A three-dimensional numerical model for the simulation of electrohydrodynamic varicose and whipping instabilities

Electrohydrodynamic (EHD) cone-jet is a widely studied research topic owing to its variable outcomes and broad applications in micro/nanoscopic additive manufacturing. Among all the EHD cone-jet outcomes, the varicose and whipping instabilities attract much attention because of their rich underlying physics involved in the unstable downward stretching jet. However, the mechanisms behind these two instabilities are still not adequately understood, partly due to the lack of a full three-dimensional numerical model for the simulation of these two instabilities. In this paper, we propose a three-dimensional numerical model and perform a numerical study on EHD varicose and whipping instabilities under variations of electric voltage, liquid electrical conductivity, and liquid flow rate. The numerical model is devised with a two-phase finite-volume-method based flow solver in the open-source CFD program OpenFOAM. From the numerical results, we obtain the distributions of electric charge density and liquid velocity at the liquid surface and around the liquid jet at critical instants and locations. This enables us to analyze the differences in Taylor cone formation with the variation of the three parameters and compare the differences between liquid jet emission of the varicose and whipping instabilities. Finally, we compare our numerical results with the experimental findings in recently published papers, with very good agreement achieved regarding the scaling laws for the transition between these two instabilities. To the best of our knowledge, this is the first three-dimensional numerical work on the transition between EHD varicose and whipping instabilities, from which we intend to provide a new approach to analyzing these interesting but complicated phenomena.