  Gravitational settling of a droplet in air onto a soft substrate is a
ubiquitous event relevant to many natural processes and engineering
applications. We study this phenomenon by developing a three-phase lubrication
model of droplet settling onto a solid substrate coated by a thin soft layer
represented by a viscous film, an elastic compressible layer and an elastic
sheet supported by a viscous film. By combining scaling analysis, analytical
methods, and numerical simulations we elucidate how the resulting droplet
dynamics is affected by the nature of the soft layer. We show that these soft
layers can significantly affect the droplet shape during gravitational
settling. When there is a linear response of the deformations of the soft
layer, the air layer takes longer to drain as compared to the case of a droplet
settling onto a rigid substrate. Our results provide new insight into the
coupled interactions between droplets and solids coated by a thin film of a
soft material.
