A universal shape function for rising jets

TL;DR

This paper introduces a universal shape function for rising jets, showing that their shape remains consistent over time and is governed by fluid parameters and deceleration, with implications for understanding splash dynamics.

Contribution

It proposes a universal, shape-preserving model for rising jets based on surface tension and deceleration, validated by numerical solutions and experimental data.

Findings

The jet shape is universal and determined by three physical parameters.

The shape scales with fluid parameters and deceleration.

Deceleration from shape matches that from height-time data.

Abstract

A small drop that splashes into a deep liquid sometimes reappears as a small rising jet, for example when a water drop splashes into a pool or when coffee drips into a cup. Here we describe that the growing and rising jet continuously redistributes its fluid to maintain a universal shape originating from a surface tension based deceleration of the jet; the shape is universal in the sense that the shape of the rising jet is the same at all times; only the scaling depends on fluid parameters and deceleration. An inviscid equation of motion for the jet is proposed assuming a time dependent but uniform deceleration; the equation of motion is made dimensionless by using a generalized time-dependent capillary length ${\lambda_c}$ and is solved numerically. As a solution a concave shape function is found that is fully determined by three measurable physical parameters: deceleration, mass density and surface tension; it is found that the surface tension based deceleration of the jet scales quadratic with the size of the jet base. Deceleration values derived from the jet shape are in good agreement with deceleration values calculated from the time plot of the height of the rising jet.