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

A new analytical solution of the linearized Navier–Stokes equations and the diffusion equation is proposed that allows to relate the intensity of Marangoni flow to the surface tension gradient in a droplet of binary solvent and to study the associated mass transfer and self-organization of solvated solutes (nanoparticles, molecules). When deriving the equations, the smallness of the Reynolds number was assumed, which corresponds to the smallness of the drop size and the liquid flow velocity. Evaporation was assumed to be slow enough for the quasi-stationary approximation to be valid. The smallness of the Peclet number was also accepted, which corresponds to relatively small velocity of convective flow in relation to the velocity of diffusion transfer of the impurity. In this case, the Marangoni number can have a value in the range from 1 to several tens. The model was tested on water—ethanol and octanol—hydrogen peroxide systems. The streamlines of convective flows were visualized, and the conditions for their occurrence were considered.