Diffusion-Controlled Evaporating Stationary Meniscus in a Channel

Isochemical liquid evaporates into a mixture of its own vapor and an inert component. On one wall, the contact line is pinned; the other wall is perfectly wetted. These walls are at uniform temperature $T_o$ equalling that of the distant gas. Liquid evaporates because the partial pressure $p_\infty^v$ of the distant vapor is less than the saturation pressure $\textsl {P}$ evaluated at $T_o$ and pressure $p_\infty^\ell$ of the distant liquid. Evaporation draws liquid into the contact region; near the wetted wall, the resulting pressure differences distorts the interface, creating an apparent contact angle. $\theta$ is a flow property and increases with the control parameter $\textsl{P}(p_\infty^\ell,T_o) - p_\infty^v$. As a preliminary to finding $\theta$, we prove the following; $(a)$ The system is effectively isothermal; though evaporation induces liquid temperature differences, they are kinetically negligible. $(b)$ Whenever the continuum approximation holds within the gas, diffusion is rate-limiting. As a result, liquid and vapor at the interface are in local thermodynamic equilibrium; the vapor partial pressure is related to liquid pressure by Kelvin's equation $p^v = \textsl {P}(p^\ell,T_o)$. Given $(a)$ and $(b)$, the film thickness $h (x)$, is determined by a system comprising of the steady state diffusion equation for $p^v(x,y)$, the lubrication equation for $p^\ell(x)$, and the augmented Young-Laplace equation for $h$. These equations are coupled by Kelvin's equation. We use our solution to address the corresponding problem for a droplet on a substrate.