Biomedical Engineering Reference
In-Depth Information
substrate roughness and consequently contribute to enhance or weaken the intensity
of contact line adhesion forces.
Besides interfaces tracking, a second important problem in evaporating droplets
is the numerical description of the liquid-gas surface tension. This task can be quite
naturally achieved in Lagrangian methods (for example FEM) whereas for the VOF
techniques, it first requires to map interfacial forces to volume ones. As already
discussed, Lagrangian methods are better adapted here, but only as long as droplets
keep a simple geometry for all the duration of their evaporation. As a result, cusps
or topological changes like the formation of smaller droplets after splitting of a
large one cannot be simply assessed. In VOF approaches, this becomes possible
naturally since all the phases are treated on the same basis, but the computation of
the surface tension contributions requires more CPU efforts. In fact, the pressure
jump across the interface is proportional to its curvature and needs the computation
of a large number of spatial derivatives of the colour function [53] imposing the use
of higher order schemes. This task can be accurately achieved with weighted, es-
sentially non oscillatory schemes [54] that are well adapted to hyperbolic problems
like colour function advection and sharpness preservation of interfaces with only
little additional CPU time. But, still due to numerical finiteness, unwanted numeri-
cal artifacts consisting in spurious velocities (the so called parasitic currents) show
up in the vicinity of the interface. Fortunately, these latter become non negligible
only for slowly evolving interfaces (slowly with respect to thermocapillary con-
vection time scales). This means that contact line regions of pinned droplet regimes
will be strongly affected by such numerical artifacts. The magnitude of the parasitic
currents can be reduced by both the inertial and viscous terms in the Navier-Stokes
equations but not with increased mesh refinement or decreased computational time
step [55]. Efforts are still devoted to interface sharpness conservation with improved
interface reconstruction algorithms (PLIC-VOF) [56], second gradient techniques
[57] or hybrid methods [57-59]. Despite much work, modeling interfacial tension
and capillary phenomena is still a maturing domain in computer science. Many new
improvements in the numerical algorithms are still to come and will provide in the
next future new techniques to overcome parasitic currents and hence allow VOF
description of both fast and slow evolving free interfaces.
Progress is also expected when incorporating soluble or insoluble surface ac-
tive agents (surfactants) in the theoretical models. While very important literature
has focused on the numerical investigation of clean interfaces, only a very limited
number of studies attempted to include the effect of surfactants. For small concen-
trations, surfactants are all collected on the interfaces and form molecular layers
at the origin of capillary instabilities due to the decreasing surface tensions. They
generate similar phenomena to the ones resulting from temperature changes and
lead to non uniform Marangoni stresses that significantly modify not only droplets
hydrodynamics but also evaporative rates [60]. Like for the previously discussed
interfacial tracking or control of parasitic currents, computation of interfacial flows
with surfactants still raises challenging questions on both algorithmic and theoreti-
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