Convection in a Small Hemispherical Droplet of Binary Solvent: Analytical Solution and Applications

A new analytical solution has been proposed for the linearized Navier–Stokes equations and the diffusion equation. The solution makes it possible to relate the intensity of the Marangoni flow to the surface tension gradient in a droplet of a binary solvent and to study the relevant mass transfer and self-organization of solvates (nanoparticles, molecules, etc.). When deriving the equations, the smallness of the Reynolds number has been assumed, which corresponds to the smallness of the droplet size and the liquid flow velocity. The evaporation has been assumed to be slow sufficiently for ensuring the validity of the quasi-stationary approximation. The smallness of the Peclet number has also been accepted, which corresponds to low velocities of the convective flows as compared with the velocity of the diffusion transfer of an impurity. In this case, the Marangoni number may have a value from unity to several tens. The model has been tested using water–ethanol and octanol–hydrogen peroxide systems. Streamlines have been plotted for the convective flows, and the conditions for their appearance have been analyzed.