The mathematical models of rotating droplets with charge or subject to electric fields: Analysis and numerical simulation
Author
Garcia Garrido, Víctor JoséEntity
UAM. Departamento de Matemáticas; Instituto de Ciencias Matemáticas (ICMAT)Date
20131219Subjects
Dinámica de fluidos Tesis doctorales; MatemáticasNote
Tesis doctoral inédita. Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Matemáticas. Fecha de lectura: 19122013Esta obra está bajo una licencia de Creative Commons ReconocimientoNoComercialSinObraDerivada 4.0 Internacional.
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 2fold symmetry or break up into several drops
with a 2 or 3fold 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, twolobed structures form, where a pincho 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 NavierStokes equations have on the stability of the system.
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