# Shape of a recoiling liquid filament

## TL;DR

This paper analytically investigates the capillary retraction of a Newtonian liquid filament, revealing detailed dynamics including wave formation, length scales, and the influence of viscosity, extending classical understanding of filament retraction.

## Contribution

It provides a comprehensive analytical model for the retraction process, identifying wave characteristics and scale regions, and compares well with numerical simulations and previous studies.

## Key findings

- Capillary waves have a wavelength about 3.63 times the filament radius.
- Wavelength, decay length, and neck size depend on the Ohnesorge number.
- Good agreement with numerical simulations and prior literature.

## Abstract

We study the capillary retraction of a Newtonian semi-infinite liquid filament through analytical methods. We derive a long-time asymptotic-state expansion for the filament profile using a one-dimensional free-surface slender cylindrical flow model based on the three-dimensional axisymmetric Navier-Stokes equations. The analysis identifies three distinct length and time scale regions in the retraction domain: a steady filament section, a growing spherical blob, and an intermediate matching zone. We show that liquid filaments naturally develop travelling capillary waves along their surface and a neck behind the blob. We analytically prove that the wavelength of the capillary waves is approximately 3.63 times the filament's radius at the inviscid limit. Additionally, the waves' asymptotic wavelength, decay length, and the minimum neck size are analysed in terms of the Ohnesorge number. Finally, our findings are compared with previous results from the literature and numerical simulations in Basilisk obtaining a good agreement. This analysis provides a full picture of the recoiling process going beyond the classic result of the velocity of retraction found by Taylor and Culick.