Stochastic Motion Model of an Adhered Droplet at the Edge of a Splashing Liquid Film

Abstract In the process of splashing induced by the impact of a spherical droplet on a wall, the characteristics of the motion of an adhered droplet at the liquid-film edge directly influence the velocity, the spatial distribution, and the size distribution of splashing droplets. Particularly, when the diameter of the adhered droplets falls within the submicron or micron range, they are subjected to numerous stochastic collision forces from surrounding air molecules, leading to a pronounced stochastic motion, which, in turn, affects the splashing process. To investigate the fundamental scientific question of the stochastic motion mechanism of the adhered droplet, a stochastic dynamics approach is adopted. Based on the Itô stochastic differential equation, a stochastic differential equation describing the motion of the adhered droplet is first formulated. Subsequently, an expression for the lift coefficient of the adhered droplet is proposed, and the diffusion coefficient is derived using the π theorem. Finally, a stochastic motion model for the adhered droplet at the liquid-film edge is established. The computational results indicate that the proposed stochastic motion model can accurately capture the stochastic behavior of the adhered droplet, and the model predictions agree well with experimental measurements, thereby validating the model.