Birth of a bubble: Drop impact onto a thin liquid film for an immiscible three-fluid system

TL;DR

This study addresses bubble birth during drop impact onto a thin immiscible liquid film by extending a three-fluid Volume-of-Fluid approach in Basilisk to air–water–oil systems. The authors quantify how gas-layer lubrication and deformation of the oil film govern the entrapment time and geometry, developing a new scaling that includes film deformation through an effective Stokes number, $St_{eff}=\sqrt{1+\delta\epsilon/h_c}\,St$. The deformation $\delta\epsilon$ is described by a Padé fit to capture both inviscid and viscous regimes, enabling collapse of solid-substrate and film-case data when expressed in terms of $St_{eff}$. The work demonstrates that while gas-layer cushioning dominates the dynamics, the immiscible film delays entrapment via its deformation, with implications for applications in printing, de-icing, and emulsification where control of bubble formation is important.

Abstract

When a drop impacts a solid substrate or a thin liquid film, a thin gas disc is entrapped due to surface tension, the gas disc retracts into one or several bubbles. While the evolution of the gas disc for impact on solid substrate or film of the same fluid as the drop have been largely studied, little is known on how it varies when the liquid of the film is different that of the drop. We study numerically the latter unexplored area, focussing on the contact between the drop and the film, leading to the formation of the air bubble. The volume of fluid method was adapted to three fluids in the framework of Basilisk solver. The numerical simulations show that the deformation of the liquid film due to the air cushioning plays a crucial role in the bubble entrapment. A new model for the contact time and the entrapment geometry was deduced from the case of the impact on a solid substrate. This was done by considering the deformation of the thin immiscible liquid layer during impact depending mainly on its thickness and viscosity. The lubrication of the gas layer was found to be the major effect governing the bubble entrapment. However the film viscosity was also identified as having a critical role in bubble formation and evolution; the magnitude of its influence was also quantified.

Figures (12)

• Figure 1: Left: impact geometry with the different parameters of interest. Right: graphical description of the deformed droplet prior to the contact, illustrating the different measured quantities of interest, the thickness of the gas layer under the center of the droplet $h_{d}$, the minimal distance between the droplet and the substrate $h_{min}$ and the radius of the dimple $r_{d}$. Right and left part of the figure indicate the difference between the impact on solid or liquid substrate with the introduction of $\delta \epsilon$ to describe he deformation of the oil film.
• Figure 2: Snapshots of the impact at several time step (indicated on the images), with different geometrical sizes. The top image shows the initial condition, while the pyramidal images below it present the dynamics at scale two. Two zoom images (scale 1/10) on the left and right of the pyramid illustrate the formation and retraction of the air film entrapped by the impact. The first contact between the water droplet (red) and the oil substrate (yellow) occurs around the $46$$\mu s$. Then a small disc of gas (in blue) is entrapped and retracts, while a corona made of a mix of oil and water emerges from the impact at larger times.
• Figure 3: (left) Dimensionless time of contact $\bar{t}_c$ of a drop impacting on a solid as function of the Stokes number $St$. The data set are obtained for two surface tension, grey $\sigma_{wg}=7.2 \cdot 10^{-2}$ and red $\sigma_{wg}=7.2*10^{-4}$$N \cdot m^{-1}$, varying both the impact velocity and the gas viscosity. The expected scaling law $St^{2/3}$ is indicated (blue line); (right) the dimensionless radius $\bar{r}_c$ (upper curve) and central height $\bar{h}_c$ (lower curve). Both are in agreement with the predicted scaling laws $St^{1/3}$ (resp $St^{2/3}$).
• Figure 4: Snapshot at $t=0.034$$ms$ (a) Impact over a solid substrate. (b) Impact on a liquid film. The two initial conditions are exactly the same. $R_0=3$$mm$, $h_{0}=0.3$$mm$, $U_0=1$$m/s$, $\mu_w=0.002$$Pa.s$, $\mu_o=0.02$$Pa.s$, $\sigma_{wg}=0.07$, $\sigma_{wo}=0.039$, $\sigma_{og}=0.032$$N/m$, $Re=1500$, $We_{wg}=43$, $St=5.3*10^{-5}$.
• Figure 5: Two snapshots ($t=0.034$$ms$) for two similar impacts on the different oil (left $\mu_o=\mu_w=0.002$$Pa.s$, right $\mu_o=0.02$$Pa.s$). The bubble size is dependent on the oil viscosity. Simulation realized with the following parameters. $St=5.3*10^{-5}$, $U_0=1$$m/s$, $D_{0}=3$$mm$, $h_{0}=0.3$$mm$, $\sigma_{wg}=0.032$, $\sigma_{wo}=0.039$, $\sigma_{og}=0.07$$N.m^{-1}$.