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

Résumé

The main goal of this thesis is to give an answer to the question: How does rotation influence the evolution of a conducting fluid droplet that could be charged or subject to an electric field parallel to the rotation axis? 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 and/or it is immersed in a sufficiently strong electric field. From the cone tips, a thin jet of microdroplets is eventually emitted, 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. 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 fixed, 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 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 and simulated with a Boundary Element Method (BEM) that has the capability of mesh adaption. 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 fluid, that rotates about a fixed 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 fluid domain. We also describe the breakup mechanisms. When evolution is at constant angular speed, unstable drops can take the form of a toroidal rim with an inner thin film whose thickness goes to zero in finite time or an elongated filament that extends indefinitely in finite time. On the other hand, if evolution takes place at constant angular momentum, and axial symmetry is imposed, thin films surrounded by a toroidal rim can develop in a self-similar way, but now the film thickness does not vanish in finite time. In the absence of axial symmetry, and for sufficiently large angular momentum, drops attain an equilibrium configuration with a 2-fold symmetry or develop 2- or 3-fold symmetries that break up into droplets. After describing the evolution of rotating drops, this thesis analyzes the effects that rotation has on the evolution of a conducting and viscous drop, contained in another viscous and insulating fluid, when it holds an amount of charge on its surface or is immersed in an external electric field 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 fissibility ratio at which charged drops become unstable decreases with angular momentum, whereas for neutral drops subject to an electric field the critical value of the field 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-off occurs in finite time, or dynamic Taylor cones develop. An interesting feature about these cones is that for small angular momentum, their seamiangle remains the same as if there was no rotation in the system. Finally, and as part of the work 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 influence that the inertial terms present in Navier-Stokes equations have on the stability of the system.