Splashing on elastic membranes: The importance of early-time dynamics

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Original entry: Scott Tsai, APPHY 226, Spring 2009

"Splashing on elastic membranes: The importance of early-time dynamics"

Rachel E. Pepper, Laurent Courbin, and Howard A. Stone

Physics of Fluids 20, 082103 (2008)

Soft Matter Keywords

Splashing, membrane, droplet, lamella, surface tension, interface, wettability, soft substrates.

Overview

(From paper)

We study systematically the effect of substrate compliance on the threshold for splashing of a liquid drop using an elastic membrane under variable tension. We find that the splashing behavior is strongly affected by the tension in the membrane and splashing can be suppressed by reducing this tension. The deflection of the membrane upon droplet impact is measured using a laser sheet, and the results allow us to estimate the energy absorbed by the film upon drop impact. Measurements of the velocity and acceleration of the spreading drop after impact indicate that the splashing behavior is set at very early times after, or possibly just before, impact, far before the actual splash occurs. We also provide a model for the tension dependence of the splashing threshold based on the pressure in the drop upon impact that takes into account the interplay between membrane tension and drop parameters.

Soft Matter Examples

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Fig.3

The authors of this paper describe their study of liquid droplet impacting on a soft substrate. They emphasize that it is important to understand splashing on soft substrates because there are many common industrial applications such as spray cooling of flexible surfaces, and examples in nature, such as raindrops falling on leaves, that can be modeled as liquid drops falling on a soft substrate.

Splashing is often defined as when satellite droplets are formed and ejected out of the rim that formed after a primary droplet impacts a solid surface. In the splashing literature an empirical relationship between the Weber and Reynolds numbers is often cited for the transition from spreading to splashing upon droplet impact.

<math> We^{1/2}Re^{1/4}=K </math>

Where the Weber number, We is defined as <math> 2 \rho R_{o} V_{o}^2 / \gamma </math>. And the Reynolds number, Re is defined as <math> 2 \rho R_{o} V_{o} / \mu </math>.

The velocity term <math> V_{o} </math> in this relationship is often referred to as the "threshold velocity", such that when the normal impact velocity of the drop is larger than the threshold velocity, splashing occurs, and when the normal impact velocity is lower, then the drop spreads on the surface.

The authors did a series of experiments here and showed that as the tension in the substrate (substrate tension) decreases, the threshold velocity for splashing increases [Fig. 1a and 1b].

They also conducted energy balances for late-time, and early-time upon droplet contact with the substrate.

For late-time, the initial kinetic energy associated with the impact of the spherical drop on the substrate is balanced with the surface energy of the drop after it has been spread to its maximum diameter. They did this analysis assuming droplet impact onto a solid substrate. By comparing the results of the analysis with experimental data from impact onto soft substrates at different tensions, they show that all of the data from 6 different substrate tensions collapse onto the same curve [Fig. 2]. Thus, they conclude that late-time dynamics does not provide insight into the splashing threshold changes that result from changing the tension in the soft substrate.

For early-time, the authors found the energy of the spherical drop at impact and the elastic energy stored in the soft membrane at maximum deflection at threshold velocities. By subtracting the two, they were able to obtain the kinetic energy in the fluid at the maximum deflection of the substrate. This can be viewed as the energy of the drop that determines whether a splash occurs or not. The authors used, this, along with further analysis regarding the timing of splashes, to develop a model that describes the splashing threshold for a drop impacting a soft substrate.

<math> V_{T} - \frac {\rho R_{o}^{3/2} V_{T}^{5/2}} {Tt^{*1/2}} = V_{TS} </math>

Where, <math> V_{T} </math> is the velocity threshold for splashing on the soft substrate, <math> V_{TS} </math> is the velocity threshold for splashing on an equivalent solid substrate system, T is the tension in the soft substrate, <math> R_{o} </math> is the drop radius, and <math> t^* </math> is the critical time when instabilities are important.

They compared the experimental values of threshold velocity on a soft substrate and found trend and order of magnitude matches with the model they developed, which shows that their model takes into account the major parameters controlling splashing on a soft substrate. [Fig. 3]