| Abstract: |
| Thin liquid films with contact lines are common in nature and in engineering applications. The existence of free boundaries and singularities poses significant challenges for both modeling and computation. In this talk, we present a unified framework for the derivation and numerical approximation of a fourth-order thin film equation with mesoscopic dynamic boundary conditions. The reduced model is systematically derived using the Onsager variational principle in conjunction with lubrication theory, yielding a thermodynamically consistent formulation that accounts for capillarity, gravity, and external force. To solve the resulting free boundary problem, we develop adaptive moving mesh methods based on a discrete Onsager variational principle, including a stabilized semi-implicit scheme to improve computational efficiency. Numerical results confirm the optimal convergence of the proposed methods and accurately capture key wetting behaviors, including contact angle hysteresis on rough substrates. This work offers a robust framework for simulating thin film flows with complex geometries. |
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