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

Abstract

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.

Edge effect: Liquid sheet and droplets formed by drop impact close to an edge

“…• could be explained by the slight hydrophilicity of the substrate [59,60] or by an imperfect transfer of momentum from vertical to horizontal during initial crushing and was also noted for symmetric impacts on poles [42]. A linear fit of the range d/R 0 ∈ [0, 1] gives…”

“…(7) for each impact. The normal extension is expected to depend both on We and on the history of the liquid on the surface prior to reaching the edge [42]. To investigate this latter dependence, the maximum normal extension l nM is shown as a function of the dimensionless offset δ in Fig.…”

“…The impact near an edge involves spreading on a solid followed by expansion in the air, an intermediate between two axisymmetric configurations already investigated: an impact on an infinite solid [20,67] and a centered impact on a circular target of size comparable to the drop [38,39,42] [ Fig. 11(a)].…”

“…Figure 11(b) shows that the liquid sheet for large We and small offsets goes further than the liquid only spreading on the solid. Indeed, spreading on a solid dissipates energy through viscous friction well captured by a Blasius-type boundary layer [42], which reduces liquid spreading. The experimental values of R sM are in good agreement with Eq.…”

“…Previous studies focused on the crown dynamics and droplet ejection in axisymmetric configurations [33][34][35][36][37]. Increasing attention has also been paid to impacts on small solid targets of size comparable to that of the drop diameter [38][39][40][41][42] or binary drop collision [43,44]. In the crescent-moon impact scenario, an intrinsic horizontal asymmetry leads to a nonaxisymmetric liquid sheet [19].…”