Validity of the Reynolds equation for miscible fluids in microchannels

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May 2012, 17(3): 801-834. doi: 10.3934/dcdsb.2012.17.801

Validity of the Reynolds equation for miscible fluids in microchannels

Mathieu Colin ^1, , Thierry Colin ^1, and Julien Dambrine ^2,

1.	Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France, France
2.	LMA, Téléport 2- BP 30179, Boulevard Pierre et Marie Curie, 86962 Futuroscope Chasseneuil Cedex, France

Received March 2011 Revised September 2011 Published January 2012

Abstract
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In this paper, we consider asymptotic models for miscible flows in microchannels. The characteristics of the flows in microfluidics imply that usually the Hele-Shaw approximation is valid. We present asymptotic models in the Hele-Shaw regime for flows of miscible fluids in a channel in the case where the bottom and the top of the channels have been modified in two different ways. The first case concerns a flat bottom with slip boundary conditions obtained by chemical patterning. The second one is a non-flat bottom with a non-slipping surface. We derive in both cases 2.5D and 2D asymptotic models. We prove global well-posedness of the 2D model. We also prove that both approaches are asymptotically equivalent in the Hele-Shaw regime and we present direct 3D simulations showing that for passive mixing strategy, the Hele-Shaw approximation is not valid anymore.

Keywords: Hele-Shaw approximation., Reynolds equation.

Mathematics Subject Classification: Primary: 76D07, 76D27; Secondary: 76D0.

Citation: Mathieu Colin, Thierry Colin, Julien Dambrine. Validity of the Reynolds equation for miscible fluids in microchannels. Discrete and Continuous Dynamical Systems - B, 2012, 17 (3) : 801-834. doi: 10.3934/dcdsb.2012.17.801

References:

[1]	G. Bayada and M. Chambat, New models in the theory of the hydrodynamic bifurcation of rough surfaces, J. Tribol., 110 (1988), 402-407. doi: 10.1115/1.3261642.
[2]	F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptotic Analysis, 20 (1999), 175-212.
[3]	F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin), 52, Springer-Verlag, Berlin, 2006.
[4]	D. Bresch, C. Choquet, L. Chupin, T. Colin and M. Gisclon, Roughness-induced effect at main order on the Reynolds approximation, Multiscale Modeling and Simulation, 8 (2010), 997-1017. doi: 10.1137/090754996.
[5]	J. Dambrine, "Modélisation et Étude Numérique de Quelques Écoulements de Fluides Complexes en Microfluidiques," Thèse de l'Université Bordeaux 1, 2009.
[6]	J. Dambrine, B. Géraud and J. B. Salmon, Interdiffusion of liquids of different viscosities in a microchannel, New Journal of Physics, 2009.
[7]	J. Fernandez, P. Kurowski, P. Petitjean and E. Meiburg, Density-driven unstable flows of miscible fluids in a Hele-Shaw cell, J. Fluid. Mech., 451 (2002), 239-260.
[8]	C. G. Gal and M. Grasselli, Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system, Physica D, 240 (2011), 629-635. doi: 10.1016/j.physd.2010.11.014.
[9]	D. Gérard-Varet and N. Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary, Comm. Math. Phys., 295 (2010), 99-137.
[10]	A. Günther, K.-F. Jensen, Multiphase microfluidics: From flow characteristics to chemical and material synthesis, Lab on a Chip, 2006.
[11]	D. Joseph and Y. Renardy, "Fundamentals of Two Fluid Dynamics. Part I. Mathematical Theory and Applications," Interdisciplinary Applied Mathematics, 3, Springer-Verlag, New York, 1993.
[12]	G. Karniadakis and A. Beskok, "Micro Flows: Fundamental and Simulation," Springer-Verlag, 2002.
[13]	O. Kuksenok and A. C. Balazs, Simulating the dynamic behavior of immiscible binary fluids in three-dimensional chemically patterned microchannels, Physical Review E, 2003.
[14]	O. Kuksenok and A. C. Balazs, Structures formation in binary fluids driven through patterned microchannels: Effect of hydrodynamics and arrangement of surface patterns, Physica D, 2004.
[15]	S. Li, J. Lowengrub and P. Leo, A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell, J. Comp. Phys., 225 (2007), 534-567.
[16]	X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115 (1994), 200-212. doi: 10.1006/jcph.1994.1187.
[17]	N.-T. Nguyen and Z. Wu, Micromixers-a review, Journal of Micromechanics and Microengineering, 2010.
[18]	P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. of the Roy. Soc. London Ser A, 245 (1958), 312-329. doi: 10.1098/rspa.1958.0085.
[19]	D. Schafroth, N. Goyal and E. Meiburg, Miscible displacements in Hele-Shaw cells: Nonmonotonic viscosity profiles, European Journal of Mechanics B Fluids, 26 (2007), 444-453. doi: 10.1016/j.euromechflu.2006.09.001.
[20]	J. Simon, Compacts sets in the space $L^p(0,T;B)$, Annali. Mat. Pura. Applicata. (4), 146 (1987), 65-96.
[21]	A. D. Stroock, S. K. W. Dertinger, A. Adjari, I. Mezić, H. A. Stone and G. M. Whitesides, Chaotic mixers in microchannels, Science, 2002.
[22]	A. D. Stroock, S. K. W. Dertinger, G. M. Whitesides and A. Adjari, Patterning flows using grooved surfaces, Analytical Chemistry, 2002.

