Simulations of inertial liquid-lens coalescence with the pseudopotential lattice Boltzmann method

TL;DR

This work addresses inertial coalescence of liquid lenses at a liquid-fluid interface across a wide range of contact angles using the pseudopotential multicomponent lattice Boltzmann method. By simulating both 2D and 3D configurations with three equal-density fluids, the authors validate equilibrium lens shapes against analytical Neumann-law expressions and explore dynamic coalescence, revealing self-similar bridge evolution in 2D that agrees with experiments for small angles ($\\theta \\lesssim 40^{\\circ}$) and highlighting the breakdown of thin-sheet theory at larger angles. In 3D, the initial bridge-radius growth is found to be independent of the initial contact angle, while the bridge-height growth increases with $\\theta$, and the bridge-cross-section angle evolves nonlinearly before approaching equilibrium. The results provide a computationally efficient framework for predicting coalescence timescales in multi-component systems with potential applications in inkjet printing and fog harvesting, and they identify limits of current thin-sheet models, suggesting avenues for future theoretical development.

Abstract

The coalescence of liquid lenses is relevant in various applications, including inkjet printing and fog harvesting. However, the dynamics of liquid-lens coalescence have been relatively underexplored, particularly in the case of liquid lenses with larger contact angles. We numerically investigate the coalescence of low-viscosity liquid lenses by means of the pseudopotential multi-component lattice Boltzmann method over a wide range of contact angles. In two-dimensional simulations, our numerical results on the growth of the bridge height are in quantitative agreement with experimental measurements for small contact angles. In addition, by comparing our simulation results with a theoretical approach based on the thin-sheet equations for liquid lenses, we find that the thin-sheet equations accurately capture the bridge-growth dynamics up to contact angles of approximately $θ< 40^{\circ}$. For the three-dimensional case, the growth of the bridge radius is independent of the equilibrium contact angle of the liquid lenses at the initial stage of growth. The dependency between the growth of the bridge height and the bridge radius exhibits a non-linear to linear transition.

Figures (8)

• Figure 1: Schematic of a single liquid lens (fluid $3$) at an interface between fluid $1$ and fluid $2$ in the equilibrium state.
• Figure 2: Snapshots of a single liquid lens in equilibrium state obtained in simulations for different combinations of surface tensions: a) $\gamma_{12}:\gamma_{13}:\gamma_{23}= 1:1:1$ , b) $\gamma_{12}:\gamma_{13}:\gamma_{23}= 1.5:1:1$ , c) $\gamma_{12}:\gamma_{13}:\gamma_{23}= 1:1.5:1$. The color represents the density difference between lens liquid $2$ and the lower liquid $3$, $\Delta \rho_{23}=\rho_2-\rho_3$.
• Figure 3: The height profile of the upper half of a single liquid lens in equilibrium state for different surface tension ratios: $\gamma_{12}:\gamma_{13}:\gamma_{23}= 1:1:1$ (red), $\gamma_{12}:\gamma_{13}:\gamma_{23}= 1.5:1:1$ (green) and $\gamma_{12}:\gamma_{13}:\gamma_{23}= 1:1.5:1$ (blue). The simulation results (symbols) agree quantitatively with the analytical solution Eq. (\ref{['eq:height1']}) (solid lines).
• Figure 4: Schematic of the side view of two coalescing lenses at a fluid-fluid interface. The lenses are up-down symmetric, and the contact angle is $\theta$. The maximal bridge height is $H$, and the bridge height at the center is $h_0$. $L$ is the distance between the far ends of the two lenses.
• Figure 5: Time sequence of the coalescence process of liquid lenses in a 2D system obtained in our simulations. The lenses merge upon contact, connected by a rapidly growing bridge, and ultimately form a larger lens that relaxes into its equilibrium shape. The color represents the density difference between lens liquid $2$ and the lower liquid $3$, $\Delta \rho_{23}=\rho_2-\rho_3$.