Computational Mechanics

, Volume 51, Issue 2, pp 171–185 | Cite as

SUPG and discontinuity-capturing methods for coupled fluid mechanics and electrochemical transport problems

  • Pablo A. Kler
  • Lisandro D. Dalcin
  • Rodrigo R. Paz
  • Tayfun E. Tezduyar
Original Paper


Electrophoresis is the motion of charged particles relative to the surrounding liquid under the influence of an external electric field. This electrochemical transport process is used in many scientific and technological areas to separate chemical species. Modeling and simulation of electrophoretic transport enables a better understanding of the physicochemical processes developed during the electrophoretic separations and the optimization of various parameters of the electrophoresis devices and their performance. Electrophoretic transport is a multiphysics and multiscale problem. Mass transport, fluid mechanics, electric problems, and their interactions have to be solved in domains with length scales ranging from nanometers to centimeters. We use a finite element method for the computations. Without proper numerical stabilization, computation of coupled fluid mechanics, electrophoretic transport, and electric problems would suffer from spurious oscillations that are related to the high values of the local Péclet and Reynolds numbers and the nonzero divergence of the migration field. To overcome these computational challenges, we propose a stabilized finite element method based on the Streamline-Upwind/Petrov-Galerkin (SUPG) formulation and discontinuity-capturing techniques. To demonstrate the effectiveness of the stabilized formulation, we present test computations with 1D, 2D, and 3D electrophoretic transport problems of technological interest.


Electrophoresis Fluid mechanics Electrochemical transport Finite element computation SUPG stabilization Discontinuity capturing 


