Biomedical Engineering Reference
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tion columns. Besides Marangoni driven convection, Rayleigh or gravity controlled
convection can also occur as a result of the same temperature and concentration
gradients which give rise to Marangoni convection. Temperature and concentration
gradients can indeed cause density stratification within the fluid which can gener-
ate a fluid circulation balancing Marangoni convection. Recent works suggests that
in situations in which the characteristic length is greater than the capillary length,
Rayleigh convection can be the dominant mechanism. In contrast, Marangoni con-
vection is dominant in situations in which the characteristic length is smaller than
the capillary one.
Ruiz and Black modelled Marangoni convection in a hemispherical droplet
placed in non-flow surroundings [38]. Their model, allowed the prediction of the
evaporation rate in small diameter hemispherical droplets with pinned contact line.
As a droplet evaporates, the contact line between solid and liquid travels over the
heated surface making the contact line vary with time. Ruiz and Black also point
to the fact that when the contact angle is less than 90 the evaporation kinetics can
be split into two main regimes. For the most of the droplet lifetime, the contact line
remains pinned and the contact angle decreases. In the second period the contact
line recedes but only once the contact angle has become small. As the contact angle
decreases the droplet approaches a thin film configuration and three dimensional
motion of the droplet begins to occur. Craft and Black [34], have shown that the
time taken for a droplet to de-pin is influenced by the temperature of the substrate.
These authors also highlight the fact that after the drop begins to de-pin there is
a step change in the rate of evaporation, which is not accurately modelled in the
model of Ruiz and Black [38]. The authors themselves are quick to highlight the
inadequacies in some of the assumptions used in their model. In particular, the as-
sumption that the mass transfer coefficient from liquid to vapour can be estimated
using the Reynolds analogy proposed by Incropera et al. [18]. This analogy as-
sumes the rate of mass transfer from liquid to vapour can be estimated from the
heat transfer characteristics. Using analogies in this manner is somewhat inaccu-
rate as they are generally based on intuition rather than rigorous analysis. Anyhow,
this model produced results with reasonable agreement with experimental observa-
tions for small diameter evaporating droplets and clearly indicates that the physics
of mass transfer into the vapour phase can be better described using the diffusion
equation as an alternative of the Reynolds analogy currently employed. Hu and
Larson [39], have shown that as a droplet evaporates, the vapour concentration dis-
tribution satisfies the Laplace equation with time varying droplet surface. Hu and
Larson's numerical simulations agree well with experimental data, implying that
the rate of evaporation (i.e., mass transfer from the liquid into vapour) can be accu-
rately predicted using a simple diffusion equation.
Despite still on-going important theoretical modelling and experimental efforts,
it is clear that the best way to gain insight in the evaporation kinetics and hydro-
dynamics of volatile droplets is through accurate and robust numerical techniques.
Developing such techniques is however far from being fully achieved and many
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