Numerical simulation of the solutal Marangoni flow and capillary flow in a pair of binary droplets

Applications like e.g. inkjet printing rely on the evaporation of multiple sessile multi-component droplets. Due to the diffusive vapor transport in the gas phase, each droplet interacts with its neighbors, which results e.g. in a diminished evaporation rate at spots facing towards a neighboring droplet, whereby the flow towards the pinned contact line (cf. coffee-stain effect) becomes non-axisymmetric. In mixture droplets, the altered evaporation rate also influences the compositional gradients and in turn the Marangoni flow. Therefore, analysis of an isolated droplet is usually insufficient to predict the drying time, flow and compositional dynamics in such applications.

We numerically analyze a pair of binary droplets, which is the most trivial scenario allowing to investigate the effect of a neighboring droplet. The analysis is carried out by both full 3d Navier-Stokes simulations and computationally cheap simulations based on lubrication theory, both transiently and in quasi-stationary approximation, where we enhance the accuracy of the lubrication theory by the consideration of Taylor dispersion, i.e. by an generalization of the model by Ramírez-Soto and Karpitschka [Phys. Rev. Fluids 7, L022001, (2022)].

In a minimal model, the entering parameters can be reduced to a Peclet number for the total evaporation rate (i.e. for the outward flow towards the contact line), a Marangoni number, the contact angle and the nondimensionalized distance to the neighboring droplet. Contrarily to naive expectations, the maximum concentration of the less volatile component is not always located at the distant poles of the contact lines, which can be attributed to enhanced replenishment from the center driven by the total evaporation rate. When Marangoni flow comes into play, the interaction with the asymmetric net bulk transport towards the contact line can enhanced or reduce the asymmetry in the flow and composition.