Drop impact on wetted walls: An analytical solution for modelling the crown spreading based on stagnation-point flow

Abstract

An analytical solution is proposed to predict the crown propagation, generated by a single droplet impact on wetted walls. This approach enables a smooth transition from the inertia-driven to the viscous-controlled regime of crown propagation. The modelling strategy is based on the stagnation-point flow, because it resembles closely the hydrodynamic flow in the lamella and offers two main advantages. First, it allows a simple estimation of the wall-film thinning rate, caused by the impulse transfer from the impacting droplet to the wall film. Second, thanks to the self-similarity of the solution, it enables a straightforward estimation of momentum losses during film spreading along the wall. By incorporating this estimation into existing inviscid models, an excellent agreement with experiments is found during the entire crown elevation phase. In general, the analysis shows that momentum losses due to viscous effects cannot be neglected during a significant portion of crown propagation, particularly for thin wall films. The proposed methodology paves the way for predicting the inception of crown bottom breakup (CBB). In this case, the crown lamella disintegrates directly at its base due to the spontaneous creation of holes that create a web-like structure in the lamella prior to its break-up. Our theoretical analysis shows that this premature break-up of the crown lamella is associated to local instability effects, caused by the unbalance between inertial forces and surface tension.