Biomedical Engineering Reference
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lar contributions (liquid-solid adhesion forces, surfactant layers kinetics, etc.). This
has led to the development of specific numerical methods for intensive CPU calcu-
lations to simulate either the hydrodynamic phenomena [43] with computational
fluid dynamics (CFD) techniques or the mechanisms at the molecular scale with
molecular dynamics (MD) [43]. In most of the investigations of evaporation pro-
cesses, quasi-steady hypotheses are assumed. They have the important advantage to
allow the dissociation of hydrodynamic time scales from the molecular ones. This
approximation is usually well justified here since molecular processes are fast when
compared to hydrodynamic ones and hence do not need to be described explicitly
as done in mesoscopic approaches [44] or in MD simulations [45]. In continuum
models, all molecular properties are averaged out and contribute only through equa-
tions of state. However, although justified in most of the studies, such a description
is no more adapted when investigating contact line dynamics. Depending on the
wetting properties of the droplet liquid on its substrate, we have already mentioned
that de-pinning phenomena can be observed for sufficiently small contact angles.
They are at the origin of complex contact line motions for which adhesion forces
have to be accounted for. These forces act at the molecular level and should be
investigated, strictly speaking, with a self-consistent molecular force field describ-
ing the interactions between the molecules of the liquid and the substrate (atoms
or molecules) at the atomic scale. Despite intense algorithmic research to speed up
MD simulations in particular with the use of averaged or effective self-consistent
force fields, the available computing power is still too limited to investigate molec-
ular systems with more than few tens of thousands of (small) molecules for more
than few milliseconds [45]. This is still too limited to be efficiently coupled with
CFD simulations in the specific case of evaporating systems. Indeed, the objective
of any numerical algorithms is to make possible undertaking simulations which are
as accurate as possible with a minimal CPU time. This goal is usually well achieved
for CFD and MD when considered separately. But when studying evaporating sys-
tems, the question is how to associate both simulation techniques when such large
scale separations have to be described. This still remains challenging in the simu-
lation of evaporating systems, although models are available in the literature [46]
and similar multi-scale problems (but at the nuclear and atomic scales) have been
resolved for more than two decades in numerical chemistry [47].
The pinned droplet evaporative regime can be described with CFD using the
Navier-Stokes and heat transfer equations coupled with continuity and specific
boundary conditions equations. The key is then the development of adapted solvers
for their integration in order to obtain well converged and stable hydrodynamic so-
lutions. Important efforts are still devoted to solvers optimization but, among the
actual challenges in computer science, optimal methods for the continuum formu-
lation of sharp free moving interfaces are still one of the main concerns. The first
difficulty in this context is to track interfaces with fast and robust algorithms in a
way to limit numerical diffusion and to avoid their artificial smearing. Several meth-
ods, having each specific advantages and limitations, are available in the literature.
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