Involutive formulation and simulation for electroneutral microfluids
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 901-913.

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.

DOI : 10.1051/m2an/2011001
Classification : 35K55, 58J10, 65M60, 76W05
Mots-clés : microfluids, electrophoresis, stacking, overdetermined PDE systems, involution 
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     title = {Involutive formulation and simulation for electroneutral microfluids},
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Mohammadi, Bijan; Tuomela, Jukka. Involutive formulation and simulation for electroneutral microfluids. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 5, pp. 901-913. doi : 10.1051/m2an/2011001. http://www.numdam.org/articles/10.1051/m2an/2011001/

[1] 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. | MR | Zbl

[2] R. Brahadwaj, B. Mohammadi and J. Santiago, Design and optimization of on-chip capillary electrophoresis. Electrophoresis J. 23 (2002) 2729-2744.

[3] G. Bruin, Recent developments in electrokinetically driven analysis of microfabricated devices. Electrophoresis 21 (2000) 3931-3951.

[4] A. Douglis and L. Nirenberg, Interior estimates for elliptic systems of partial differential equations. Comm. Pure Appl. Math. 8 (1955) 503-538. | MR | Zbl

[5] 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. | MR | Zbl

[6] S.D. Eidelman, Parabolic equations, in Partial differential equations VI, M.A. Shubin Ed., Encyclopaedia of Mathematical Sciences 63, Springer (1994) 201-313. | MR | Zbl

[7] M.G. El Hak, The MEMS Handbook, Handbook series for Mechanical Engineering 7. CRC Press (2002). | Zbl

[8] K. Krupchyk, W. Seiler and J. Tuomela, Overdetermined elliptic systems. Found. Comp. Math. 6 (2006) 309-351. | MR | Zbl

[9] K. Krupchyk and J. Tuomela, Completion of overdetermined parabolic PDEs. J. Symb. Comput. 43 (2008) 153-167. | MR | Zbl

[10] H. Lin, B. Storey, M. Oddy, C.-H. Chen and J. Santiago, Instability of electrokinetic microchannel flows with conductivity gradients. Phys. Fluids 16 (2004) 1876-1899. | Zbl

[11] M. Marden, On the zeros of certain rational functions. Trans. Amer. Math. Soc. 32 (1930) 658-668. | JFM | MR

[12] B. Mohammadi and J. Tuomela, Simplifying numerical solution of constrained PDE systems through involutive completion. ESAIM: M2AN 39 (2005) 909-929. | Numdam | MR | Zbl

[13] B. Mohammadi and J. Tuomela, Involutive upgrades of Navier-Stokes solvers. Int. J. Comput. Fluid Dyn. 23 (2009) 439-447. | MR | Zbl

[14] M. Oddy and J. Santiago, Multiple-species model for electrokinetic instability. Phys. Fluids 17 (2005) 1245-1278. | Zbl

[15] B. Perthame, Transport equations in biology, Frontiers in Mathematics. Birkhäuser, Basel (2007). | MR | Zbl

[16] T. Poinsot and D. Veynante, Theoretical and Numerical Combustion. 2nd edn., R.T. Edwards, Inc. (2005).

[17] J.F. Pommaret, Systems of Partial Differential Equations and Lie Pseudogroups. Mathematics and its applications 14. Gordon and Breach Science Publishers (1978). | MR | Zbl

[18] R.F. Probstein, Physicochemical Hydrodynamics. Wiley (1995).

[19] W. Seiler, Involution - The Formal Theory of Differential Equations and its Applications in Computer Algebra. Algorithms and Computation in Mathematics 24. Springer, 2010. | MR | Zbl

[20] V.A. Solonnikov, On boundary value problems for linear parabolic systems of differential equations of general form. Trudy Mat. Inst. Steklov. 83 (1965) 3-163 (in Russian). | MR | Zbl

[21] D. Spencer, Overdetermined systems of linear partial differential equations. Bull. Amer. Math. Soc. 75 (1969) 179-239. | MR | Zbl

[22] T. Squires and S.R. Quake, Instability of electrokinetic microchannel flows with conductivity gradients. Rev. Mod. Phys. 77 (2005) 977-1026. | Zbl

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