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  1. 1.
    Albrecht JW, El-Ali J, Jensen KF (2007) Cascaded free-flow isoelectric focusing for improved focusing speed and resolution. Anal Chem 79: 9364–9371CrossRefGoogle Scholar
  2. 2.
    Arnaud I, Josserand J, Rossier J, Girault H (2002) Finite element simulation of off-gel buffering. Electrophoresis 23: 3253–3261CrossRefGoogle Scholar
  3. 3.
    Barz DP (2009) Comprehensive model of electrokinetic flow and migration in microchannels with conductivity gradients. Microfluid Nanofluidics 7(2): 249–265CrossRefGoogle Scholar
  4. 4.
    Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2007) YZβ discontinuity-capturing for advection-dominated processes with application to arterial drug delivery. Int J Numer Methods Fluids 54: 593–608. doi: 10.1002/fld.1484 MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bercovici M, Lele SK, Santiago JG (2009) Open source simulation tool for electrophoretic stacking, focusing, and separation. J Chromatogr A 1216: 1008–1018CrossRefGoogle Scholar
  6. 6.
    Berli CLA (2008) Equivalent circuit modeling of electrokinetically driven analytical microsystems. Microfluid Nanofluidics 4(5): 391–399CrossRefGoogle Scholar
  7. 7.
    Berli CLA, Piaggio M, Deiber J (2003) Modeling the zeta potential of silica capillaries in relation to the background electrolyte composition. Electrophoresis 24: 1587–1595CrossRefGoogle Scholar
  8. 8.
    Brooks AN, Hughes TJR (1982) Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations. Comput Methods Appl Mech Eng 32: 199–259MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Brunet E, Adjari A (2004) Generalized Onsager relations for electrokinetic effects in anisotropic and heterogeneous geometries. Phys Rev E 69(1): 016306CrossRefGoogle Scholar
  10. 10.
    Chatterjee A (2003) Generalized numerical formulations for multi-physics microfluidics-type applications. J Micromech Microeng 13: 758–767CrossRefGoogle Scholar
  11. 11.
    Chau M, Spiteri P, Guivarich R, Boisson H (2008) Parallel asynchronous iterations for the solution of a 3d continuous flow electrophoresis problem. Comput Fluids 37(9): 1126–1137zbMATHCrossRefGoogle Scholar
  12. 12.
    Corsini A, Iossa C, Rispoli F, Tezduyar TE (2010) A DRD finite element formulation for computing turbulent reacting flows in gas turbine combustors. Comput Mech 46: 159–167. doi: 10.1007/s00466-009-0441-0 MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Corsini A, Menichini C, Rispoli F, Santoriello A, Tezduyar TE (2009) A multiscale finite element formulation with discontinuity capturing for turbulence models with dominant reactionlike terms. J Appl Mech 76: 021211. doi: 10.1115/1.3062967 CrossRefGoogle Scholar
  14. 14.
    Corsini A, Rispoli F, Santoriello A, Tezduyar TE (2006) Improved discontinuity-capturing finite element techniques for reaction effects in turbulence computation. Comput Mech 38: 356–364. doi: 10.1007/s00466-006-0045-x MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Corsini A, Rispoli F, Tezduyar TE (2011) Stabilized finite element computation of NOx emission in aero-engine combustors. Int J Numer Methods Fluids 65: 254–270. doi: 10.1002/fld.2451 MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Corsini A, Rispoli F, Tezduyar TE (2012) Computer modeling of wave-energy air turbines with the SUPG/PSPG formulation and discontinuity-capturing technique. J Appl Mech 79: 010910. doi: 10.1115/1.4005060 CrossRefGoogle Scholar
  17. 17.
    Craven TJ, Rees JM, Zimmerman WB (2008) On slip velocity boundary conditions for electroosmotic flow near sharp corners. Phys Fluids 20(4): 043603CrossRefGoogle Scholar
  18. 18.
    Ermakov S, Jacobson S, Ramsey J (1998) Computer simulations of electrokinetic transport in microfabricated channel structures. Anal Chem 70(21): 4494–4504CrossRefGoogle Scholar
  19. 19.
    Ganjoo DK, Tezduyar TE (1987) Petrov-Galerkin formulations for electrochemical processes. Comput Methods Appl Mech Eng 65: 61–83. doi: 10.1016/0045-7825(87)90183-6 zbMATHCrossRefGoogle Scholar
  20. 20.
    Ganjoo DK, Tezduyar TE, Goodrich WD (1989) A new formulation for numerical solution of electrophoresis separation processes. Comput Methods Appl Mech Eng 75: 515–530. doi: 10.1016/0045-7825(89)90045-5 MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Hessel V, Lowe H, Schönfeld F (2005) Micromixers—a review on passive and active mixing principles. Chem Eng Sci 60(8–9): 2479–2501Google Scholar
  22. 22.
    Hruška V, Jaros M, Gaš B (2006) Simul 5—free dynamic simulator of electrophoresis. Electrophoresis 27: 984–991CrossRefGoogle Scholar
  23. 23.
