sci. 317, 631–636 (2008)CrossRefGoogle Scholar
  24. 24.
    Park H.M., Lee W.M.: Effect of viscoelasticity on the flow pattern and the volumetric flow rate in electroosmotic flows through a microchannel. Lab Chip 8, 1163–1170 (2008)CrossRefGoogle Scholar
  25. 25.
    Zhao C., Yang C.: Exact solutions for electro-osmotic flow of viscoelastic fluids in rectangular micro-channels. Appl. Math. Comput. 211, 502–509 (2009)CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Afonso A.M., Alves M.A., Pinho F.T.: Analytical solution of mixed electro-osmotic pressure driven flows of viscoelastic fluids in microchannels. J. Non-Newton. fluid Mech. 159, 50–63 (2009)CrossRefMATHGoogle Scholar
  27. 27.
    Liu Q.S., Jian Y.J., Yang L.G.: Time periodic electroosmotic flow of the generalized Maxwell fluids between two micro-parallel plates. J. Non-Newton. fluid Mech. 166, 478–486 (2011)CrossRefMATHGoogle Scholar
  28. 28.
    Jian Y.J., Liu Q.S., Yang L.G.: AC electroosmotic flow of generalized Maxwell fluids in a rectangular microchannel. J. Non-Newton. fluid Mech. 166, 1304–1314 (2011)CrossRefMATHGoogle Scholar
  29. 29.
    Berli C.L.A., Olivares M.L.: Electrokinetic flow of non-Newtonian fluids on microchannels. J. Colloid Interface Sci. 320, 582–589 (2008)CrossRefGoogle Scholar
  30. 30.
    Sousa J.J., Afonso A.M., Pinho F.T.: Effect of the skimming layer on electro- osmotic-Poiseuille flows of viscoelastic fluids. Microfluid Nanofluid 10, 107–122 (2011)CrossRefGoogle Scholar
  31. 31.
    Liu Q.S., Jian Y.J., Yang L.G.: Alternating current electroosmotic flow of the Jeffreys fluids through a slit microchannel. Phys. fluids 23, 102001 (2011)CrossRefGoogle Scholar
  32. 32.
    Volkov A.G., Deamer D.W., Tanelian D.L., Markin V.S.: Electrical double layers at the oil/water interface. Prog. Surf. Sci. 53, 1–134 (1996)CrossRefGoogle Scholar
  33. 33.
    Choi W., Sharma A., Qian S.Z., Lim G., Joo S.W.: On steady two-fluid electroosmotic flow with full interfacial electrostatics. J. Colloid Interface Sci. 357, 521–526 (2008)CrossRefGoogle Scholar
  34. 34.
    Mayur M., Amiroudine S., Lasseux D.: Free-surface instability in electro-osmotic flows of ultrathin liquid films. Phys. Rev. E 85, 046301 (2012)CrossRefGoogle Scholar
  35. 35.
    De Hoog F.R., Knight J.H., Stokes A.N.: An improved method for numerical inversion of Laplace transforms. SIAM J. Sci. Stat. Comput. 3, 357–366 (1982)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Yongjun Jian
    • 1
  • Jie Su
    • 1
  • Long Chang
    • 1
    • 2
  • Quansheng Liu
    • 1
  • Guowei He
    • 3
  1. 1.School of Mathematical ScienceInner Mongolia UniversityHohhotChina
  2. 2.School of Mathematics and StatisticsInner Mongolia University of Finance and EconomicsHohhotChina
  3. 3.The State Key Laboratory of Nonlinear Mechanics (LNM)Institute of Mechanics, Chinese Academy of SciencesBeijingChina

Personalised recommendations