Impacting spheres: from liquid drops to elastic beads

Abstract

A liquid drop impacting a non-wetting rigid substrate spreads laterally, then retracts, and finally jumps off again. An elastic solid, by contrast, undergoes a slight deformation, contacts briefly, and bounces. The impact force on the substrate – crucial for engineering and natural processes – is classically described by Wagner's (liquids) and Hertz's (solids) theories. This work bridges these limits by considering a generic viscoelastic medium. Using direct numerical simulations, we study a viscoelastic sphere impacting a rigid, non-contacting surface and quantify how the elasticity number (El, dimensionless elastic modulus) and the Weissenberg number (Wi, dimensionless relaxation time) dictate the impact force. We recover the Newtonian liquid response as either El → 0 or Wi → 0, and obtain elastic-solid behavior in the limit Wi → ∞ and El ≫ 1. In this elastic-memory limit, three regimes emerge – capillary-dominated, Wagner scaling, and Hertz scaling – with a smooth transition from the Wagner to the Hertz regime. Sweeping Wi from 0 to ∞ reveals a continuous shift from materials with no memory to materials with permanent memory of deformation, providing an alternate, controlled route from liquid drops to elastic beads. The study unifies liquid and solid impact processes and offers a general framework for the liquid-to-elastic transition relevant across systems and applications.