    Hsu MC, Bazilevs Y, Calo VM, Tezduyar TE, Hughes TJR (2010) Improving stability of stabilized and multiscale formulations in flow simulations at small time steps. Comput Methods Appl Mech Eng 199: 828–840. doi: 10.1016/j.cma.2009.06.019 MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hughes T, Mallet M (1986) A new finite element formulation for computational fluid dynamics: IV a discontinuity-capturing operator for multidimensional advective-diffusive systems. Comput Methods Appl Mech Eng 58(3): 329–336MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Hughes TJR, Franca LP, Balestra M (1986) A new finite element formulation for computational fluid dynamics: V. Circumventing the Babuška–Brezzi condition: a stable Petrov–Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput Methods Appl Mech Eng 59: 85–99MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Hughes TJR, Mallet M, Mizukami A (1986) A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Comput Methods Appl Mech Eng 54: 341–355MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Hughes TJR, Tezduyar TE (1984) Finite element methods for first-order hyperbolic systems with particular emphasis on the compressible Euler equations. Comput Methods Appl Mech Eng 45: 217–284. doi: 10.1016/0045-7825(84)90157-9 MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Hunter R (2001) Foundations of colloid science, 2nd edn. Oxford University Press, OxfordGoogle Scholar
  29. 29.
    Karniadakis G, Beşkök A, Aluru N (2005) Microflows and nanoflows: fundamentals and simulation. Springer, New YorkzbMATHGoogle Scholar
  30. 30.
    Kler P (2010) Modeling and simulation of microfluidic chips for analytical applications. Ph.D thesis, UNIVERSIDAD NACIONAL DEL LITORALGoogle Scholar
  31. 31.
    Kler P, Berli C, Guarnieri F (2011) Modeling and high performance simulation of electrophoretic techniques in microfluidic chips. Microfluid Nanofluid 10(1): 187–198CrossRefGoogle Scholar
  32. 32.
    Kler PA, López EJ, Dalcín LD, Guarnieri FA, Storti MA (2009) High performance simulations of electrokinetic flow and transport in microfluidic chips. Comput Methods Appl Mech Eng 198(30–32): 2360–2367zbMATHCrossRefGoogle Scholar
  33. 33.
    Li D (2004) Electrokinetics in microfluidics. Elsevier Academic Press, LondonGoogle Scholar
  34. 34.
    MacInnes JM (2002) Computation of reacting electrokinetic flow in microchannel geometries. Chem Eng Sci 57(21): 4539–4558CrossRefGoogle Scholar
  35. 35.
    Park YJ, Deans HA, Tezduyar TE (1990) Finite element formulation for transport equations in a mixed coordinate system: an application to determine temperature effects on the single-well chemical tracer test. Int J Numer Methods Fluids 11: 769–790. doi: 10.1002/fld.1650110605 CrossRefGoogle Scholar
  36. 36.
    Park YJ, Deans HA, Tezduyar TE (1991) Thermal effects on single-well chemical tracer tests for measuring residual oil saturation. Soc Petroleum Eng Form Eval 190: 401–408. doi: 10.2118/19683-PA Google Scholar
  37. 37.
    Patankar N, Hu H (1998) Numerical simulation of electroosmotic flow. Anal Chem 70(9): 1870–1881CrossRefGoogle Scholar
  38. 38.
    Peng Y, Pallandre A, Tran NT, Taverna M (2008) Recent innovations in protein separation on microchips by electrophoretic methods. Electrophoresis 29(1): 157–178CrossRefGoogle Scholar
  39. 39.
    Probstein R (2003) Physicochemical hydrodynamics. An Introduction, 2nd edn. Wiley-Interscience, New YorkGoogle Scholar
  40. 40.
    Rispoli F, Corsini A, Tezduyar TE (2007) Finite element computation of turbulent flows with the discontinuity-capturing directional dissipation (DCDD). Comput Fluids 36: 121–126. doi: 10.1016/j.compfluid.2005.07.004 zbMATHCrossRefGoogle Scholar
  41. 41.
    Rispoli F, Saavedra R, Corsini A, Tezduyar TE (2007) Computation of inviscid compressible flows with the V-SGS stabilization and YZβ shock-capturing. Int J Numer Methods Fluids 54: 695–706. doi: 10.1002/fld.1447 MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Rispoli F, Saavedra R, Menichini F, Tezduyar TE (2009) Computation of inviscid supersonic flows around cylinders and spheres with the V-SGS stabilization and YZβ shock-capturing. J Appl Mech 76: 021209. doi: 10.1115/1.3057496 CrossRefGoogle Scholar
  43. 43.
    Saville D, Palusinski O (1986) Theory of electrophoretic separations. Part I: formulation of a mathematical model. AIChE J 32(2): 207–214CrossRefGoogle Scholar
  44. 44.
    Schönfeld F, Hessel V, Hofmann C (2004) An optimised split-and-recombine micro-mixer with uniform ‘chaotic’ mixing. Lab Chip 4(1): 65–69CrossRefGoogle Scholar
  45. 45.
    Shim J, Dutta P, Ivory C (2007) Modeling and simulation of IEF in 2-D microgeometries. Electrophoresis 28: 572–586CrossRefGoogle Scholar
  46. 46.
