Self-Similar Liquid Lens Coalescence

A basic feature of liquid drops is that they can merge upon contact to form a larger drop. In spite of its importance to various applications, drop coalescence on prewetted substrates has received little attention. Here, we experimentally and theoretically reveal the dynamics of drop coalescence on a thick layer of a low viscosity liquid. It is shown that these so-called “liquid lenses” merge by the self-similar vertical growth of a bridge connecting the two lenses. Using a slender analysis, we derive similarity solutions corresponding to the viscous and inertial limits. Excellent agreement is found with the experiments without any adjustable parameters, capturing both the spatial and temporal structures of the flow during coalescence. Finally, we consider the crossover between the two regimes and show that all data of different lens viscosities collapse on a single curve capturing the full range of the coalescence dynamics.

Figure 1

(a) Schematic view of two coalescing liquid lenses connected by a bridge of height $h_0(t)$. The lenses float on a pool with a depth that is much larger than the size of the lenses. The zoomed region shows a typical snapshot of the bridge region. (b) Measurements of the bridge height $h_0$ as a function of time $t$ for several viscosities. Two distinct power laws are identified.

Figure 2

Coalescence in the viscous regime. (a) Height of the bridge $h_0$ as a function of time after contact $t$ (mineral oil lenses, $\theta =33°$, $\eta =115487\mathrm{mPa}·\mathrm{s}$, initial height $\approx 0.5\mathrm{mm}$). The solid line is the prediction from (6). The error bars are only shown for one in every ten datapoints for clarity. The horizontal dashed line indicates the resolution limit. (b) Rescaled experimental profiles at different times, $\mathcal{H}=h(x,t)/h_0(t)$ versus $\xi =x\theta /h_0(t)$. The collapse of the profiles indicates self-similar dynamics. The solid line is the similarity solution obtained from (4), (5). (c) Rescaled velocity profile. The solid line is the similarity solution. The colored lines are numerical simulations for different values of $h_0/R$.

Figure 3

Coalescence in the inertial regime. (a) Height of the bridge $h_0$ as a function of time after contact $t$ (dodecane lenses, $\theta =29°$, $\eta =1.36\mathrm{mPa}·\mathrm{s}$, initial height $\approx 0.5\mathrm{mm}$). The solid line is the prediction from (9). The horizontal dashed line indicates the resolution limit. (b) Rescaled experimental profiles at different times, $\mathcal{H}=h(x,t)/h_0(t)$ versus $\xi =x\theta /h_0(t)$. The collapse of the profiles indicates self-similar dynamics. The solid line is the similarity solution obtained from (4), (8). (c) Rescaled velocity profile. The solid line is the similarity solution. The colored lines are numerical simulations for different values of $h_0/R$.

Figure 4

Crossover between the viscous and inertial regimes, shown by a collapse of all experimental data on a master curve. Dashed line: viscous theory. Dotted-dashed line: inertial theory. Solid line: interpolation based on (11). The lime-colored data points are with larger lens size ($R\approx 4.1\mathrm{mm}$, compared to $R\approx 2.5\mathrm{mm}$ for all other data, showing that the dynamics do not depend on the drop size).