Analytical solution for inviscid flow inside an evaporating sessile drop

Inviscid flow within an evaporating sessile drop is analyzed. The field equation $E^2\psi =0$ is solved for the stream function. The exact analytical solution is obtained for arbitrary contact angle and distribution of evaporative flux along the free boundary. Specific results and computations are presented for evaporation corresponding to both uniform flux and purely diffusive gas phase transport into an infinite ambient. Wetting and nonwetting contact angles are considered, with flow patterns in each case being illustrated. The limiting behaviors of small contact angle and droplets of hemispherical shape are treated. All of the above categories are considered for the cases of droplets whose contact lines are either pinned or free to move during evaporation.

Figure 1

Lines of constant $\alpha$ and $\theta$ and positive direction of velocity vector components in toroidal coordinates.

Figure 2

(a) Nondimensional diffusive evaporative flux for contact angles of 120°, 90°, and 60°. Inset: flux at $r∕R=0$ versus contact angle. (b) Hemispherical $({\theta }_c={90}^{\circ })$; contours of nondimensional stream function (${\psi }^{*}=0.005$, 0.03, 0.08, 0.15, and 0.22) for pinned contact and uniform diffusive evaporative flux.

Figure 3

Contours of nondimensional stream function (${\psi }^{*}=0.005$, 0.03, 0.08, 0.15, and 0.22) for a pinned contact line, diffusive evaporative flux, and contact angles of (a) 120° and (b) 60°.

Figure 4

Contours of nondimensional stream function for a freely moving contact line, diffusive evaporative flux, and contact angles of (a) 120° (${\psi }^{*}=-0.007$, $-0.001$, 0.0001, 0.002, 0.008, 0.017, and 0.027) and (b) 60° (${\psi }^{*}=-0.0001$, 0.0001, 0.001, 0.006, and 0.02).

Figure 5

Contours of nondimensional stream function for a pinned contact line, uniform evaporative flux, and contact angles of (a) 120° (${\psi }^{*}=0.01$, 0.05, 0.125, 0.24, and 0.38) and (b) 60° (${\psi }^{*}=0.005$, 0.022, 0.05, 0.09, and 0.13).

Figure 6

Contours of nondimensional stream function for a freely moving contact line, uniform evaporative flux, and contact angles of (a) 120° (${\psi }^{*}=0.005$, 0.022, 0.05, 0.09, and 0.135) and (b) 60° (${\psi }^{*}=-0.001$, $-0.005$, $-0.012$, $-0.021$, and $-0.03$).

Figure 7

Nondimensional radial velocities versus vertical position at different radial positions $r∕R=0.1$, 0.3, 0.5, 0.7, and 0.9, for a pinned contact line, diffusive evaporative flux, and contact angles of (a) 40° and (b) 10°. The dash-dotted lines are from the small-contact-angle approximation, the solid lines are from the exact solution, and the dashed lines are from the vertically averaged radial velocity.

Figure 8

Nondimensional vertical velocities versus vertical position at different radial positions $r∕R=0.1$, 0.3, 0.5, 0.7, and 0.9, for a pinned contact line, diffusive evaporative flux, and contact angles of (a) 40° and (b) 10°. The dashed lines are from the small-contact-angle approximation and the solid lines are from the exact solution.