The mathematical models of rotating droplets with charge or subject to electric fields: Analysis and numerical simulation

Abstract

The main goal of this thesis is to give an answer to the question: How does rotation influence the evolution of a charged or neutral droplet that could also be subject to an external electric eld along its axis of rotation? It is well known from experiments that a drop can develop singularities in the form of Taylor cones when it holds an amount of charge larger than Rayleigh's limit on its surface [58] and/or it is immersed in a su ciently strong electric eld [68]. From the cone tips, a thin jet of microdroplets is eventually emitted [20], which is of crucial importance and has many applications in industrial processes such as electrospraying, elecronic printing, Field Induced Droplet Ionization mass spectrometry and Field Emission Electric Propulsion thrusters among others. An intriguing and yet not completely understood problem is the discrepancy existent between the results given for the opening semiangle of these cones by theoretical calculations, experiments and numerical simulations [27]. This thesis tries to give an insight into this problem by a complete description on how the stability of a conducting and viscous drop changes when rotation is considered as a force acting on the system. When dealing with rotating bodies, there are two possible situations: one where the angular speed ! remains xed, i.e. imagine a constant force turning the system at a constant rate, or another where the system is initally set into rotation and is left to evolve without further interaction with it, so its angular momentum L is conserved. This work discusses both cases. The free boundary problem arising from the modeling of rotating droplets is described, in the limit of large Ekman number and small Reynolds number, by Stokes equation [56] and simulated with a Boundary Element Method (BEM) that has the capability of mesh adaption [17]. With this approach, we can analyze with precision the regions of the drop's interface where singularities (Taylor cones or drop breakup) develop and their formation process. We begin by studying the evolution of a viscous drop, contained in another viscous uid, that rotates about a xed axis at constant angular speed or angular momentum. The analysis is carried out by combining asymptotic analysis and full numerical simulation, focusing on the stability/instability of equilibrium shapes and the formation of singularities that change the topology of the uid domain. When evolution is at constant !, unstable drops can take the form of a at lm whose thickness goes to zero in nite time or an elongated lament that extends inde nitely in nite time. On the other hand, if evolution takes place at constant L, and axial symmetry is imposed, thin lms surrounded by a toroidal rim can develop, but the lm thickness does not vanish in nite time. In the absence of axial symmetry, and for su ciently large L, drops break axial symmetry and reach an equilibrium con guration with a 2-fold symmetry or break up into several drops with a 2 or 3-fold symmetry. The mechanism of breakup is also described. After describing the evolution of rotating drops, this thesis analyzes the e ects that rotation has on the evolution of a conducting and viscous drop, contained in another viscous and insulating uid, when it holds an amount of charge Q on its surface or is immersed in an external electric eld of magnitude E✽ parallel to the rotation axis. We pay special attention to the case where rotation is at constant angular momentum because of its physical relevance. Numerical simulations and stability analysis show that the Rayleigh ssibility ratio at which charged drops become unstable decreases with angular momentum, whereas for neutral drops subject to an electric eld the critical value of the eld at which the droplet destabilizes increases with rotation. Concerning equilibrium shapes, approximate spheroids and ellipsoids are obtained and the transition between these two families of solutions is established with an energy minimization argument. When drops become unstable, two-lobed structures form, where a pinch-o occurs in nite time, or dynamic Taylor cones in the sense of [7] develop. An interesting feature about these cones is that for small L, their seamiangle remains the same as if there was no rotation in the system. Finally, and as part of the work I developed during a research stay at the University of Cambridge, the evolution problem is solved with the Finite Element Method (FEM). This approach, which validates the axisymmetric results obtained in this thesis for rotating drops using BEM, will allow us in the future to study the in uence that the inertial terms present in Navier-Stokes equations have on the stability of the system.