Modeling Approaches and Challenges of Evaporating Sessile Droplets - Drops and Bubbles in Contact with Solid Surfaces - page 152

Figure 9. Marangoni velocity profile v mar at the surface of the droplet as a function of r ∗ for two

different contact angles (see legend) with T s = 50 ◦ Cand L s /R = 0 . 1.

Figure 10. Streamlines for = 0 . 1 (a) and = 1 . 0(b)when θ = 20 ◦ with T s = 50 ◦ C. Black arrows
indicate the orientation of water circulation.

of , only one such maximum shows up for the entire evaporation time and, in the
last stage of the evaporation, it drifts towards the apex of the droplet. This drift is ex-
plained by the large deformation of the convective cells presented in Fig. 7 and Fig.
8 as the droplet volume decreases. When the droplet height h becomes too small, the
available space for convective cells shrinks and the corresponding Reynolds num-
ber follows the same trend. Consequently, for not too fast evaporating droplets, the
hydrodynamics evolves from convective to diffusive. This transition is usually not
observable experimentally due to receding contact line regimes that finally prevails
in the very last phase of droplet evaporation. Back to 0 . 1, Fig. 9 indicates a

relatively complex Marangoni circulation with the appearance of two maxima for
decreasing contact angles. Although no stagnation point ( v mar =

=

0 m/s) can be ob-
served here, these two maxima are reminiscent of bifurcations in the hydrodynamics
of the droplets similar to the one illustrated in Fig. 10(a) that is a snapshot of the
water streamlines when θ 20 ◦ 1 . 0 is also displayed

(Fig. 10(b)). The comparison between these two figures indicate qualitatively simi-
lar water clockwise circulation in the droplet part where r ∗ > 0 . 5. But for r ∗ < 0 . 5,
water circulation is completely different with a counter clockwise rotating convec-

= and = 0 . 1. The case =