Droplet formation simulation using mixed finite elements

Droplet formation happens in finite time due to the surface tension force. The linear stability analysis is useful to estimate the size of a droplet but fails to approximate the shape of the droplet. This is due to a highly nonlinear flow description near the point where the first pinch-off happens. A one-dimensional axisymmetric mathematical model was first developed by Eggers and Dupont [ Drop formation in a one-dimensional approximation of the Navier-Stokes equation, J. Fluid Mech. 262, 205-221 (1994)] 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 [Eggers and Dupont, Drop formation in a one-dimensional approximation of the Navier-Stokes equation, J. Fluid Mech. 262, 205-221 (1994)] and finite element method [Ambravaneswaran et al., Drop formation from a capillary tube: Comparison of one-dimensional and two-dimensional analyses and occurrence of satellite drops, Phys. Fluids 14, 2606-2621 (2002)]. 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. The code is verified using the Method of Manufactured Solutions and validated using previous experimental studies done by Zhang and Basaran [ An experimental study of dynamics of drop formation, Phys. Fluids 7, 1184-1203 (1995)]. 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. Published under an exclusive license by AIP Publishing.

Automated summary

Droplet formation occurs in a finite amount of time due to surface tension force. Linear stability analysis is useful in estimating the droplet size but fails to approximate its shape. A one-dimensional axisymmetric mathematical model developed through asymptotic analysis provides a universal scaling that explains the self-similar nature of the droplet formation solution.