Modeling Approaches and Challenges of Evaporating Sessile Droplets - Drops and Bubbles in Contact with Solid Surfaces - page 143

where U is the characteristic velocity of the system. Equations (1)-(3) then write:

1

(v ∗ • grad )v ∗ =

Re v ∗ ,

(6)

1

(v ∗ • grad )T ∗ =

Re Pr T ∗ ,

(7)

T ∗ = 0 . (8)

Due to the axi-symmetry of the system, these equations are expressed in di-
mensionless cylindrical coordinates ( r ∗ ,z ∗ ). The detailed construction of equations
(6), (7) and (8) is described in Ref. [26]. They constitute the basic equations that
will be simulated in the following Paragraph. It is important to note here that due
to the approximations that are used for their construction and the fact that the
substrate is assumed to be ideal, these equations describe the evaporating droplet
problem on the basis of three quasi-steady subsystems: the droplet, the substrate
and the surrounding air. The coupling between these subsystems is determined by
the properties at each of the interfaces and in particular at the water/air one where
evaporation takes place. This separation into subsystems is justified as long as the
substrate can be assumed to be an ideal heat conductor and evaporation remains
slow when compared to the setting of local equilibrium in both air and water. As
the typical time in this problem is the time for the complete evaporation of the
droplet, this hypothesis holds as long as T s remains small compared to the water
vaporization temperature. This is the reason why T s is always set to values smaller
than 70 ◦ C in all the numerical results presented hereafter. Another important point
to be discussed is the use of Eq. (8) for temperature in air. This equation implies that
hydrodynamics is neglected but the use of this equation also ignores the fact that
air is a gaseous mixture in which vapor water is expected to diffuse from regions
where the relative humidity is high (i.e., near the droplet interface) towards regions
where it is smaller (i.e., droplet far field). The characteristic time of this diffusive
phenomenon is given by water concentration gradients in air. The larger they are,
the faster will be evaporation when the droplet is isothermal [11, 19, 39, 69]. In
non isothermal situations, temperature gradients will contribute as an addition and
reinforce concentration gradients due to changes in the value of the local saturated
vapor concentration. The vapor water diffusion in air is hence driven by temperature
gradients in the neighborhood of the droplet interface where they will be enhanced
due to the small thermal conductivity of air.
In all the above equations time does not appear any more due to the quasi-steady
hypotheses that have been adopted. However, evaporative phenomena are clearly
time evolving and explicit time dependence has to be reintroduced in the model.
This is done using the same approach as the one for isothermal systems when ne-
glecting evaporative cooling [8, 70]. In nonsaturated air and in isothermal situations,
the driving mechanism for evaporation is the nonhomogeneity of relative humidity
in air. Air is saturated with water vapor at the droplet interface and in the far field
relative humidity takes a value denoted by H . In this context it is actually possible to