Three-Dimensional Self-Similarity of Coalescing Viscous Drops in the Thin-Film Regime

Coalescence and breakup of drops are classic problems in fluid physics that often involve self-similarity and singularity formation. While the coalescence of suspended drops is axisymmetric, the coalescence of drops on a substrate is inherently three-dimensional. Yet, studies so far have only considered this problem in two dimensions. In this Letter, we use interferometry to reveal the three-dimensional shape of the interface as two drops coalescence on a substrate. We unify the known scaling laws in this problem within the thin-film approximation and find a three-dimensional self-similarity that enables us to describe the anisotropic shape of the dynamic interface with a universal curve.

Figure 1

Three-dimensional reconstruction of the shape of the interface using interferometry. (a) Schematic showing the drop geometry and the interferometry setup used in the experiments. The base radius $R_0$, height $H_0$, and the radius of the spherical cap $R$ that describes the shape of the drop at the time of coalescence are labeled. (b) Interferometry images showing two silicone oil drops ($\mu =102\mathrm{mPa}\mathrm{s}$, $V=2.53\mu \mathrm{l}$) at $t\approx 0$, 4, and 6 s after coalescence. Dashed yellow lines are guides for the eyes and show the macroscopic contact line of the drops. The half-width of the meniscus bridge is labeled $r_m(t)$. Scale bar represents $100\mu \mathrm{m}$. (c) Side view image of the drops near the time of contact. Scale bar represents 1 mm. (d) Reconstructed 3D shape of the interface at $t\approx 5\mathrm{s}$.

Figure 2

The height profile in the $xz$ plane at any arbitrary $y$ location is self-similar. (a) Schematic of the height profile along this plane at $y=y_0$, where $h_{y_0}(t)$ is the height at the coalescence point in that plane. (b) Data from an experiment using silicone oil drops with $\mu =102\mathrm{mPa}\mathrm{s}$ and $V=2.53\mu \mathrm{l}$ showing the time evolution of the height profile along the $xz$ plane with $y_0=168\mu \mathrm{m}$. The lines connecting the markers are just guides for the eyes. (c) The height profile rescaled with the height at the initial coalescence point $h_0(t)=vt$. (d) The height profile rescaled with the height at the coalescence point in this specific plane, $h_{y_0}(t)$, shows agreement with the similarity solution (black line [10]). The red dot in the inset is a schematic representation of the location of $h_{y_0}$.

Figure 3

The height profile along the meniscus bridge in the $yz$ plane is a self-similar parabola. (a) Schematic of the height profile, which resembles a circular segment of radius $a$. (b) Data from an experiment using silicone oil drops with $\mu =102\mathrm{mPa}\mathrm{s}$ and $V=2.53\mu \mathrm{l}$ showing the time evolution of the height profile. The lines connecting the markers are just guides for the eyes. (c) Rescaled height profile collapses the data and agrees with the proposed model [Eq. (1)]. (d) The value of the length scale $a$ used to collapse all of the experimental data plotted against $V/{\theta }^4$ follows a $1/3$ power law (black line). The fitted slope was 0.29.

Figure 4

The dynamic 3D interface can be mapped onto a universal curve. (a) The 3D shape of the interface at four different times during coalescence, $t=4.8$, 7.0, 13.7, 19.6 s, with the lighter shade of blue corresponding to later times. (b) The 3D shapes of the interface collapse onto a single surface when the axes are rescaled as shown. Data are from an experiment using silicone oil drops with $\mu =102\mathrm{mPa}\mathrm{s}$, $V=2.53\mu \mathrm{l}$, and $a=0.01\mathrm{m}$. (c) Rescaling as shown in Eq. (3) gives the universal shape of the 3D interface. The experimental data are from eight experiments varying viscosity and volume of the drops as shown in the legend. The darker colored markers correspond to early times and the lighter colors markers correspond to later times during coalescence for a given experiment. For each experiment, interface profiles at four or five different times each are shown at three different $y$ locations between 0 and $200\mu \mathrm{m}$, represented by circle, square, and diamond markers. The black line is the numerical solution to Eqs. (4) and (5).