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An Immersed Interface Method for Axisymmetric Electrohydrodynamic Simulations in Stokes flow

Published online by Cambridge University Press:  30 July 2015

H. Nganguia ,
Y.-N. Young ,
A. T. Layton ,
W.-F. Hu  and
M.-C. Lai
H. Nganguia
Affiliation:
Department of Biological Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
Y.-N. Young*
Affiliation:
Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA
A. T. Layton
Affiliation:
Department of Mathematics, Duke University, Box 90320, Durham, NC 27708, USA
W.-F. Hu
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, 1001, Ta Hsueh Road, Hsingchu 300, Taiwan, R.O.C.
M.-C. Lai
Affiliation:
Department of Applied Mathematics, National Chiao Tung University, 1001, Ta Hsueh Road, Hsingchu 300, Taiwan, R.O.C.
*
*Corresponding author. Email addresses: hn9@njit.edu (H. Nganguia), yyoung@njit.edu (Y.-N. Young), alayton@math.duke.edu (A. T. Layton), weifanhu.am95g@g2.nctu.edu.tw (W.-F. Hu), mclai@math.nctu.edu.tw (M.-C. Lai)
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Abstract

A numerical scheme based on the immersed interface method (IIM) is developed to simulate the dynamics of an axisymmetric viscous drop under an electric field. In this work, the IIM is used to solve both the fluid velocity field and the electric potential field. Detailed numerical studies on the numerical scheme show a second-order convergence. Moreover, our numerical scheme is validated by the good agreement with previous analytical models, and numerical results from the boundary integral simulations. Our method can be extended to Navier-Stokes fluid flow with nonlinear inertia effects.

