Effective Macroscopic Stokes-Cahn-Hilliard Equations for Periodic Immiscible Flows in Porous Media

  • Markus SchmuckEmail author
  • Grigorios A. Pavliotis
  • Serafim Kalliadasis
Part of the Springer Proceedings in Complexity book series (SPCOM)


Using thermodynamic and variational principles we study a basic phase field model for the mixture of two incompressible fluids in strongly perforated domains. We rigorously derive an effective macroscopic phase field equation under the assumption of periodic flow and a sufficiently large Péclet number with the help of the multiple scale method with drift and our recently introduced splitting strategy for Ginzburg-Landau/Cahn-Hilliard-type equations (Schmuck et al., Proc. R. Soc. A, 468:3705–3724, 2012). As for the classical convection-diffusion problem, we obtain systematically diffusion-dispersion relations (including Taylor-Aris-dispersion). In view of the well-known versatility of phase field models, our study proposes a promising model for many engineering and scientific applications such as multiphase flows in porous media, microfluidics, and fuel cells.


Homogenization Diffusion-dispersion relations Porous structures Stokes-Cahn-Hilliard equations 



We acknowledge financial support from EPSRC Grant No. EP/H034587, EU-FP7 ITN Multiflow and ERC Advanced Grant No. 247031.


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Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Markus Schmuck
    • 1
    • 2
    Email author
  • Grigorios A. Pavliotis
    • 2
  • Serafim Kalliadasis
    • 1
  1. 1.Department of Chemical EngineeringImperial College LondonLondonUK
  2. 2.Department of MathematicsImperial College LondonLondonUK

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