Droplet formation simulation using mixed finite elements

TL;DR

This paper develops a robust computational model using mixed finite elements to simulate droplet formation, capturing complex shapes and dynamics near pinch-off, validated against experiments and previous models.

Contribution

It introduces a new finite element based simulation framework for axisymmetric droplet formation, improving robustness and accuracy over previous methods.

Findings

Validated against experimental data

Successfully simulated droplets of water, glycerol, and paraffin wax

Provides a foundation for simulating complex pinch-off phenomena

Abstract

Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate droplet size but fails to approximate droplet shape. This is due to a highly non-linear flow description near the point where the first pinch-off happens. A one-dimensional axisymmetric mathematical model was first developed by Eggers and Dupont using asymptotic analysis. This asymptotic approach to the Navier-Stokes equations leads to a universal scaling explaining the self-similar nature of the solution. Numerical models for the one-dimensional model were developed using the finite difference and finite element method. The focus of this study is to provide a robust computational model for one-dimensional axisymmetric droplet formation using the Portable, Extensible Toolkit for Scientific Computation (PETSc). The code is verified using the Method of Manufactured Solutions (MMS) and validated using previous experimental studies done by Zhang and Basaran. The present model is used for simulating pendant drops of water, glycerol, and paraffin wax, with an aspiration of extending the application to simulate more complex pinch-off phenomena.