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The non-linear interaction between air and a water droplet just prior tohigh-speed impingement on a surface is a phenomenon that has been researchedextensively and occurs in a number of industrial settings. The role thatsurface deformation plays in an air cushioned impact of a liquid droplet isconsidered here. In a two-dimensional framework, assuming small density andviscosity ratios between the air and the liquid, a reduced system ofintegro-differential equations is derived governing the liquid dropletfree-surface shape, the pressure in the thin air film and the deformation ofthe surface, assuming the effects of surface tension, compressibility andgravity to be negligible. The deformation of the surface is first described ina rather general form, based on previous membrane-type models. The coupledsystem is then investigated in two cases: a soft viscoelastic case where thesubstrate stiffness and (viscous) damping are considered and a more generalflexible surface where all relevant parameters are retained. Numericalsolutions are presented, highlighting a number of key consequences of substratedeformability on the pre-impact phase of droplet impact, such as reduction inpressure buildup, increased air entrapment and considerable delay to touchdown.Connections (including subtle dependence of the size of entrapped air on thedroplet velocity, reduced pressure peaks and droplet gliding) with recentexperiments and a large deformation analysis are also presented.

Pre-impact dynamics of a droplet impinging on a deformable surface