Biomedical Engineering Reference
In-Depth Information
attempts are still under way. This chapter critically appraises the merits of models
developed to predict the evaporation of small sessile droplets with pinned contact
line placed upon a heated surface. Furthermore, the results of a Finite Element
Method (FEM) approach applied to specific evaporation conditions are presented
and discussed in Paragraph D. This example will illustrate how far numerical CFD
simulations can address droplet evaporation mechanisms and will simultaneously
emphasize the inherent limitations of FEM.
B. Challenges and Limitations in CFD Simulations
This Paragraph discusses some still open questions in the numerical simulations of
evaporating droplets deposited on substrates. An evaporating droplet is a typical ex-
ample of unsteady multiphase system with free interfaces for which the liquid phase
can be assumed as viscous and incompressible. Boussinesq approximation can be
used here since temperature gradients in liquids are small due to their large heat
conductivity (when compared to the one of gases). The explicit time dependence of
the equations have, however, to be accounted for in the numerical schemes as well
as heat and mass transfer mechanisms generated in particular by the phase change.
Moreover, subsequent droplet volume reduction requires adapted front localization
techniques to track the moving water/air free interface that, from the numerical
point of view, corresponds to discontinuities where derivatives have to be handled
and computed carefully. This is one of the first important challenges in the simula-
tion of unsteady, multiphase systems with an additional complexity in evaporating
heated droplets since liquid/air, liquid/solid and solid/air interfaces are all simulta-
neously involved. In free interface systems, it is of paramount importance to capture
interfaces numerically, to keep their sharpness finite and, while processing that way,
satisfy overall mass and volume conservation laws. As mentioned above, in the case
where droplets are wetting the substrate, pinned evaporating regime is first observed
and the only free moving boundary is the liquid-gas interface. Such moving inter-
faces are the most studied cases in two phase systems where no solid phase has to
be accounted for, like sprays, atomization problems or emulsions (in this last case
gas being replaced by the continuous liquid phase). A second important challenge
in the simulation of droplets on substrates arises when contact angles become small
enough for the fast contact line de-pinning regime to occur. All the aforementioned
interfaces, and not only liquid-air, are then rapidly time-evolving. The interplay be-
tween the interactions involved in this evaporative regime still raises a number of
open numerical questions that range from mass conservation constraints to a reli-
able description of all moving free interfaces. Also still challenging is the accurate
description of interfacial properties, such as surface tension or adhesion forces of
the liquid on the substrate [41, 42].
A complete description of droplet evaporation dynamics involves almost all pos-
sible spatial and temporal scales from macroscopic down to microscopic: macro-
scopic, for hydrodynamics (convection, heat transfer) and microscopic, for molecu-
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