The impulsive motion of a small cylinder at an interface

Abstract

We study the unsteady motion caused by an impulse acting at time t=0 on a small cylinder floating horizontally at a liquid–gas interface. This is a model for the impact of a cylinder onto a liquid surface after the initial splash. Following the impulse, the motion of the cylinder is determined by its weight per unit length (pulling it into the bulk liquid) and resistance from the liquid, which acts to keep the cylinder at the interface. The range of cylinder radii r and impact speeds U considered is such that the resistance from the liquid comes from both the interfacial tension and hydrodynamic pressures. We use two theoretical approaches to investigate this problem. In the first, we apply the arbitrary Lagrangian Eulerian (ALE) method developed by Li et al. [“An arbitrary Lagrangian Eulerian method for moving-boundary problems and its application to jumping over water,” J. Comput. Phys. 208, 289 (2005)] to compute the fluid flow caused by the impulse and the (coupled) motion of the cylinder. We show that at early times the interfacial deformation is given by a family of shapes parametrized by r/t2/3. We also find that for a given density and radius there is a critical impulse speed below which the cylinder is captured by the interface and floats but above which it pierces the interface and sinks. Our second theoretical approach is a simplified one in which we assume that the interface is in equilibrium and derive an ordinary differential equation for the motion of the cylinder. Solving this we again find the existence of a critical impulse speed for sinking giving us some quantitative understanding of the results from the ALE simulations. Finally, we compare our theoretical predictions with the results of experiments for cylinder impacts by Vella and Metcalfe [“Surface tension dominated impact,” Phys. Fluids 19, 072108 (2007)]. This comparison suggests that the influence of contact line effects, neglected here, may be important in the transition from floating to sinking.