Evaporation and fluid flow near the boundary of a stationary dry patch

We consider an isolated circular dry patch formed in an evaporating liquid layer and investigate local viscous flows in both liquid and air near the contact line which is the boundary of the dry patch. Flow patterns in the liquid deviate significantly from the predictions of single-phase models even when the air-to-liquid dynamic viscosity ratio is small. In particular, the separatrices in the liquid flow patterns at large contact angles disappear completely for a range of realistic viscosity ratios when the shear stress on the air side of the interface is accounted for. Experimentally observed motion of microdroplets near the contact line under combined action of gravity and moist air flow is described using our local flow model. We demonstrate that analysis of droplet trajectories leads to unambiguous determination of the local evaporative flux profile. A numerical finite-element approach for the steady diffusion equation is then used to extract the same flux profile from the global solution for concentration field in the limiting case of very thin liquid layer and small contact angle. The local evaporation rate is underpredicted by the numerical method, most likely due to the neglect of convective mass transfer in the air.

Figure 1

Sketch of the geometric configuration of a circular dry patch of radius $R$ on a solid surface covered by a liquid film. The enlarged view shows the region near the contact line where local solutions are valid; the contact angle $\theta$ is assumed constant here.

Figure 2

Streamlines of the local solutions for liquid (red color) and air (blue color) near the contact line at $\theta ={130}^{\circ }$ and $\rho =8\times {10}^{-4}$ for three different viscosity ratios, top to bottom: $\mu ={10}^{-6}, \mu ={10}^{-4}, \mu ={10}^{-2}$.

Figure 3

Streamlines of the local solutions for $\mu =0.049, \rho =8\times {10}^{-4}$, and different contact angles, top to bottom: $\theta ={22}^{\circ }, \theta ={45}^{\circ }, \theta ={60}^{\circ }$.

Figure 4

The exponent of the local solution for the eigenmode (dashed line) and the evaporation induced flow (solid line) (top); Typical flow pattern at $\mu =0.049, \theta ={150}^{\circ }$ in the regime where the eigenmode dominates (bottom).

Figure 5

(Top) A frame from the video illustrating droplet fly-over near the contact line. From Ref. [25], reprinted with permission. (Bottom) Comparison between experimental (filled squares) and theoretical (solid line) results for a typical droplet trajectory. The droplet size is $8\mu \mathrm{m}$, substrate temperature is $79{}^{\circ }\mathrm{C}$. Black solid line shows the approximate location of the liquid-air interface.

Figure 6

Comparison with experiments for two droplets of different sizes at $T=75{}^{\circ }\mathrm{C}$. Filled squares are experimental recordings from Ref. [21]. Solid lines correspond to theoretically predicted trajectories.

Figure 7

(Top) Droplet trajectories under the same conditions as in Fig. 5 but with initial velocity matching the experiment (blue line) and increased to the value of $2U$ (red line); (Bottom) Droplet trajectories with the same initial conditions but different values of the constant that characterizes the evaporation rate: $K=0.2$ (blue line), $K=0.25$ (black line), $K=0.3$ (red line).

Figure 8

Sketch of the geometry of the domain used in the simulation (top) and solution for concentration near the dry patch (bottom).

Figure 9

Comparison of concentration along the surface of dry patch found from the numerical solution (solid line) and predicted by the local analysis (dashed line).