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The application of Wagner theory to idealised two-dimensional inertially dominated droplet impacts is extended to incorporate non-flat substrates that are continuous functions of distance along the surface. Mixed boundary value problems are solved for the displacement and velocity potentials for both a single impact with an asymmetric substrate and a pair of impacts. The droplet free-surface position and the pressure on the wetted surface are calculated, along with the load and moment on the substrate. For double impacts a void may be formed between the substrate and the droplet free surface. Double impacts are compared with a single asymmetric impact with one half of the equivalent substrate geometry to assess how the free surface, loads and moments are affected by the separation between impact sites. Interactions between symmetric double impacts enhance the liquid penetration between the impact sites and increase the load and moment on each substrate element compared with the corresponding single impact. The time taken for the void between droplet and substrate to become saturated is found assuming the gas pressure build-up and capillary forces are negligible, giving an estimate for the transition time from a partially wetted to a fully wetted surface close to the initial impact site. After the void becomes saturated, the subsequent free-surface evolution is determined and the effect of periodic roughness on the contact line evolution is calculated. For surfaces formed of an array of asperities, secondary impacts which both traps further voids and completely wet the surface are found. [ABSTRACT FROM AUTHOR]

Droplet deformation by air cushioning prior to impact is considered. A model is presented coupling the free-surface deformation of a droplet with the pressure field in the narrow air layer generated as a droplet approaches an impact. The model is based upon the density and viscosity in the air being small compared with those in the liquid. Additionally, the Reynolds number, defined using the droplet radius R and approach velocity Wl, is such that lubrication forces dominate in the air layer. In the absence of significant surface tension or compressibility effects, these assumptions lead to coupled nonlinear integro-differential equations describing the evolution of a droplet free surface approaching a solid wall through air, with or without topography. The problem is studied numerically with a boundary-element method in the inviscid droplet coupled with a finite-difference method in the lubricating air. In normal impacts, air cushioning will be shown to deflect the free surface upwards, delaying the moment of touchdown and trapping a bubble. The volume of the bubble is found to be (μg4/3 R5/3/ρl4/3 Wl4/3) V̂̂, where μg is the gas viscosity and ρl is the liquid density and the numerically computed pre-factor V̂̂ = 94.48. Bubble volumes predicted by this relationship are shown to be in good agreement with experimental observations. In oblique impact or impact with a moving surface with sufficient horizontal motion a bubble is not trapped beneath the approaching droplet. In this case, the region of touchdown is initially crescent shaped with air effects accelerating the moment of touchdown. [ABSTRACT FROM AUTHOR]