= Droplet evaporation near a planar reservoir
For a spherical droplet at fixed center height $h>R$ above a flat equilibrium reservoir, the <diffusion-capacitance analogy> and <sphere-plane capacitance> give
$$
\dot R=-\frac{M\gamma}{2\phi_B^2R^2}\,c(R/h),
\qquad
\tau=\frac{2\phi_B^2}{M\gamma}\int_0^{R_0}\frac{R^2}{c(R/h)}\,dR.
$$
At large separation, $c(R/h)=1+R/(2h)+O((R/h)^2)$, so
$$
\tau=\frac{2\phi_B^2R_0^3}{3M\gamma}
\left[1-\frac{3R_0}{8h}+O((R_0/h)^2)\right].
$$
The nearby reservoir increases the total flux and shortens the lifetime. The spherical-shape assumption controls the geometry; the local current density is not uniform over the surface.
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