Skip to main content
SpringerLink
Log in

A Coupled Lattice Boltzmann Method to Solve Nernst–Planck Model for Simulating Electro-osmotic Flows

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

In this paper, we focus on the nonlinear coupling mechanism of the Nernst–Planck model and propose a coupled lattice Boltzmann method (LBM) to solve it. In this method, a new LBM for the Nernst–Planck equation is developed, a multi-relaxation-time (MRT)-LBM for flow field and an LBM for the Poisson equation are used. And then, we discuss the choice of the model and found that the MRT-LBM is much more stable and accurate than the LBGK model. A reasonable iterative sequence and evolution number for each LBM are proposed by considering the properties of the coupled LBM. The accuracy and stability of the presented coupled LBM are also discussed through simulating electro-osmotic flows (EOF) in micro-channels. Furthermore, to test the applicability of it, the EOF with non-uniform surface potential in micro-channels based on the Nernst–Planck model is simulated. And we investigate the effects of non-uniform surface potential on the pattern of the EOF at different external applied electric fields. Finally, a comparison of the difference between the Nernst–Planck model and the Poisson–Boltzmann model is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Gad-el-Hak, M.: The MEMS Handbook. CRC Press, Boca Raton, FL (2002)

    MATH  Google Scholar 

  2. Stone, H.A., Stroock, A.D., Ajdari, A.: Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411 (2004)

    Article  Google Scholar 

  3. Chai, Z.H., Shi, B.C.: Simulation of electro-osmotic flow in microchannel with lattice Boltzmann method. Phys. Lett. A 364, 183–188 (2007)

    Article  MATH  Google Scholar 

  4. Chai, Z.H., Shi, B.C., Zheng, L.: Lattice Boltzmann simulation of viscous dissipation in electro-osmotic flow in microchannels. Int. J. Mod. Phys. C 18(7), 1119–1131 (2007)

    Article  MATH  Google Scholar 

  5. Fu, L.M., Lin, J.Y., Yang, R.J.: Analysis of electroosmotic flow with step change in zeta potential. J. Colloid Interface Sci. 258(2), 266–275 (2003)

    Article  Google Scholar 

  6. Park, H.M., Lee, J.S., Kim, T.W.: Comparison of the Nernst-Planck model and the Poisson-Boltzmann model for electroosmotic flows in microchannels. J. Colloid Interface Sci. 315(2), 731–739 (2007)

    Article  MathSciNet  Google Scholar 

  7. Bhattacharyya, S., Nayak, A.K.: Electroosmotic flow in micro/nanochannels with surface potential heterogeneity: An analysis through the Nernst-Planck model with convection effect. Colloid Surf. A 339(1–3), 167–177 (2009)

    Article  Google Scholar 

  8. Yang, R.J., Fu, L.M., Hwang, C.C.: Electro-osmotic entry flow in a microchannel. J. Colloid Interface Sci. 224(1), 173–179 (2001)

    Article  Google Scholar 

  9. Ng, E.Y.K., Tan, S.T.: Study of EDL effect on 3-D developing flow in microchannel with Poisson-Boltzmann and Nernst-Planck models. Int. J. Numer. Methods Eng. 71(7), 818–836 (2007)

    Article  MATH  Google Scholar 

  10. Chen, Z.: Comparison of the mobile charge distribution models in mixed ionic-electronic conductors. J. Electrochem. Soc. 151(10), A1576–A1583 (2004)

    Article  Google Scholar 

  11. Benzi, R., Succi, S., Vergassola, M.: The lattice Boltzmann equation: theory and applications. Phys. Rep. 222, 145–197 (1992)

    Article  Google Scholar 

  12. Aidun, C.K., Clausen, J.R.: Lattice Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 42, 439–472 (2010)

    Article  MathSciNet  Google Scholar 

  13. Melchionna, S., Succi, S.: Electrorheology in nanopores via lattice Boltzmann simulation. J. Chem. Phys. 120(9), 4492–4497 (2004)

