Hostname: page-component-74656f9884-8r8nv Total loading time: 0 Render date: 2026-11-09T19:55:22.098Z Has data issue: false hasContentIssue false
  • English
  • Français

Involutive formulation and simulation for electroneutral microfluids

Published online by Cambridge University Press:  15 April 2011

Bijan Mohammadi  and
Jukka Tuomela
Bijan Mohammadi
Affiliation:
Institut de Mathématiques et Modélisation de Montpellier, Université Montpellier II, France. bijan.mohammadi@univ-montp2.fr
Jukka Tuomela
Affiliation:
Department of Physics and Mathematics, University of Eastern Finland, Finland. jukka.tuomela@uef.fi

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

We study a microfluidic flow model where the movement of several charged species is coupled with electric field and the motion of ambient fluid. The main numerical difficulty in this model is the net charge neutrality assumption which makes the system essentially overdetermined. Hence we propose to use the involutive and the associated augmented form of the system in numerical computations. Numerical experiments on electrophoresis and stacking show that the completed system significantly improves electroneutrality constraint conservation and recovers analytical results while a direct implementation of the initial model fails.

Keywords

Microfluidselectrophoresisstackingoverdetermined PDE systemsinvolution

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 901 - 913
Copyright
© EDP Sciences, SMAI, 2011

References

M.S. Agranovich, Elliptic boundary problems, in Partial differential equations IX, M.S. Agranovich, Y.V. Egorov and M.A. Shubin Eds., Encyclopaedia of Mathematical Sciences 79, Springer (1997) 1–144.
Brahadwaj, R., Mohammadi, B. and Santiago, J., Design and optimization of on-chip capillary electrophoresis. Electrophoresis J. 23 (2002) 2729–2744. 3.0.CO;2-I>CrossRef
Bruin, G., Recent developments in electrokinetically driven analysis of microfabricated devices. Electrophoresis 21 (2000) 3931–3951. 3.0.CO;2-M>CrossRef
Douglis, A. and Nirenberg, L., Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8 (1955) 503–538. CrossRef
P. Dudnikov and S. Samborski, Linear overdetermined systems of partial differential equations, in Partial Differential Equations VIII, M. Shubin Ed., Encyclopaedia of Mathematical Sciences 65, Springer (1996) 1–86.
S.D. Eidelman, Parabolic equations, in Partial differential equations VI, M.A. Shubin Ed., Encyclopaedia of Mathematical Sciences 63, Springer (1994) 201–313.
M.G. El Hak, The MEMS Handbook, Handbook series for Mechanical Engineering 7. CRC Press (2002).
Krupchyk, K., Seiler, W. and Tuomela, J., Overdetermined elliptic systems. Found. Comp. Math. 6 (2006) 309–351. CrossRef
Krupchyk, K. and Tuomela, J., Completion of overdetermined parabolic PDEs. J. Symb. Comput. 43 (2008) 153–167. CrossRef
Lin, H., Storey, B., Oddy, M., Chen, C.-H. and Santiago, J., Instability of electrokinetic microchannel flows with conductivity gradients. Phys. Fluids 16 (2004) 1876–1899. CrossRef
Marden, M., On the zeros of certain rational functions. Trans. Amer. Math. Soc. 32 (1930) 658–668. CrossRef
Mohammadi, B. and Tuomela, J., Simplifying numerical solution of constrained PDE systems through involutive completion. ESAIM: M2AN 39 (2005) 909–929. CrossRef
Mohammadi, B. and Tuomela, J., Involutive upgrades of Navier-Stokes solvers. Int. J. Comput. Fluid Dyn. 23 (2009) 439–447. CrossRef
Oddy, M. and Santiago, J., Multiple-species model for electrokinetic instability. Phys. Fluids 17 (2005) 1245–1278. CrossRef
B. Perthame, Transport equations in biology, Frontiers in Mathematics. Birkhäuser, Basel (2007).
T. Poinsot and D. Veynante, Theoretical and Numerical Combustion. 2nd edn., R.T. Edwards, Inc. (2005).
J.F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups. Mathematics and its applications 14. Gordon and Breach Science Publishers (1978).
R.F. Probstein, Physicochemical Hydrodynamics. Wiley (1995).
W. Seiler, Involution – The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics 24. Springer, 2010.
Solonnikov, V.A., On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov. 83 (1965) 3–163 (in Russian).
Spencer, D., Overdetermined systems of linear partial differential equations. Bull. Amer. Math. Soc. 75 (1969) 179–239. CrossRef
Squires, T. and Quake, S.R., Instability of electrokinetic microchannel flows with conductivity gradients. Rev. Mod. Phys. 77 (2005) 977–1026. CrossRef