Drop impact on a solid surface: short time self-similarity

The early stages of drop impact onto a solid surface are considered. Detailed numerical simulations and detailed asymptotic analysis of the process reveal a self-similar structure both for the velocity field and the pressure field. The latter is shown to exhibit a maximum not near the impact point, but rather at the contact line. The motion of the contact line is furthermore shown to exhibit a 'tank treading' motion. These observations are apprehended at the light of a variant of Wagner theory for liquid impact. This framework offers a simple analogy where the fluid motion within the impacting drop may be viewed as the flow induced by a flat rising expanding disk. The theoretical predictions are found to be in very close agreement both qualitatively and quantitatively with the numerical observations for about three decades in time. Interestingly the inviscid self-similar impact pressure and velocities are shown to depend solely on the self-similar variables $(r/\sqrt{t},z/\sqrt{t})$. The structure of the boundary layer developing along the wet substrate is investigated as well, and is proven to be formally analogous to that of the boundary layer growing in the trail of a shockwave. Interestingly, the boundary layer structure only depends on the impact self-similar variables. This allows to construct a seamless uniform analytical solution encompassing both impact and viscous effects. The depiction of the different dynamical fields enables to quantitatively predict observables of interest, such as the evolution of the integral viscous shearing force and of the net normal force.