Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Validity of the Reynolds equation for miscible fluids in microchannels

Pages: 801 - 834, Volume 17, Issue 3, May 2012      doi:10.3934/dcdsb.2012.17.801

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Mathieu Colin - Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France (email)
Thierry Colin - Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France (email)
Julien Dambrine - LMA, Téléport 2- BP 30179, Boulevard Pierre et Marie Curie, 86962 Futuroscope Chasseneuil Cedex, France (email)

Abstract: 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:  Reynolds equation, Hele-Shaw approximation.
Mathematics Subject Classification:  Primary: 76D07, 76D27; Secondary: 76D03.

Received: March 2011;      Revised: September 2011;      Available Online: January 2012.