Comparison of the potential energy for different equilibrium configurations of symmetric and asymmetric floating drops

TL;DR

The paper addresses the equilibrium of three immiscible fluids in a bounded capillary by formulating a free-boundary variational problem and solving the resulting Young-Laplace equations with a Newton-Chebyshev collocation approach on generating curves. It reveals pervasive non-uniqueness of both Euler-Lagrange solutions and energy minimizers, with symmetry breaking observed in 2D and, to a lesser extent, in 3D; energy landscapes are explored across a nine-parameter space including volume, container size, densities, surface tensions, and wetting energies. Key contributions include explicit volume-matching formulas for central and wall-bound drops, robust numerical strategies for solving free-boundary problems, and detailed parameter studies showing scenarios where central, wall-bound, or asymmetric configurations minimize energy, as well as cases with multiple equal-energy states. The results have implications for interpreting experiments with floating drops and provide a computational framework and data to guide future investigations into capillary interfaces and energy minimization in multi-fluid systems, with code available at the provided GitHub repository.

Abstract

We provide a numerical method for computing solutions to a free boundary problem arising from the equilibrium state of a floating drop. This numerical method is based on a Newton's method for the underlying nonlinear boundary value problems, and at each iterative step a Chebyshev spectral collocation method is employed. The problems considered here are those that can be described by using generating curves, and include problems in $\mathbb{R}^2$ and $\mathbb{R}^3$. The resulting nine-dimensional space of physical parameters is explored, and examples are given that highlight the potential energy of centrally located drops, wall-bound drops, and asymmetrical configurations in $\mathbb{R}^2$. Non-uniqueness of solutions to the corresponding Euler-Lagrange equations is displayed, and also strong evidence of non-uniqueness of energy minimizers is given.

Figures (24)

• Figure 1: Schematically, a drop of oil floats on the surface of water inside a capillary tube. Shown is a vertical section of an axially symmetric configuration in $\mathbb{R}^3$. The capillary tube is indicated by the two vertical lines, and the tube walls will be omitted in all other figures in this work.
• Figure 2: The surface tensions are shown in a force balance form as vectors, and the contact angles associated contact angles are shown. The plate angles are also shown. The floating drop pictured here is a wall-bound drop in $\mathbb{R}^2$.
• Figure 3: A typical centrally located floating drop. The physical parameters that determine this drop are a volume of 1, tube radius of $R=2$, densities $\rho_1 = 1$ and $\rho_2 = 15$, surface tensions $\sigma_{01} = 6, \sigma_{02} = 2.0001$, and $\sigma_{12} = 7.9999$, and plate angle $\gamma^2_{0p} = 0.5$.
• Figure 4: A typical wall-bound floating drop. The physical parameters that determine this drop are a volume of 1, tube radius of $R=2$, densities $\rho_1 = 14$ and $\rho_2 = 15$, surface tensions $\sigma_{01} = 6.01, \sigma_{02} = 2.99$, and $\sigma_{12} = 7$, and plate angles $\gamma^1_{0p} = \pi - 0.01$ and $\gamma^2_{1p} = 1$.
• Figure 5: Displayed are analogues of the symmetric floating drops in $\mathbb{R}^2$. On the left is a centrally located drop with the physical parameters that determine this drop being a volume of 1, tube radius of $X=2$, densities $\rho_1 = 0.4$ and $\rho_2 = 15$, surface tensions $\sigma_{01} = 7.9, \sigma_{02} = 2$, and $\sigma_{12} = 6.1$, and plate angle $\gamma^2_{0p} = \pi - 0.01$. On the right is a wall-bound drop with the physical parameters that determine this drop being a volume of 1.5 split evenly on each wall, tube radius of $X=2$, densities $\rho_1 = 7.5$ and $\rho_2 = 15$, surface tensions $\sigma_{01} = 7.5, \sigma_{02} = 2.5$, and $\sigma_{12} = 6$, and plate angles $\gamma^1_{0p} = 0.01$ and $\gamma^2_{1p} = 0.01$.