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Global Weak Solutions to a Diffuse Interface Model for Incompressible Two-Phase Flows with Moving Contact Lines and Different Densities

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Abstract

In this paper, we analyze a general diffuse interface model for incompressible two-phase flows with unmatched densities in a smooth bounded domain \(\Omega \subset {\mathbb {R}}^d\) (\(d=2,3\)). This model describes the evolution of free interfaces in contact with the solid boundary, namely the moving contact lines. The corresponding evolution system consists of a nonhomogeneous Navier–Stokes equation for the (volume) averaged fluid velocity \({\mathbf {v}}\) that is nonlinearly coupled with a convective Cahn–Hilliard equation for the order parameter \(\varphi \). Due to the nontrivial boundary dynamics, the fluid velocity satisfies a generalized Navier boundary condition that accounts for the velocity slippage and uncompensated Young stresses at the solid boundary, while the order parameter fulfils a dynamic boundary condition with surface convection. We prove the existence of a global weak solution for arbitrary initial data in both two and three dimensions. The proof relies on a combination of suitable approximations and regularizations of the original system together with a novel time-implicit discretization scheme based on the energy dissipation law.

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Acknowledgements

The authors would like to thank the anonymous referees for their careful reading of an initial version of the manuscript and for several helpful comments that allowed us to improve the paper. This work was commenced in May 2017 when the first two authors were visiting Key Laboratory of Mathematics for Nonlinear Science (Fudan University), Ministry of Education and School of Mathematical Sciences at Fudan University, whose hospitality and support is gratefully acknowledged. M. Grasselli is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). H. Wu was partially supported by NNSFC Grant No. 11631011 and the Shanghai Center for Mathematical Sciences at Fudan University.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Florida International University, Miami, FL, 33199, USA

    Ciprian G. Gal

  2. Dipartimento di Matematica, Politecnico di Milano, Milan, 20133, Italy

    Maurizio Grasselli

  3. School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai, 200433, China

    Hao Wu

  4. Key Laboratory of Mathematics for Nonlinear Science (Fudan University), Ministry of Education, Shanghai, 200433, China

    Hao Wu

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  1. Ciprian G. Gal
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Gal, C.G., Grasselli, M. & Wu, H. Global Weak Solutions to a Diffuse Interface Model for Incompressible Two-Phase Flows with Moving Contact Lines and Different Densities. Arch Rational Mech Anal 234, 1–56 (2019). https://doi.org/10.1007/s00205-019-01383-8

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