One-dimensional modeling of pressure-driven droplet pinch-off

The aim of this research is to develop a mathematical model to simulate the effects of shear force on droplet formation. Droplet formation has been an interest of study since the seventeenth-century [L. Rayleigh, "On the instability of jets," Proceedings of the London mathematical society, vol. 1, no. 1, pp. 4–13, 1878]. Droplets are seen in applications like emulsion, spraying, ink-jet printing, atomization and entrainment, and many more. Understanding the droplet dynamics is crucial to improve the efficacy of these processes. The self-similar nature of the flow with a singularity at the pinch-off requires accurate modeling of the curvature. We present a one-dimensional model to simulate the droplet pinch-off in a quiescent environment using front tracking [D. K. Nathawani and M. G. Knepley, "Droplet formation simulation using mixed finite elements", Physics of Fluids, vol. 34, no. 6, p. 064105, 2022]. The model is verified using the method of manufactured solution and validated against laboratory experiments. We propose a self-consistent algorithm with an adaptively refined mesh. Furthermore, we expand the model for droplets in a pressure-driven continuous outer flow to incorporate the effects of shear force on droplet evolution.