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Development of a High-Resolution Scheme for Solving the PNP-NS Equations in Curved Channels

Part of: Incompressible viscous fluids Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  01 February 2016

Tony W. H. Sheu ,
Yogesh G. Bhumkar ,
S. T. Yuan  and
S. C. Syue
Tony W. H. Sheu*
Affiliation:
Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan, 10617 Center of Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei, Taiwan, 10617 Institute of Applied Mathematical Sciences, National Taiwan University, Taipei, Taiwan, 10617
Yogesh G. Bhumkar
Affiliation:
Center of Advanced Study in Theoretical Sciences (CASTS), National Taiwan University, Taipei, Taiwan, 10617 School of Mechanical Sciences, IIT Bhubaneswar, Odisha, India751013
S. T. Yuan
Affiliation:
Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan, 10617
S. C. Syue
Affiliation:
Department of Engineering Science and Ocean Engineering, National Taiwan University, Taipei, Taiwan, 10617
*
*Corresponding author. Email addresses:twhsheu@ntu.edu.tw (T. W. H. Sheu), bhumkaryogesh@gmail.com (Y. G. Bhumkar), r99525055@ntu.edu.tw (S. T. Yuan), r02525068@ntu.edu.tw (S. C. Syue)
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Abstract

A high-order finite difference scheme has been developed to approximate the spatial derivative terms present in the unsteady Poisson-Nernst-Planck (PNP) equations and incompressible Navier-Stokes (NS) equations. Near the wall the sharp solution profiles are resolved by using the combined compact difference (CCD) scheme developed in five-point stencil. This CCD scheme has a sixth-order accuracy for the second-order derivative terms while a seventh-order accuracy for the first-order derivative terms. PNP-NS equations have been also transformed to the curvilinear coordinate system to study the effects of channel shapes on the development of electroos-motic flow. In this study, the developed scheme has been analyzed rigorously through the modified equation analysis. In addition, the developed method has been computationally verified through four problems which are amenable to their own exact solutions. The electroosmotic flow details in planar and wavy channels have been explored with the emphasis on the formation of Coulomb force. Significance of different forces resulting from the pressure gradient, diffusion and Coulomb origins on the convective electroosmotic flow motion is also investigated in detail.

Keywords

PNPNSfive-point stencilcombined compact differenceelectroosmotic flowCoulomb force

MSC classification

Secondary: 76D05: Navier-Stokes equations 76D55: Flow control and optimization 65M06: Finite difference methods 65M22: Solution of discretized equations
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 2 , February 2016 , pp. 496 - 533
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Jensen, K. F., Microchemical systems; status, challenges, and opportunities, AIChE J., 45, 1999, 2051–2054.CrossRefGoogle Scholar
[2]Manz, A., Graber, N., Widmer, H. M., Miniaturized total chemical-analysis systems: A novel concept for chemical sensing. Sensors Actuators, B-Chemical, 1(1-6): 244–248, 1990.Google Scholar
[3]Reyes, D. R., Lossifidis, D., Auroux, P. A., Manz, A., Micro total analysis systems, I. Introduction, theory, and technology, Anal. Chem., 74, 2623–2636, 2002.Google Scholar
[4]Cao, J., Cheng, P., Hong, F. J., Applications of electrohydrodynamics and Joule heating effects in microfluidic chips: A review, Science in China Series E: Technological Sciences, vol. 52(12), 3477–3490, 2009.CrossRefGoogle Scholar
[5]Squires, T. M., Bazant, M. Z., Induced-charge electroosmosis, J. Fluid Mech., 2004, 509: 217–252.Google Scholar
[6]Probstein, R. F., Physiochemical Hydrodynamics, John Wiley and Sons, New York, 1994.Google Scholar
[7]Castellanos, A., Electrohydrodynamics, New York: Springer Wien, 1998.CrossRefGoogle Scholar
[8]Griffiths, D.J., Introduction to Electrodynamics. NJ: Prentice Hall, 1999.Google Scholar
[9]Ramos, A., Morgan, H., Greenetal, N. G.., AC electrokinetics: A review of forces in micro-electrode structures, J. Phys D: Appl. Phys., 1998, 31(18): 2338–2353.CrossRefGoogle Scholar
[10]Li, D. Q., Electrokinetics in Microfluidics, New York: Academic Press, 2004.Google Scholar
[11]Melcher, J. R., Taylor, G. I., Electrohydrodynamics: A review of the role of interfacial shear stresses, Annu. Rev. Fluid Mech., 1, 111–146, 1969.Google Scholar
[12]Patankar, N. A., Hu, H. H., Numerical simulation of electroosmotic flow, Anal. Chem., 1998, 70, 1870–1881.CrossRefGoogle ScholarPubMed
[13]Chiu, P. H., Sheu, T. W. H., On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation, J. Comput. Phys., 228, 2009, 3640–3655.Google Scholar
[14]Tam, C. K. W., Webb, J. C.. Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys., 1993, 107, 262–281.Google Scholar
[15]Johnson, R. W., Mackinnon, R. J., Equivalent versions of the QUICK scheme for finite-difference and finite-volume numerical methods, Commun. Appl. Numer. Methods, 8, 1992, 841–847.Google Scholar
[16]Sengupta, T. K., High Accuracy Computing Methods: Fluid Flows and Wave Phenomena, Cambridge Univ. Press., New York, USA, 2013.Google Scholar
[17]Sengupta, T. K., Dipankar, A., Sagaut, P., Error dynamics: Beyond von Neumann analysis, J. Comput. Phys. 226 (2007), 1211–1218.Google Scholar
[18]Sengupta, T. K., Ganeriwal, G., De, S., Analysis of central and upwind schemes, J. Comput. Phys., 192 (2003), 677–694.Google Scholar
[19]Sengupta, T. K., Lakshmanan, V., Vijay, V. V. S. N., A new combined stable and dispersion relation preserving compact scheme for non-periodic problems, J. Comput. Phys. 228 (2009), 3048–3071.CrossRefGoogle Scholar
[20]Sheu, T. W. H., Chiu, P. H., A divergence-free-condition compensated method for incompressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 196, 2007, 4479–4494.Google Scholar
[21]Dutta, P., Breskok, A., Analytical solution of combined electroosmotic/pressure driven flows in two-dimensional straight channels: Finite Debye layer effects, Anal. Chem., 2001, 73, 1979–1986.Google Scholar
[22]Burgreen, D., Nakache, F. R., Electro kinetic flow in ultrafine capillary slits, J. Phy. Chem., 1964, 68(5), 1084–1091.Google Scholar
[23]Qian, S., Bau, H. H., Theoretical investigation of electro-osmotic flows and chaotic stirring in rectangular cavities, Appl. Math. Modelling, 29 (2005), 726–753.CrossRefGoogle Scholar