Quantifying evaporative-driven flow inside drying droplets with pinned contact lines

A mathematical model is presented to predict the shape of an evaporating droplet with a pinned contact line, and the axisymmetric time-dependent fluid flow inside. We consider the diffusion-limited evaporation regime, where the evaporative flux diverges at the contact line. To regularize this singularity, the droplet domain is divided into two regions: (i) a wedge at the contact line and (ii) the remaining drop. Near the contact line, the flow is characterized by solving Stokes equations analytically in a wedge. Away from the contact line region, lubrication theory is used. The drop and wedge regions are connected via boundary conditions preserving continuities of mass and drop shape. We show how the proposed model reasonably predicts experimentally observed evaporation rates and contact angle dynamics for small initial contact angles. In addition, the work demonstrates that removal of the diverging evaporative flux near the contact line, as done in prior works to address the singularity, can significantly affect the predicted drying droplet dynamics.