# American Institute of Mathematical Sciences

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

## Validity of the Reynolds equation for miscible fluids in microchannels

 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

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.
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.

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##### 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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