Biomedical Engineering Reference
In-Depth Information
The model and the treatment of evaporation under consideration here are adapted
to only small pinned droplets keeping a spherical cap shape and evaporating slowly.
This excludes all the movements of the interfaces that could lead to complex geome-
tries and renders the front tracking FEM approach inadequate. This is in particular
true for contact line dynamics where molecular forces have to be accounted for
when describing receding contact angle regimes. Although they are known to play
a key role in evaporation of liquids on substrates, they only come into play either
in the very early or very last evolution stage of droplets evaporation and hence run
out of the scope of the illustration of FEM's we aim to give here. The quasi-steady
approximations used to obtain Eqs (6), (7) and (8) and the FEM on which numerics
will be based on are hence applicable only to deposited droplets on substrates with
pinned contact lines that is to say to moderate substrate temperatures. This is why
T
s
will be restricted to the range 30
◦
C-70
◦
C in the following and with a far field
temperature that will always be set to 20
◦
C. Although restrictive, we will illustrate
in the next Paragraph that these conditions and the FEM approach allows a precise
study of the circulation of water in the droplets and the description of the proper-
ties of thermocapillary effects. These conditions have also the advantage to avoid
the implementation of unsteady multiphase algorithms for which interface track-
ing and numerical diffusion control require much more efforts [73] than for FEM
techniques in particular when phase changes have to be accounted for.
E. Simulation of the Hydrodynamics of Heated Water Droplets
When performing numerical simulations, whatever the numerical technique is, one
first problem to tackle is the initial conditions of the system. For evaporating
droplets investigations, some variables are fixed for the complete duration of the
simulations (
T
s
,
T
∞
,
H
,
L
s
and
R
) whereas others evolve in time. Among these,
some are initially free (
θ
,
T
,
v
) and others (
m
) are deduced from these free vari-
ables. In the following
T
∞
=
20
◦
C,
H
1 mm for all the simulations.
Initial values of
θ
,
T
and
v
have also to be set for the simulations to start with. They
will be the same for all the runs and are given in the table below:
Pressure will be set to the atmospheric one. With these conditions the only free
parameters will be associated with the substrate properties
T
s
and
L
s
. Finally as
=
20% and
R
=
Ta b l e 1 .
Initial values of the free variables
θ
,
T
,
v
Variable
Initial value
=
80
◦
θ
T
=
T
s
everywhere inside the droplet
=
T
∞
everywhere outside the droplet
v
=
0m/s
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