An Eulerian-PFEM Approach to Model Transient Capillary Forces on Moving Contact Lines of Spreading Liquid Drops: Applications in Hydraulic and Fluid Dynamics

This study addresses the complexities of simulating multiphase flows, particularly the dynamics of droplet behaviour on solid substrates, which are influenced by intricate capillary interactions. Despite advancements in fixed-grid and moving-grid methodologies, there remains a critical gap in accurately modelling transient phenomena and dissipative forces at the solid-liquid interface. We introduce a novel Eulerian-particle finite element method (PFEM) framework that captures the transient capillary forces associated with the movement of contact lines in liquid droplet spreading scenarios. Through a numerical assessment of three established dissipative force models—Jiang’s, Bracke’s, and Seeberg’s we identify Jiang's model as the most effective and stable in representing transient behaviours across various wettability conditions. Experimental and numerical investigations into water droplet impacts on hydrophilic triethoxysilane and hydrophobic triethoxysilybutraldehyde substrates reveal strong correlations between experimental results and numerical predictions. Specifically, observed spreading radius ratios for a 2.14 mm droplet demonstrate promising alignment with numerical results, showcasing an 11.4% and 8.2% error margin at 1.0 ms and 1.5 ms, respectively, for triethoxysilane, and an 11.1% and 8.5% error for triethoxysilybutraldehyde under similar conditions. This validation underscores the model's efficacy in elucidating the relationships between spreading rates and time. By integrating a dynamic contact angle condition, our approach mitigates mesh dependency and enhances simulation accuracy, paving the way for improved modeling of interface dynamics in multiphase flows. The implications of these findings extend to various industrial applications, including fuel cells and cooling systems, emphasizing the necessity of accurately representing dissipative phenomena for realistic simulations.