Capillary liquid channels in cylindrical support surfaces: stability and bifurcation

TL;DR

The paper studies capillary surfaces on cylindrical supports under negligible gravity and translational symmetry, formulating the energy and constant-mean-curvature conditions with a fixed contact angle $\gamma$. It develops a rigorous stability framework via the Jacobi operator $\mathcal{L}=\Delta+|A|^2$ and the boundary term, establishing how stability (via the quadratic form $Q$ and Morse indices) governs the behavior of planar strips and circular-cylinder segments through Plateau-Rayleigh-type instabilities. Building on this, it applies Crandall–Rabinowitz bifurcation theory to show that, when $0$ is a simple eigenvalue of $\mathcal{L}$, new families of cmc liquid channels bifurcate from a given circular cylinder, with explicit examples on parabolic and catenary cylindrical supports demonstrating the mechanism. The explicit computations for $\gamma=\pi/2$ reveal precise bifurcation radii and configurations, highlighting how curvature of the support at contact points controls stability and the emergence of novel morphologies in microchannel-like systems.

Abstract

Planes and circular cylinders are models of interfaces of a fluid when the support surface is translationally invariant in a direction of the space. After a study of the eigenvalues of the Jacobi operator, it is investigated when planar strips and sections of circular cylinders are stable in cylindrical symmetric support surfaces. This analysis depends on the curvature of the support at the contact points with the interface. The Plateau-Rayleigh instability phenomenon is studied finding the critical value $h_0>0$ such that rectangular pieces of planar strips or circular cylinders of length greater than $h_0$ are necessarily unstable. It is also studied when new morphologies of capillary surfaces can emerge from given circular cylinders. Using the method of bifurcation by simple eigenvalues, we establish conditions on the support surface that prove that when $0$ is a simple eigenvalue of the Jacobi operator, there is bifurcation from explicit circular cylinders. It will be presented examples of supports (parabolic and catenary cylinders) where this bifurcation appears.

Figures (7)

• Figure 1: Cross section of a liquid channel $\Omega$ deposited on a cylindrical support $\mathcal{S}(\mathbf{c})$. It is assumed that $\mathcal{S}(\mathbf{c})$ is symmetric about a vertical plane. The contact angle is determined by the unit normal vectors $N$ and $\widetilde{N}$ of $\Sigma$ and $\mathcal{S}(\mathbf{c})$, respectively.
• Figure 2: A plane on $\mathcal{S}(\mathbf{c})$ with different contact angles (left). A capillary strip on $\mathcal{S}(\mathbf{c})$ (middle). A capillary plane outside of $\mathcal{S}(\mathbf{c})$ (right).
• Figure 3: A capillary planar strip on $\mathcal{S}(\mathbf{c})$. The different types are: $\widetilde{A}>0$ and $\gamma\in (0,\frac{\pi}{2})$ (a); $\widetilde{A}>0$ and $\gamma\in ( \frac{\pi}{2},\pi)$ (b); $\widetilde{A}<0$ and $\gamma\in (0,\frac{\pi}{2})$ (c); $\widetilde{A}<0$ and $\gamma\in (\frac{\pi}{2},\pi)$ (d).
• Figure 4: Case $A(\nu,\nu)=-1/r$. The different types are: $\widetilde{A}>0$ and $\gamma\in (0,\frac{\pi}{2})$ (a); $\widetilde{A}>0$ and $\gamma\in ( \frac{\pi}{2},\pi)$ (b); $\widetilde{A}<0$ and $\gamma\in (0,\frac{\pi}{2})$ (c); $\widetilde{A}<0$ and $\gamma\in (\frac{\pi}{2},\pi)$ (d).
• Figure 5: Case $A(\nu,\nu)=1/r$. The different types are: $\widetilde{A}>0$ and $\gamma\in (0,\frac{\pi}{2})$ (a); $\widetilde{A}>0$ and $\gamma\in ( \frac{\pi}{2},\pi)$ (b); $\widetilde{A}<0$ and $\gamma\in (0,\frac{\pi}{2})$ (c); $\widetilde{A}<0$ and $\gamma\in (\frac{\pi}{2},\pi)$ (d).

Theorems & Definitions (18)

• Example 2.1: Planar strips
• Example 2.2: Sections of circular cylinders
• Remark 3.1
• Theorem 3.2
• proof
• Example 3.3
• Example 3.4
• Theorem 4.1
• Corollary 4.2
• proof