show all references

References:

[1]	G. Bayada and M. Chambat, New models in the theory of the hydrodynamic bifurcation of rough surfaces, J. Tribol., 110 (1988), 402-407. doi: 10.1115/1.3261642.
[2]	F. Boyer, Mathematical study of multi-phase flow under shear through order parameter formulation, Asymptotic Analysis, 20 (1999), 175-212.
[3]	F. Boyer and P. Fabrie, "Éléments d'Analyse pour l'Étude de Quelques Modèles d'Écoulements de Fluides Visqueux Incompressibles," Mathématiques & Applications (Berlin), 52, Springer-Verlag, Berlin, 2006.
[4]	D. Bresch, C. Choquet, L. Chupin, T. Colin and M. Gisclon, Roughness-induced effect at main order on the Reynolds approximation, Multiscale Modeling and Simulation, 8 (2010), 997-1017. doi: 10.1137/090754996.
[5]	J. Dambrine, "Modélisation et Étude Numérique de Quelques Écoulements de Fluides Complexes en Microfluidiques," Thèse de l'Université Bordeaux 1, 2009.
[6]	J. Dambrine, B. Géraud and J. B. Salmon, Interdiffusion of liquids of different viscosities in a microchannel, New Journal of Physics, 2009.
[7]	J. Fernandez, P. Kurowski, P. Petitjean and E. Meiburg, Density-driven unstable flows of miscible fluids in a Hele-Shaw cell, J. Fluid. Mech., 451 (2002), 239-260.
[8]	C. G. Gal and M. Grasselli, Instability of two-phase flows: A lower bound on the dimension of the global attractor of the Cahn-Hilliard-Navier-Stokes system, Physica D, 240 (2011), 629-635. doi: 10.1016/j.physd.2010.11.014.
[9]	D. Gérard-Varet and N. Masmoudi, Relevance of the slip condition for fluid flows near an irregular boundary, Comm. Math. Phys., 295 (2010), 99-137.
[10]	A. Günther, K.-F. Jensen, Multiphase microfluidics: From flow characteristics to chemical and material synthesis, Lab on a Chip, 2006.
[11]	D. Joseph and Y. Renardy, "Fundamentals of Two Fluid Dynamics. Part I. Mathematical Theory and Applications," Interdisciplinary Applied Mathematics, 3, Springer-Verlag, New York, 1993.
[12]	G. Karniadakis and A. Beskok, "Micro Flows: Fundamental and Simulation," Springer-Verlag, 2002.
[13]	O. Kuksenok and A. C. Balazs, Simulating the dynamic behavior of immiscible binary fluids in three-dimensional chemically patterned microchannels, Physical Review E, 2003.
[14]	O. Kuksenok and A. C. Balazs, Structures formation in binary fluids driven through patterned microchannels: Effect of hydrodynamics and arrangement of surface patterns, Physica D, 2004.
[15]	S. Li, J. Lowengrub and P. Leo, A rescaling scheme with application to the long-time simulation of viscous fingering in a Hele-Shaw cell, J. Comp. Phys., 225 (2007), 534-567.
[16]	X.-D. Liu, S. Osher and T. Chan, Weighted essentially non-oscillatory schemes, Journal of Computational Physics, 115 (1994), 200-212. doi: 10.1006/jcph.1994.1187.
[17]	N.-T. Nguyen and Z. Wu, Micromixers-a review, Journal of Micromechanics and Microengineering, 2010.
[18]	P. G. Saffman and G. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. of the Roy. Soc. London Ser A, 245 (1958), 312-329. doi: 10.1098/rspa.1958.0085.
[19]	D. Schafroth, N. Goyal and E. Meiburg, Miscible displacements in Hele-Shaw cells: Nonmonotonic viscosity profiles, European Journal of Mechanics B Fluids, 26 (2007), 444-453. doi: 10.1016/j.euromechflu.2006.09.001.
[20]	J. Simon, Compacts sets in the space $L^p(0,T;B)$, Annali. Mat. Pura. Applicata. (4), 146 (1987), 65-96.
[21]	A. D. Stroock, S. K. W. Dertinger, A. Adjari, I. Mezić, H. A. Stone and G. M. Whitesides, Chaotic mixers in microchannels, Science, 2002.
[22]	A. D. Stroock, S. K. W. Dertinger, G. M. Whitesides and A. Adjari, Patterning flows using grooved surfaces, Analytical Chemistry, 2002.

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