Keywords

76W0565M0665M12Immersed interface methodleast squares interpolationelectrohydrodynamics
Type
Research Article
Information
Communications in Computational Physics , Volume 18 , Issue 2 , August 2015 , pp. 429 - 449
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Ajayi, O. O.. A note on Taylor’s electrohydrodynamic theory. Proc. R. Soc. Lond. A, 364:499–507, 1978.Google Scholar
[2]Aubry, N.. Simple recipe for electroosmotic mixing in microchannels. In Herold, K. E. and Rasooly, A., editors, Lab-on-a-Chip Technology, pages 201–209. Horizon Scientific Press, 2009.Google Scholar
[3]Beale, T. and Layton, A. T.. A velocity decomposition approach for moving interfaces in viscous fluids. J. Comput. Phys., 228:3358–3367, 2009.CrossRefGoogle Scholar
[4]Bentenitis, N. and Krause, S.. Droplet deformation in DC electric fields: The extended leaky dielectric model. Langmuir, 21:6194–6209, 2005.Google Scholar
[5]Brady, P. and Desjardins, O.. A sharp, robust, conservative cut-cell immersed boundary technique for stationary and moving structures. In International Conference on Multiphase Flow, 2013.Google Scholar
[6]Dash, R.K., Borca-Tasciuc, T., Purkayastha, A., and Ramanath, G.. Electrowetting on dielectric-actuation of microdroplets of aqueous bismuth telluride nanoparticle suspensions. Nanotechnol., 18:475711, 2007.Google Scholar
[7]Dubash, N. and Mestel, A. J.. Behavior of a conducting drop in a highly viscous fluid less conducting than that of the ambient fluid. J. Fluid Mech., 581:469–493, 2007.Google Scholar
[8]Feng, J.Q.. Electrohydrodynamic behavior of a drop subjected to a steady uniform electric field at finite electric Reynolds number. Proc. Math. Phys. Eng. Sci., 455:2245–2269, 1999.CrossRefGoogle Scholar
[9]Feng, J. Q. and Beard, K. V.. Three-dimenionsional oscillation characteristics of electrostatistically deformed drops. J. Fluid Mech., 227:429–447, 1991.Google Scholar
[10]Feng, J.Q. and Scott, T. C.. A computational analysis of electrohydrodynamics of a leaky dielectric drop in an electric field. J. Fluid Mech., 311:289–326, 1996.CrossRefGoogle Scholar
[11]Fowler, J., Moon, H., and Kim, C. J.. Enhancement of mixing of droplet-based microfluidics. Proc. IEEE Conf. MEMS, pages 97–100, 2002.Google Scholar
[12]Griffith, B. and Peskin, C. S.. On the order of accuracy of the immersed boundary method: higher order convergence rates for sufficiently smooth problems. J. Comput. Phys., 208:75–105, 2005.Google Scholar
[13]Ha, J.-W. and Yang, S.-M.. Deformation and breakup of Newtonian and non-Newtonian conducting drops in an electric field. J. Fluid Mech., 405:131–156, 2000.Google Scholar
[14]Ha, J.-W. and Yang, S.-M.. Electrohydrodynamics and electrorotation of a drop with fluid less conducting than that of the ambient fluid. Phys. Fluids, 12:764, 2000.Google Scholar
[15]Hu, W.-F., Lai, M.-C. and Young, Y.-N.. A hybrid immersed boundary and immersed interface method for electrohydrodynamic simulations. J. Comput. Phys., 282:47–61, 2015.Google Scholar
[16]Kolahdouz, E. M. and Salac, D.. The Semi Implicit Gradient Augmented Level Set Method. ArXiv e-prints, February 2012.Google Scholar
[17]Lac, E. and Homsy, G. M.. Axisymmetric deformation and stability of a viscous drop in a steady electric field. J. Fluid Mech., 590:239–264, 2007.Google Scholar
[18]Lai, M.-C., Huang, C.-Y., and Huang, Y.-M.. Simulating the axisymmetric interfacial flows with insoluble surfactant by immersed boundary method. Int. J. Numer. Anal. Modeling, 8:105–117, 2011.Google Scholar
[19]Lai, M.-C. and Peskin, C. S.. An immersed boundary method with formal second-order accuracy and reduced numerical viscosity. J. Comput. Phys., 160:705–719, 2000.Google Scholar
[20]Leveque, R.J. and Li, Z.. The immersed interface method for elliptic equations with discontinuous coefficients and singular sources. SIAM J. Numer. Anal., 31:1019–1044, 1994.CrossRefGoogle Scholar
[21]Leveque, R. J. and Li, Z.. Immersed interface methods for Stokes flow with elastic boundaries or surface tension. SIAM J. Sci. Comput., 18:709–735, 1997.Google Scholar
[22]Li, Y.. Numerical methods for simulating fluid motion driven by immersed interfaces. PhD thesis, Duke University, 2012.Google Scholar
[23]Li, Y., Sgouralis, I., and Layton, A. T.. Computing viscous flow in an elastic tube. To appear in Numer. Math. Theor. Meth. Appl.Google Scholar
[24]Li, Y., Williams, S. A., and Layton, A. T.. A hybrid immersed interface method for driven Stokes flow in an elastic tube. Numer. Math. Theor. Meth. Appl., 6:600–616, 2013.Google Scholar
[25]Li, Z.. Fast iterative algorithm for elliptic interface problems. SIAM J. Numer. Anal., 35:230–254, 1998.CrossRefGoogle Scholar
[26]Li, Z., Lai, M.-C., He, G., and Zhao, H.. An augmented method for free boundary problems with moving contact lines. Comp. & Fluids, 39:1033–1040, 2010.Google Scholar
[27]McConnell, L. C., Miksis, M. J., and Vlahovska, P. M.. Vesicle electrohydrodynamics in DC electric fields. IMA J. Appl. Math., pages 1–21, 2013.CrossRefGoogle Scholar
[28]Melcher, J. R. and Taylor, G. I.. Electrohydrodynamics: A review of the role of interfacial shear stresses. Annu. Rev. Fluid Mech., 1:111–146, 1969.Google Scholar
[29]Mori, Y.. Convergence proof of the velocity field for a Stokes flow immersed boundary method. Comm. Pure and App. Math., LXI:1213–1263, 2008.Google Scholar
[30]Mori, Y. and Peskin, C. S.. Implicit second-order immersed boundary methods with boundary mass. Comput. Methods Appl. Mech. Eng., 197:2049–2067, 2008.Google Scholar
[31]Nganguia, H., Young, Y.-N., Vlahovska, P. M., Bławzdziewcz, J., Zhang, J., and Lin, H.. Equilibrium electro-deformation of a surfactant-laden viscous drop. Phys. Fluids, 25:092106, 2013.Google Scholar
[32]Nudurupati, S., Janjua, M., Aubry, N., and Singh, P.. Concentrating particles on drop surfaces using external electric fields. Electrophoresis, 29:1164–1172, 2008.Google Scholar
[33]Paknemat, H., Pishevar, A. R., and Pournaderi, P.. Numerical simulation of drop deformations and breakup modes caused by direct current electric fields. Phys. Fluids, 24:102101, 2012.Google Scholar
[34]Peskin, C. S.. Numerical analysis of blood flow in the heart. J. Comput. Phys., 25:220–252, 1977.Google Scholar
[35]Van Poppel, B. P., Desjardins, O., and Daily, J. W.. A ghost fluid, level set methodology for simulating multiphase electrohydrodynamic flows with application to liquid fuel injection. J. Comput. Phys., 229:7977–7996, 2010.Google Scholar
[36]Ramos, A.. Electrokinetics and electrohydrodynamics in microsystems, volume 530 of CISM International Centre for Mechanical Sciences. Springer, 2011.Google Scholar
[37]Saville, D. A.. Electrohydrodynamics: The Taylor-Melcher leaky dielectric model. Annu. Rev. Fluid Mech., 29:27–64, 1997.CrossRefGoogle Scholar
[38]Sherwood, J. D.. Breakup of fluid droplets in electric and magnetic fields. J. Fluid Mech., 188:133–146, 1998.Google Scholar
[39]Taylor, G.. Studies in electrohydrodynamics I. The circulation produced in a drop by electric field. Proc. R. Soc. Lond. A, 291:159–166, 1966.Google Scholar
[40]Teigen, K. E. and Munkejord, S. T.. Influence of surfactant on drop deformation in an electric field. Phys. Fluids, 22:112104, 2010.Google Scholar
[41]Tomar, G., Gerlach, D., Biswas, G., Alleborn, N., Sharma, A., Durst, F., Welch, S. W. J., and Delgado, A.. Two-phase electrohydrodynamic simulations using a volume-of-fluid approach. J. Comput. Phys., 227:1267–1285, 2007.CrossRefGoogle Scholar
[42]Vizika, O. and Saville, D. A.. The electrohydrodynamic deformation of drops suspended in liquids in steady and oscillatory electric fields. J. Fluid Mech., 239:1–21, 1992.Google Scholar
[43]Wang, C., Wang, J., Cai, Q., Li, Z., Zhao, H.-K., and Luo, R.. Exploring accurate Poisson-Boltzman methods for biomolecular simulations. Comput. Theor. Chem., 1024:34–44, 2013.Google Scholar
[44]Ward, T. and Homsy, G. M.. Phys. Fluids, 15:2987–2994, 2003.Google Scholar
[45]Woodson, H. H. and Melcher, J. R.. Electromechanical dynamics. John Wiley and Sons, 1968.Google Scholar
[46]Zhang, J., Zahn, J. D., Tan, W., and Lin, H.. A transient solution for vesicle electrodeformation and relaxation. Phys. Fluids, 25:071903, 2013.Google Scholar