Kinematic Closure of Drop Impact

Abstract

Predicting how far a liquid droplet spreads upon impacting a solid surface is governed by a complex competition between inertia, surface tension, and viscous dissipation. Existing models typically estimate this maximum spreading diameter by prescribing the spreading time or velocity from a decoupled asymptotic limit, restricting predictive accuracy across regimes. Here, we formulate a reduced kinematic description by resolving both the characteristic spreading time and spreading velocity from an energy balance coupling capillary storage and viscous dissipation. Unifying these two asymptotically constrained closures via the kinematic relation $D_{\max }=V_s{t}_s$ yields a compact, parameter-free approximation for the maximum spreading ratio that bridges the inertio-capillary and inertio-viscous regimes. We quantify the accuracy of this reduced closure against the exact implicit energy relation and demonstrate that it quantitatively collapses our experimental measurements alongside extensive literature data across wide ranges of Weber and Ohnesorge numbers, droplet sizes, and surface wettabilities without adjustable prefactors.

arXiv comment6 pages, 4 figures, to be submitted for publication as a letter