    Sounart TL, Baygents JC (2007) Lubrication theory for electro-osmotic flow in a non-uniform electrolyte. J Fluid Mech 576: 139–172MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Takizawa K, Tezduyar TE (2011) Multiscale space–time fluid–structure interaction techniques. Comput Mech 48: 247–267. doi: 10.1007/s00466-011-0571-z MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    Tezduyar TE (1992) Stabilized finite element formulations for incompressible flow computations. Adv Appl Mech 28: 1–44. doi: 10.1016/S0065-2156(08)70153-4 MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Tezduyar TE (2003) Computation of moving boundaries and interfaces and stabilization parameters. Int J Numer Methods Fluids 43: 555–575. doi: 10.1002/fld.505 MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    Tezduyar TE (2004) Finite element methods for fluid dynamics with moving boundaries and interfaces. In: Stein E, Borst RD, Hughes TJR (eds) Encyclopedia of computational mechanics, vol 3: Fluids, chap. 17. Wiley, New YorkGoogle Scholar
  51. 51.
    Tezduyar TE, Behr M, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces—the deforming-spatial-domain/space–time procedure: I. The concept and the preliminary numerical tests. Comput Methods Appl Mech Eng 94(3): 339–351. doi: 10.1016/0045-7825(92)90059-S MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Tezduyar TE, Behr M, Mittal S, Liou J (1992) A new strategy for finite element computations involving moving boundaries and interfaces – the deforming-spatial-domain/space–time procedure: II. Computation of free-surface flows, two-liquid flows, and flows with drifting cylinders. Comput Methods Appl Mech Eng 94(3): 353–371. doi: 10.1016/0045-7825(92)90060-W MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Tezduyar TE, Mittal S, Ray SE, Shih R (1992) Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements. Comput Methods Appl Mech Eng 95: 221–242. doi: 10.1016/0045-7825(92)90141-6 zbMATHCrossRefGoogle Scholar
  54. 54.
    Tezduyar TE, Osawa Y (2000) Finite element stabilization parameters computed from element matrices and vectors. Comput Methods Appl Mech Eng 190: 411–430. doi: 10.1016/S0045-7825(00)00211-5 zbMATHCrossRefGoogle Scholar
  55. 55.
    Tezduyar TE, Park YJ (1986) Discontinuity capturing finite element formulations for nonlinear convection-diffusion-reaction equations. Comput Methods Appl Mech Eng 59: 307–325. doi: 10.1016/0045-7825(86)90003-4 zbMATHCrossRefGoogle Scholar
  56. 56.
    Tezduyar TE, Park YJ, Deans HA (1987) Finite element procedures for time-dependent convection-diffusion-reaction systems. Int J Numer Methods Fluids 7: 1013–1033. doi: 10.1002/fld.1650071003 zbMATHCrossRefGoogle Scholar
  57. 57.
    Tezduyar TE, Ramakrishnan S, Sathe S (2008) Stabilized formulations for incompressible flows with thermal coupling. Int J Numer Methods Fluids 57: 1189–1209. doi: 10.1002/fld.1743 MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Tezduyar TE, Sathe S (2007) Modeling of fluid–structure interactions with the space–time finite elements: solution techniques. Int J Numer Methods Fluids 54: 855–900. doi: 10.1002/fld.1430 MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Tezduyar TE, Senga M (2006) Stabilization and shock-capturing parameters in SUPG formulation of compressible flows. Comput Methods Appl Mech Eng 195: 1621–1632. doi: 10.1016/j.cma.2005.05.032 MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    Tezduyar TE, Senga M (2007) SUPG finite element computation of inviscid supersonic flows with YZβ shock-capturing. Comput Fluids 36: 147–159. doi: 10.1016/j.compfluid.2005.07.009 zbMATHCrossRefGoogle Scholar
  61. 61.
    Tezduyar TE, Senga M, Vicker D (2006) Computation of inviscid supersonic flows around cylinders and spheres with the SUPG formulation and YZβ shock-capturing. Comput Mech 38: 469–481. doi: 10.1007/s00466-005-0025-6 zbMATHCrossRefGoogle Scholar
  62. 62.
    Thormann W, Caslavska J, Mosher R (2007) Modeling of electroosmotic and electrophoretic mobilization in capillary and microchip isoelectric focusing. J Chromatogr A 1155(2): 154–163CrossRefGoogle Scholar
  63. 63.
    Tsai WB, Hsieh CJ, Chieng CC (2005) Parallel computation of electroosmotic flow in L-shaped microchannels. In: 6th world congress of structural and multidisciplinary optimization, pp 4971–4980Google Scholar
  64. 64.
    Wu D, Qin J, Lin B (2008) Electrophoretic separations on microfluidic chips. J Chromatogr A 1184(1–2): 542–559Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Pablo A. Kler
    • 1
  • Lisandro D. Dalcin
    • 2
  • Rodrigo R. Paz
    • 2
  • Tayfun E. Tezduyar
    • 3
  1. 1.Central Division of Analytical ChemistryForschungszentrum JülichJülichGermany
  2. 2.Centro Internacional de Métodos Computacionales en Ingeniería Instituto de Desarrollo Tecnológico para la Industria QuímicaUniversidad Nacional del Litoral—Consejo Nacional de Investigaciones Científicas y TécnicasSanta FeArgentina
  3. 3.Mechanical EngineeringRice UniversityHoustonUSA

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