Self-similar impulsive capillary waves on a ligament

TL;DR

This paper investigates the short-time dynamics of a liquid ligament subjected to impulsive acceleration, revealing self-similar behaviors and a maximum velocity threshold linked to contact line de-pinning.

Contribution

It introduces both linear and non-linear self-similar solutions for ligament dynamics under impulsive acceleration, including the discovery of a maximum velocity limit.

Findings

The axial depth affected scales as √t.

Existence of a maximum driving velocity U* beyond which solutions do not exist.

Non-linear solution suggests contact line de-pinning at high velocities.

Abstract

We study the short-time dynamics of a liquid ligament, held between two solid cylinders, when one is impulsively accelerated along its axis. A set of one-dimensional equations in the slender-slope approximation is used to describe the dynamics, including surface tension and viscous effects. An exact self-similar solution to the linearized equations is successfully compared to experiments made with millimetric ligaments. Another non-linear self-similar solution of the full set of equations is found numerically. Both the linear and non-linear solutions show that the axial depth at which the liquid is affected by the motion of the cylinder scales like $\sqrt{t}$. The non-linear solution presents the peculiar feature that there exists a maximum driving velocity $U^\star$ above which the solution disappears, a phenomenon probably related to the de-pinning of the contact line observed in experiments for large pulling velocities.