Drop impact on small surfaces: thickness and velocity profiles of the expanding sheet in the air

We consider the radially expanding sheet formed upon impact of a drop on a surface of comparable size to that of the drop. A unified self-similar solution for the unsteady radial thickness profile of the expanding sheet is derived from first principles in the inviscid limit. This unified functional form reconciles two conflicting theoretical profiles of sheet thickness proposed in the literature and allows for the collapse on a single curve direct measurements of sheet thickness profiles reported in the literature and the detailed measurements conducted herein. We show good agreement between our proposed unified thickness profile and data from our experiments for a range of surface-to-drop size ratios. We show that there is an optimal range of surface-to-drop size ratio for which the hypothesis of inviscid thin sheet expansion in the air holds. Outside of this optimal range, either insufficient vertical momentum is transferred to horizontal momentum to form an expanding sheet or viscous effects become too important to neglect. In this latter regime, the dominant effect of surface friction is to modify the velocity profile. We elucidate this effect using a Blasius-type boundary layer model. Finally, we relate the geometry of the drop in its early phase of impact to the sheet thickness profile in the air. We show that the coefficients of the proposed unified similarity thickness profile can directly be linked to volume flux conservation at early times, and to the maximum sheet thickness at the edge of the surface. Our results thus quantitatively link the fluid history on the surface to the thickness and velocity profiles of the freely expanding sheet in the air.