An Effective Model for Droplet Impact Dynamics on Micro-Structured Surfaces: Nonlocal Theory and SPH Simulation of Pancake Bouncing

Abstract

The accurate mathematical modeling of droplet impact dynamics on micro-structured surfaces is fundamental to understanding and predicting complex fluid behaviors relevant to a wide range of engineering and scientific applications. In particular, the pancake bouncing phenomenon--systematically studied by Liu et al. (Nature Physics, 2014)--on superhydrophobic micro-structured substrates presents significant theoretical challenges. Central to these challenges is the need to construct effective mathematical models that capture the intricate influence of substrate micro/nanostructures on droplet dynamics. This requires the development of robust formulations for surface tension, contact line dynamics, and the interaction forces between fluid and solid structures. In this work, we formulate a nonlocal mathematical framework for the simulation of 3D pancake bouncing on superhydrophobic micro-cone arrays. The model incorporates intermolecular attractive forces to represent droplet surface tension, and we provide a strict theoretical derivation linking these forces quantitatively to the macroscopic surface tension coefficient, thereby circumventing the reliance on empirical parameter tuning. The complex geometry of micro-cone arrays introduces fundamental difficulties in defining local normal directions for contact algorithms. To overcome this, we develop a nonlocal contact repulsion force model that governs fluid-solid interactions and ensures numerical stability under high Weber number conditions. Based on this mathematical foundation, we implement the model using smoothed particle hydrodynamics (SPH), enabling high-precision 3D simulations. Computational experiments, validated against empirical data, confirm the model's accuracy and robustness, while underscoring the key role of numerical simulation in elucidating droplet-microstructure interactions.