Asymptotic analysis of a kinematic model for coffee-ring deposition

The method of matched asymptotic expansions is used to analyse a depth-averaged kinematic model for the coffee-ring effect in a thin evaporating colloidal droplet on a hierarchy of timescales. The evolution at small times is shown to be governed by a uniformly valid asymptotically self-similar solution in which the particle surface density jumps by a constant factor across the deposit front. The matched asymptotic analysis of the dilute regime formalizes existing results on the timescale of evaporation and extends them to analyse the rush-hour effect on the fast timescale of touchdown in which the thickness of the thinning meniscus becomes comparable to that of the growing ring deposit. A local analysis predicts the asymptotically self-similar blowup of the particle volume fraction on the axis of symmetry driven by the touchdown of the meniscus, providing a mechanism for the formation of a nascent eye deposit in the dilute regime. The asymptotic analysis is used to motivate and validate a uniformly valid approximation in the dilute limit and a tailored high-precision numerical scheme to resolve the non-uniformities. The modelling predictions are compared favourably with a wide selection of experimental results.