Bounds on the spreading radius in droplet impact: the inviscid case

Abstract

We consider the classical problem of droplet impact and droplet spread on a smooth surface in the case of an ideal inviscid fluid. We revisit the rim–lamella model of Roisman et al. (Roisman et al . 2002 Proc. R. Soc. Lond. A 458 , 1411–1430 ( doi:10.1098/rspa.2001.0923 )). This model comprises a system of ordinary differential equations (ODEs); we present a rigorous theoretical analysis of these ODEs, and derive upper and lower bounds for the maximum spreading radius. Both bounds possess a W e 1 / 2 scaling behaviour, and by a sandwich result, the spreading radius itself also possesses this scaling. We demonstrate rigorously that the rim–lamella model is self-consistent: once a rim forms, its height will invariably exceed that of the lamella. We introduce a rational procedure to obtain initial conditions for the rim–lamella model. Our approach to solving the rim–lamella model gives predictions for the maximum droplet spread that are in close agreement with existing experimental studies and direct numerical simulations.