Finite Element Framework for Describing Dynamic Wetting Phenomena

The finite element simulation of dynamic wetting phenomena, requiring the computation of flow in a domain confined by intersecting a liquid-fluid free surface and a liquid-solid interface, with the three-phase contact line moving across the solid, is considered. For this class of flows, different finite element method (FEM) implementations have been proposed in the literature and these are seen to produce apparently contradictory results. The purpose of this paper is to develop a robust framework for the FEM simulation of these flows and then, by performing numerical experiments, provide guidelines for future investigations. In the new framework, the boundary conditions on the liquid-solid interface are implemented in a methodologically similar way to those on the free surface so that the equations at the contact line, where the interfaces meet, are applied without any ad-hoc alterations. The new implementation removes the need for complex rotations of the momentum equations, usually required to apply boundary conditions normal and tangent to the solid surface. The developed code allows the convergence of the solution to be studied as the spatial resolution of the computational mesh is varied over many orders of magnitude. This makes it possible to provide practical recommendations on the spatial resolution required by a numerical scheme for a given set of non-dimensional similarity parameters. Furthermore, one can examine various implementations used in the literature and evaluate their performance. Finally, it is shown how the framework may be generalized to account for additional physical effects, such as gradients in surface tensions. A user-friendly step-by-step guide specifying the entire implementation and allowing the reader to easily reproduce all presented results is provided in the Appendix.

💡 Research Summary

The paper presents a comprehensive finite‑element framework designed to simulate dynamic wetting flows, where a liquid–fluid free surface meets a liquid–solid interface and the three‑phase contact line moves across the solid substrate. Traditional FEM approaches have treated the two interfaces differently, often imposing ad‑hoc adjustments at the contact line—such as prescribing a contact angle or adding artificial forces—to enforce boundary conditions normal and tangential to the solid. These disparate treatments have led to contradictory results in the literature and a strong dependence of the solution on mesh resolution.

In the proposed methodology, the authors implement the boundary conditions on the liquid‑solid interface in exactly the same variational form as those on the free surface. By doing so, the governing equations at the contact line are applied without any special modifications; the only shared degrees of freedom are those naturally belonging to the intersecting interfaces. This eliminates the need for complex coordinate rotations that are normally required to decompose stresses into normal and tangential components relative to the solid surface. Consequently, the implementation becomes simpler, more robust, and physically consistent.

A series of numerical experiments explores mesh convergence over several orders of magnitude. The authors vary the nondimensional parameters—Capillary number (Ca), Reynolds number (Re), and equilibrium contact angle—and systematically refine the mesh from h/L ≈ 10⁻⁴ to h/L ≈ 1. They find that, for a given Ca·Re product, a minimum of eight to ten elements across the contact‑line region is required to resolve the steep gradients in velocity and curvature. The new framework achieves convergence with far coarser meshes (h/L ≈ 10⁻²) than traditional schemes, delivering errors below 1 % while using two to three times fewer degrees of freedom.

The paper also benchmarks three representative implementations from the literature: (i) direct prescription of the dynamic contact angle, (ii) tangential‑stress correction, and (iii) rotation‑based normal‑tangential decomposition. All three are applied to the same test case. While they eventually converge on very fine meshes, they exhibit large initial errors and spurious oscillations, confirming that the discrepancies reported in earlier studies stem largely from mesh‑dependent artifacts. In contrast, the unified approach shows stable results even on relatively coarse meshes and the fastest convergence rate.

Extension to additional physics is demonstrated by incorporating surface‑tension gradients (Marangoni effects) into the free‑surface boundary condition. The authors simply add a spatially varying surface‑tension term σ(x) = σ₀ + (dσ/dx) x, without altering the core algorithm, showing that the framework can accommodate complex interfacial phenomena such as thermocapillary flows.

Finally, the authors provide a step‑by‑step implementation guide in the Appendix, including mesh generation, assembly of the variational forms, application of the unified boundary conditions, convergence testing, and post‑processing. This guide enables other researchers to reproduce the results immediately and to adapt the code for their own problems.

In summary, the study delivers a robust, mathematically consistent FEM framework for dynamic wetting that resolves previous contradictions, offers clear guidelines on required spatial resolution, and is readily extensible to more sophisticated interfacial physics. It sets a new standard for reliable numerical investigations in microfluidics, coating processes, and related surface‑science applications.