Droplet impact onto a spring-supported plate: analysis and simulations

Abstract

The high-speed impact of a droplet onto a flexible substrate is a highly non-linear process of practical importance, which poses formidable modelling challenges in the context of fluid–structure interaction. We present two approaches aimed at investigating the canonical system of a droplet impacting onto a rigid plate supported by a spring and a dashpot: matched asymptotic expansions and direct numerical simulation (DNS). In the former, we derive a generalisation of inviscid Wagner theory to approximate the flow behaviour during the early stages of the impact. In the latter, we perform detailed DNS designed to validate the analytical framework, as well as provide insight into later times beyond the reach of the proposed analytical model. Drawing from both methods, we observe the strong influence that the mass of the plate, resistance of the dashpot, and stiffness of the spring have on the motion of the solid, which undergo forced damped oscillations. Furthermore, we examine how the plate motion affects the dynamics of the droplet, predominantly through altering its internal hydrodynamic pressure distribution. We build on the interplay between these techniques, demonstrating that a hybrid approach leads to improved model and computational development, as well as result interpretation, across multiple length and time scales.

Inertio-capillary rebound of a droplet impacting a fluid bath

“…Figure 3 (accompanied by movie supplementary material) also outlines velocity field information inside each of the fluid phases (figure 3aii), as well as the aforementioned gas film thickness (figure 3b) and pressure distribution (figure 3c) across relevant flow stages. Manipulating raw pressure data in projection methods is a known challenge (Philippi, Lagrée & Antkowiak 2016;Negus et al 2021), with the oscillatory behaviour observed in figure 3(c) linked to the VOF approximation on a structured grid, and the underlying colour function sampled across various approximation points inside the cells in near vicinity of interfaces with large value contrasts in the quantities of interest. We underline in particular that the aforementioned variations oscillate around well-defined means within the contact region.…”

Pre-impact dynamics of a droplet impinging on a deformable surface

“…While the present study focuses solely on the pre-impact dynamics of droplet impact on the deformable surfaces, important questions remain about what happens next; post-impact. The problem of droplet post-impact theory onto deformable surfaces has been tackled previously 13,15 by ignoring the air cushioning phase and the entrapped air. However, these features do influence the post-impact dynamics, 26,70 and so questions remain on how this occurs in impact on deformable surfaces.…”

“…When substituting scales (15) and expansions ( 16) into the governing equations for the gas (4), for Re ) 1 the leading order equations take the form of lubrication flow,…”

“…There have been a number of experimental studies on droplet impacts with flexible, or soft deformable, substrates, considering both the pre-impact 8,9 and post-impact behaviors. [10][11][12] There has also been some analytical work on the post-impact behavior [13][14][15] and the pre-impact dynamics of droplet settling, 16 whereas the novelty in our work lies in analyzing the pre-impact behavior of a high-speed droplet impact with a deformable surface.…”

“…Using a similar approach, the effect of surface vibrations was examined in more detail by Khabakhpasheva and Korobkin. 38 Also using postimpact axisymmetric Wagner theory, Negus et al 15 recently investigated droplet impact on a spring-supported plate, where they found solutions for the composite pressure and force on the plate, and provided an excellent comparison to results obtained via direct numerical simulation. Xiong et al 14 performed numerical simulations of a droplet impacting a flexible surface using a Lattice-Boltzmann method, investigating the effect of bending stiffness on the contact time and wettability of the droplet on the surface.…”