Bounds on the Spreading Radius in Droplet Impact: The Viscous Case

TL;DR

This paper derives bounds on the maximum spreading radius of a viscous droplet impacting a surface, confirming the dominant scaling law and providing corrections aligned with experimental observations.

Contribution

It introduces a regularization of the rim-lamella model's singularity and derives bounds on spreading radius without relying on dimensional analysis.

Findings

Confirmed the $ ext{Re}^{1/5}$ scaling law for spreading radius.

Derived correction involving $ ext{We}^{-1/2} ext{Re}^{2/5}$ consistent with experiments.

Bounds agree with numerical solutions and energy-budget calculations.

Abstract

We consider the problem of droplet impact and droplet spreading on a smooth surface in the case of a viscous Newtonian fluid. We revisit the concept of the rim-lamella model, in which the droplet spreading is described by a system of ordinary differential equations (ODEs). We show that these models contain a singularity which needs to be regularized to produce smooth solutions, and we explore the different regularization techniques in detail. We adopt one particular technique for further investigation, and we use differential inequalities to derive upper and lower bounds for the maximum spreading radius $\mathcal{R}_{max}$. Without having to resort to dimensional analysis or scaling arguments, our bounds confirm the leading-order behaviour $\mathcal{R}_{max} \sim \mathrm{Re}^{1/5}$, as well as a correction to the leading-order behaviour involving the combination $\mathrm{We}^{-1/2}\mathrm{Re}^{2/5}$, in agreement with the experimental literature on droplet spread. Our bounds are also consistent with full numerical solutions of the ODEs, as well as with energy-budget calculations, once head loss is accounted for in the latter.