ICLASS-2006 Aug.27-Sept.1, 2006, Kyoto, Japan Paper ID ICLASS06-289 Splash due to rim transverse instability Ilia V. Roisman 1 and Cameron Tropea 2 1 Dr. habil., Chair of Fluid Mechanics and Aerodynamics, Darmstadt University of Technology, roisman@sla.tu-darmstadt.de 2 Professor, Chair of Fluid Mechanics and Aerodynamics, Darmstadt University of Technology, ctropea@sla.tu-darmstadt.de ABSTRACT In this theoretical study the dynamics of a rim bounding a free liquid sheet is considered. This rim is formed by capillary forces. When rim is unstable it creates cusps, jets, and secondary droplets. In this study the linear stability analysis is performed for small transverse disturbances of the rim centreline. The influence of the film thickness and rim acceleration has been investigated. The results of the study are important for the understanding of splash produced by drop impact and spray wall interaction. On the base of the theoretical analysis a length scale is proposed for the size of the secondary drops generated by spray impact. This assumed scaling is confirmed by the experimental data. Keywords: spray impact; splash, rim, transverse instability, capillary flows 1. INTRODUCTION The main motivation of the present study is the investigation of a single drop impact (see for example comprehensive reviews of the single drop impact modelling [1,2]) or spray wall interaction leading to splash. Spray impact is fascinating, rich in hydrodynamic phenomena, but extremely complicated for reliable modelling. Nevertheless, all inertially dominated flows associated with drop and spray impact onto a cold stationary rigid target can be subdivided into three main types: - transformation of a nearly spherical drop into a film - transformation of a film into a jet, formation of a rim - breakup of a jet and creation of secondary droplets These phenomena are definitely influenced by secondary effects typical for spray impacts: interaction of films, holes formation in films, etc. Other transformations are possible only when a sufficiently strong input of external energy is introduced into the system. For example, direct ejection of Faraday jets from a drop vibrating with high frequency; from the surface of a liquid reservoir (drop-to-jet transformation), [1] or explosive defragmentation of a liquid (drop-to-droplets transformation) [4]. The important elements of splash modelling are a consistent scale for the film thickness, determining the criteria for the appearance of the uprising sheet, and description of the dynamics of a rim formed at the edge of the sheet due to the capillary forces [5-8]. From the modelling point of view the splash process can be subdivided into several stages shown schematically in Fig. 1: bending of the rim centreline and development of its instability (described below), cusp formation [6], and the ejection of the finger-like jets at the cusps locations. The last stage of the splash - capillary break of a jet leading to creation of secondary droplets has also been analyzed and explained [9,10]. Analysis of the mechanism of rim bending instability leading to the cusp formation and ejection of a finger-like jet should complete the general description of splash. Among parameters influencing rim stability are the geometry of the problem, air density [11], internal stresses appearing due to stretching (when for example the radius of its centreline increases) or compressing rim, velocity gradients in the sheet, or rim acceleration. In this section a linear analysis is performed for a transverse instability of a straight rim formed by the capillary forces in a planar free sheet and rim acceleration. 2. RIM TRANSVERSE INSTABILITY 2.1. Base solution: motion of a straight rim Consider a straight rim bounding a free non-viscous planar sheet of the thickness h(y, t). The effect of viscosity is assumed to be small and neglected. This rim is formed at Fig.1: stages of rim instability, breakup and ejection of the secondary drops