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arXiv · 2503.08334

Navier-Stokes/Allen-Cahn system with moving contact line

Yinghua Li·Yuanxiang Yan·Xijun Yin
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Abstract

In this paper, we study a diffuse interface model for two-phase immiscible flows coupled by Navier-Stokes equations and mass-conserving Allen-Cahn equations. The contact line (the intersection of the fluid-fluid interface with the solid wall) moves along the wall when one fluid replaces the other, such as in liquid spreading or oil-water displacement. The system is equipped with the generalized Navier boundary conditions (GNBC) for the fluid velocity ${\boldsymbol u}$, and dynamic boundary condition or relaxation boundary condition for the phase field variable $ϕ$. We first obtain the local-in-time existence of unique strong solutions to the 2D and 3D Navier-Stokes/Allen-Cahn (NSAC) system with generalized Navier boundary conditions and dynamic boundary condition. For the 2D case in channels, we further show these solutions can be extended to any large time $T$. Additionally, we prove the local-in-time strong solutions for systems with generalized Navier boundary conditions and relaxation boundary condition in 3D channels. Finally, we establish a global unique strong solution accompany with some exponential decay estimates when the fluids are near phase separation states and the contact angle closes to 90 degrees or the fluid-fluid interface tension constant is small.

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Yinghua Li, Yuanxiang Yan, Xijun Yin. 2025-03-11. Navier-Stokes/Allen-Cahn system with moving contact line. https://arxiv.org/abs/2503.08334

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