Stationary Soap Film Bridge Subjected to an Electrostatic Force: The Balanced Case

TL;DR

This paper analytically investigates the stability and deflection behavior of a stationary soap film bridge under electrostatic forces, identifying a critical threshold where stability changes and demonstrating monotonic outward deflection with increasing electrostatic strength.

Contribution

It introduces a mathematical model for a soap film bridge under electrostatic force, constructs stationary solutions, and identifies a stability threshold, providing rigorous analysis of the electrostatic balancing effect.

Findings

Identified a sharp threshold for stability transition.

Proved monotonic outward deflection with increasing electrostatic force.

Constructed local stationary solutions for the soap film bridge.

Abstract

We analytically study a stationary free boundary problem describing a soap film bridge subjected to an electrostatic force. Starting from a cylinder that the soap film forms if its surface tension and the electrostatic force are perfectly balanced, we construct a local branch of stationary solutions. Then, we prove that there is a sharp threshold value for a characteristic model parameter at which the stability behaviour of this branch switches from stable to unstable. Finally, we show that the soap film deflects monotonically outwards if the strength of the electrostatic force is increased. Besides rigorous results, we also discuss a possible balancing effect of the electrostatic force.