Liquid films falling down a vertical fiber: modeling, simulations and experiments

TL;DR

This paper develops and compares simplified models for gravity-driven liquid films on vertical fibers, analyzing their stability, simulating droplet formation, and validating results with experiments and machine learning parameter fitting.

Contribution

It introduces two coupled nonlinear PDE models for liquid film flow on fibers, incorporating various physical effects, and validates them through experiments and machine learning optimization.

Findings

Both models are unstable to certain wave disturbances.

Simulations produce droplet chain solutions.

Good agreement between experiments and model predictions.

Abstract

We present a control-volume approach for deriving a simplified model for the gravity-driven flow of an axisymmetric liquid film along a vertical fiber. The model accounts for gravitational, viscous, inertial and surface tension effects and results in a pair of coupled one-dimensional nonlinear partial differential equations for the film profile and average downward velocity as functions of time and axial distance along the fiber. Two versions of the model are obtained, one assuming a plug-flow velocity profile and a constant thin boundary layer thickness to model the drag force on the fluid, the other approximating the drag using the fully-developed laminar velocity profile for a locally uniform film. A linear stability analysis shows both models to be unstable to long waves or short wavenumbers, with a specific wavenumber in that range having a maximal growth rate. Numerical simulations confirm this instability and lead to nonlinear periodic traveling wave solutions which can be thought of as chains of identical droplets falling down the fiber. Physical experiments are also carried out on such a system using safflower oil as the working liquid and a taut fishing line as the fiber. A machine learning scheme is used to find the best set of parameters in the laminar flow model to match the experimental results to the simulations. Good agreement is found between the two, with parameter values that are quite close to their original estimates based on the approximate values of the physical parameters.