    Article  Google Scholar 

  14. Hlushkou, D., Kandhai, D., Tallarek, U.: Coupled lattice-Boltzmann and finite-difference simulation of electroosmosis in microfluidic channels. Int. J. Numer. Methods Fluids 46(5), 507–532 (2004)

    Article  MATH  Google Scholar 

  15. He, X., Li, N.: Lattice Boltzmann simulation of electrochemical systems. Comput. Phys. Commun. 129(1–3), 158–166 (2000)

    Article  MATH  Google Scholar 

  16. Wang, M.R., Kang, Q.J.: Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann methods. J. Comput. Phys. 229(3), 728–744 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Qian, Y.H., d’Humiéres, D., Lallemand, P.: Lattice BGK model for Navier-Stokes equations. Europhys. Lett. 17(6), 479–484 (1992)

    Article  MATH  Google Scholar 

  18. Lallemand, P., Luo, L.S.: Theory of the Lattice Boltzmann method: dispersion, dissipation, isotropy, Galilean invariance, and stability. Phys. Rev. E 61(6), 6546 (2000)

    Article  MathSciNet  Google Scholar 

  19. Guo, Z.L., Zheng, C.G., Shi, B.C.: Lattice Boltzmann equation with multiple effective relaxation times for gaseous microscale flow. Phys. Rev. E 77(3), 036707 (2008)

    Article  Google Scholar 

  20. Du, R., Shi, B.C., Chen, X.W.: Multi-relaxation-time lattice Boltzmann model for incompressible flow. Phys. Lett. A 359(6), 564–572 (2006)

    Article  MATH  Google Scholar 

  21. Guo, Z.L., Zheng, C.G., Shi, B.C.: Discrete lattice effects on the forcing term in the lattice Boltzmann method. Phys. Rev. E 65(4), 046308 (2002)

    Article  Google Scholar 

  22. Chai, Z.H., Shi, B.C.: A novel lattice Boltzmann model for the Poisson equation. Appl. Math. Model. 32(10), 2050–2058 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Shi, B.C., Guo, Z.L.: Lattice Boltzmann model for nonlinear convection–diffusion equations. Phys. Rev. E 79(1), 016701 (2009)

    Article  Google Scholar 

  24. Shi, B.C., Deng, B., Du, R., Chen, X.W.: A new scheme for source term in LBGK model for convection–diffusion equation. Comput. Math. Appl. 55(7), 1568–1575 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  25. Guo, Z.L., Zheng, C.G., Shi, B.C.: Non-equilibrium extrapolation method for velocity and pressure boundary conditions in the lattice Boltzmann method. Chin. Phys. 11(4), 366 (2002)

    Article  MathSciNet  Google Scholar 

  26. Tian, F., Li, B., Kwok, D.Y.: Lattice Boltzmann Simulation of Electroosmotic Flows in Micro- and Nanochannels. International Conference on MEMS, NANO, and Smart Systems (ICMENS) of 2004, Banff, Alberta, Canada, 294 (2004)

  27. Miller, W., Succi, S.: A lattice Boltzmann model for anisotropic crystal growth from melt. J. Stat. Phys. 107(1–2), 173–186 (2002)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

We gratefully acknowledge the Prof. Jiankang Wu for his helpful discussions during this work. This work is supported by the National Science Foundation of China (Grant Nos. 51125024, 51006040, 11272132) and China Postdoctoral Science Foundation (Grant No. 2012M521424).

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan,  430074, People’s Republic of China

    Xuguang Yang, Baochang Shi & Zhenhua Chai

  2. State Key Laboratory of Coal Combustion, Huazhong University of Science and Technology, Wuhan,  430074, People’s Republic of China

    Zhaoli Guo

Authors
  1. Xuguang Yang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Baochang Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhenhua Chai
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Zhaoli Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Baochang Shi.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Yang, X., Shi, B., Chai, Z. et al. A Coupled Lattice Boltzmann Method to Solve Nernst–Planck Model for Simulating Electro-osmotic Flows. J Sci Comput 61, 222–238 (2014). https://doi.org/10.1007/s10915-014-9820-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9820-6

Keywords

Search

Navigation