Modeling Approaches and Challenges of Evaporating Sessile Droplets - Drops and Bubbles in Contact with Solid Surfaces - page 142

For the gas phase, the same assumptions as for water will be used. It is treated as
an ideal gas mixture with a vapor saturation that is known for all temperatures [68].
As liquid water and air have viscosities that differ by orders of magnitude, the shear
stress generated by Marangoni circulation on the gas phase can be neglected. Air
is therefore assumed to remain still above the droplets for the complete duration of
evaporation. It is hence not necessary to use the full Navier-Stokes equation for its
description since the convective term does not come into play. This approximation
might be restrictive considering the possible development of convective motion in
air due to the temperature differences between air adjacent to the substrate and
in the far field or to relative density differences between air and the vapor water.
This latter is lighter than air and can convect away from the droplet to the far field
because of buoyancy forces. Dragging forces due to this flow can then generate
circulating motion in air. This effect will be neglected in the following. With all
the previous approximations, description of air can finally be limited to a simple
Laplace equation [26]:

T = 0 . (3)

Equations (1)-(3) have to be further completed with the appropriate boundary con-
ditions. In the far field of the simulation domain, air is assumed to have everywhere
the same temperature denoted by T ∞ . The substrate is assumed to be constituted by
two embedded ideal heat conductors that are supposed to be thermally isolated from
one another (see Fig. 3). The inner part is at temperature T s whereas the temperature
of the outer one is fixed either by the one of the water droplet (when R>L s )orair
(when R<L s ).
Continuous temperature conditions are imposed at water/air boundaries for all
times but at the initial condition (this will be discussed below). Temperature deriva-
tives on the symmetry axis of the system are kept to zero while one of all the other
boundaries (droplet/air interface and droplet/non heated substrate region) is deter-
mined from the solutions of the Eqs (1), (2) and (3). Concerning velocity boundary
conditions, normal (respectively radial) components are set to zero in the substrate
vicinity (respectively on the symmetry axis) whereas tangential constraints are set
up at the water/air interface. Finally radial derivatives of the velocity field are fixed
tozeroonthesymmetryaxis.
For the FEM approach efficiency and to avoid numerical precision problems,
we use a dimensionless expression of the previous equations. We introduce to this
end the usual Reynolds (Re) and Prandtl (Pr) numbers, dimensionless radius r ∗ and
heigh z ∗

defined respectively by r ∗ = r/R and z ∗ = z/R as well as dimensionless

velocity ( v ∗ ) and temperature ( T ∗ ) defined by:

v
U ,

v ∗ = (4)

T − T ∞

T ∗ = , (5)

T s − T ∞