Evaporation of an Elongated Droplet

The final deposit patterns left behind on a substrate by evaporating particle-laden droplets are of great interest due to applications such as inkjet printing or agrochemical spraying of plants. In the present work we examine the evaporation of an elongated, particle-laden droplet which has a rectangular footprint, and consider how the ratio between the length and width of the rectangle affects the evaporation dynamics of the droplet. This differs from most existing studies of small sessile droplets, which are primarily focussed on droplets that have a circular footprint. We use the method of matched asymptotic expansions to formulate outer and inner problems for the total evaporative flux, where the inner problem involves solving the evaporative problem for a thin, two-dimensional droplet, and the outer problem treats the droplet as a flat rectangle at leading order. The resulting solution for the total evaporative flux allows us to determine the evolution, and hence the lifetime, of a droplet with a pinned contact line. We analyse the flow and particle transport within the droplet, and find that the flow transports all particles toward the contact line, leading to a final deposit concentrated at the edges.