Drop impact onto wetted walls: an unsteady analytical solution for modelling crown spreading

An unsteady analytical solution is proposed to predict the spreading rate of the crown generated by an impacting droplet onto wetted walls. The modelling strategy is based on the direct integration of the boundary layer correction into the potential flow solution that leads to the well-established square-root time dependence. The original potential flow has the structure of an unsteady, stagnation point flow with decaying strength. For initial strengths of the potential flow $a_0 \geqslant 100\ {\rm s}^{-1}$, we find that a self-similar solution can also be obtained for the boundary layer in the variables $\left (r \sqrt {a(t)/(\nu t)}, z \sqrt {a(t)/(\nu t)} \right )$. The self-similarity of the solution enables a straightforward estimation of momentum losses during the spreading of the liquid layer along the wall. The proposed modelling approach yields an excellent agreement with experiments during the entire spreading phase. Moreover, it enables a smooth transition from the inertia-driven to the shear-controlled regime of crown propagation. In general, the analysis shows that momentum losses arising from viscous effects cannot be neglected during a significant portion of crown propagation, particularly for thin wall films.