We propose a novel arbitrary Lagrangian--Eulerian (ALE) finite element method for simulating incompressible Navier--Stokes flows driven by surface tension on evolving free boundaries as well as for simulating the dynamics of moving contact lines at fluid--solid interfaces under a prescribed contact angle condition. The proposed method ensures energy dissipation while incorporating the following two key strategies to maintain mesh quality within the ALE framework: the artificial tangential motion strategy of Barrett, Garcke, and Nunberg (BGN) on the fluid's free surface, and the minimal-deformation-rate (MDR) approach within the fluid bulk and along the solid boundary. A central challenge in three-dimensional simulations---the ambiguity of tangential mesh velocity at the moving contact line---is resolved by enforcing a constraint that eliminates tangential motion at the contact line, thereby maintaining stability of tangential motion and mesh quality of computed surfaces. Numerical experiments in both two and three dimensions demonstrate the robustness of the proposed method in maintaining mesh quality and energy dissipation.

Publication:

SIAM JOURNAL ON SCIENTIFIC COMPUTING

http://dx.doi.org/10.1137/25M1784958

Author:

JIASHUN HU

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, HongKong

Email:jiashun.hu@polyu.edu.hk

BUYANG LI

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, HongKong

Email:buyang.li@polyu.edu.hk

RONG TANG

Department of Applied Mathematics, The Hong Kong Polytechnic University, Kowloon, HongKong

Email:claire.tang@polyu.edu.hk

NUO LEI

Corresponding author. Hua Loo-Keng Center for Mathematical Sciences, Academy of Mathe-matics and Systems Science, Chinese Academy of Sciences, Beijing, China

Email:nuo97.lei@polyu.edu.hk