Internal flow and concentration in neighbouring evaporating binary droplets and rivulets

In this paper, the evaporation of neighbouring multi-component droplets or rivulets – often found in applications such as inkjet printing, spray cooling and pesticide delivery – is studied numerically and theoretically. The proximity induces a shielding effect that reduces individual evaporation rates and disrupts the symmetry of both the concentration profile and the flow field in the liquids. We examine how the symmetry of flow and concentration fields is affected by key parameters, namely the contact angle, the inter-droplet (or inter-rivulet) distance and the magnitude of surface tension gradient forces (i.e. the Marangoni number). We focus on binary mixtures, such as water and 1,2-hexanediol, where only one component evaporates and evaporation is slow, thereby allowing simplifications to the governing equations. To manage the complexity of the full three-dimensional droplet problem, we begin with a two-dimensional model of neighbouring rivulets. Solving the complete transient equations for rivulets with pinned contact lines and fixed inter-rivulet distance reveals that the asymmetry – quantified by the position of the interfacial stagnation point of the flow – diminishes over time. Using a validated quasi-stationary model, we find, with increasing contact angle and inter-rivulet distance, that the stagnation point migrates closer to the centre, yet it remains unaffected by the Marangoni number. A simplified lubrication model applied to droplets shows similar dependencies on contact angle and distance, although here the stagnation point appears to vary with the Marangoni number. We attribute this dependence to the additional azimuthal flow in droplets, leading to a nonlinear evolution of the concentration and therefore a non-trivial dependence of the symmetry on the Marangoni number.