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Allen–Cahn–Navier–Stokes–Voigt Systems with Moving Contact Lines

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Abstract

We consider a diffuse interface model for an incompressible binary fluid flow. The model consists of the Navier–Stokes–Voigt equations coupled with the mass-conserving Allen–Cahn equation with Flory–Huggins potential. The resulting system is subject to generalized Navier boundary conditions for the (volume averaged) fluid velocity \({{\textbf {u}}}\) and to a dynamic contact line boundary condition for the order parameter \(\phi \). These boundary conditions account for the moving contact line phenomenon. We establish the existence of a global weak solution which satisfies an energy inequality. A similar result is proven for the Allen–Cahn–Navier–Stokes system. In order to obtain some higher-order regularity (w.r.t. time) we propose the Voigt approximation: in this way we are able to prove the validity of the energy identity and of the strict separation property. Thanks to this property, we can show the uniqueness of quasi-strong solutions, even in dimension three. Regularization in finite time of weak solutions is also shown.

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Data availability is not applicable to this article as no new data were created or analysed in this study.

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Acknowledgements

The authors are grateful to the anonymous referee for the careful reading of the manuscript and the valuable comments (we refer, in particular, to Corollary 3.10). The second author and the third author have been partially funded by MIUR-PRIN Grant 2020F3NCPX “Mathematics for Industry 4.0 (Math4I4)”, their research is part of the activities of “Dipartimento di Eccellenza 2023–2027”, and they are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), Istituto Nazionale di Alta Matematica (INdAM).

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

    Ciprian G. Gal

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

    Maurizio Grasselli & Andrea Poiatti

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  1. Ciprian G. Gal
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Gal, C.G., Grasselli, M. & Poiatti, A. Allen–Cahn–Navier–Stokes–Voigt Systems with Moving Contact Lines. J. Math. Fluid Mech. 25, 89 (2023). https://doi.org/10.1007/s00021-023-00829-0

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