Spreading of an inviscid drop impacting on a liquid film
Introduction
Drop impact on solid surfaces occurs in many industrial processes (e.g. spray cooling, spray coating, ink jet printing, material erosion) and in nature (e.g. impact of rain drops). With repeated drop impacts, an initially dry surface acquires a liquid film. Although drop impact on surfaces has been the topic of investigation since the very early account by Worthington (1876), the phenomenon is not fully understood. Studies have considered impact with dry surfaces, deep and shallow liquids, heated surfaces, and splash formation (Weiss & Yarin, 1999).
Although there are many studies of single drop impact with dry solid surfaces and deep liquids (Rein, 1993), a smaller but increasing number of articles are concerned with thin liquid films. These include experimental studies by Levin and Hobbs (1971), Macklin and Metaxas (1976), and Cossali, Coghe, and Marengo (1997), and a combined theoretical/experimental study by Yarin and Weiss (1995). The first numerical study was made by Harlow and Shannon (1967) using a marker-and-cell finite-difference scheme. More recently, volume-of-fluid (VOF) methods (Gueyffier & Zaleski, 1998; Vincent, Caltagirone, & Arquis, 1999; Rieber & Frohn 1999) and boundary integral (BIM) methods (Davidson 1998, Davidson 1999; Weiss & Yarin, 1999) have been used to predict the deformation of the air–liquid interface after drop impact. Both Yarin and Weiss (1995) and Weiss and Yarin (1999) summarise related work on drop impact.
When a drop impacts on a liquid film with sufficient velocity to produce splashing (large Weber number), a liquid sheet is ejected upwards and almost normal to the film from the periphery of the drop. Small jets form and breakup into secondary droplets at the top of the ejected sheet, giving it a crown-like appearance. The crown (i.e. the liquid sheet) propagates radially outwards from the centre of impact. For lower impact velocities, the drop will spread without splashing in combination with the film.
Yarin and Weiss (1995) developed a one-dimensional theory of splash in which the crown is represented as a kinematic discontinuity arising from an initial velocity distribution in the liquid film. Viscous effects were ignored and surface tension effects were taken to be small. They predicted a t1/2 dependence for the radial position of the crown at large times t, and verified the result experimentally. The result was subsequently confirmed by Cossali et al. (1997) in an experiment with water for a single Weber number and dimensionless film thickness. Since then, this result has been predicted for large Weber number by all of the VOF and BIM numerical studies mentioned above.
The BIM study of Weiss and Yarin (1999) considered in minute detail the initial deformation of the neck region where the drop meets the film. They predicted the formation of a jet in the middle of the neck for times which are very small compared with the impact time scale a/U for sufficiently large Weber number (drop of radius a impacting with velocity U). However, the existence of such a jet is yet to be confirmed either experimentally or theoretically. (While the present paper was under review, Thoroddsen (2002) reported the first experimental observations of jetting at very small times.) For times comparable to, or larger than, the impact time scale, Weiss and Yarin (1999) predicted crown propagation rates in accord with their one-dimensional theory (Yarin & Weiss, 1995). However, their crown shapes seem more rounded than VOF predictions by Rieber and Frohn (1999) and initial BIM predictions by the author (Davidson, 1999) under similar conditions. The aim of the present paper is to use the author's BIM to seek confirmation of jetting behaviour at the initial moments after impact, and to predict the shape of the evolving crown at larger times. Some additional calculations are performed using a VOF method. Results are discussed especially in relation to Yarin and Weiss (1995) and Weiss and Yarin (1999).
Section snippets
Formulation and numerical method
A spherical liquid drop of radius a impacting vertically with velocity U on a horizontal film of the same liquid is considered. The motion of the drop is investigated from a state of non-zero contact with the liquid film (Fig. 1a). Only axisymmetric deformation is considered and viscous effects are ignored. Such a representation of drop impact has been justified in detail by Weiss and Yarin (1999). Fig. 1(b) is a schematic of the liquid profile in the (r,z) plane in cylindrical polar
The neck region immediately after impact
For very small times after impact, WY predicted the formation of a radial jet in the middle of the neck region where the drop meets the film. The jet occurs after the compressible stage (Rein, 1993) which leads to the formation of contact area, and is unrelated to it. They concluded that the jet is an inevitable consequence of the step change in downwards liquid velocity at the moment of impact if the velocity is large enough for inertial effects to dominate over surface tension effects.
Conclusion
Weiss and Yarin (1999) predicted that, for sufficiently large Weber number, jetting will occur in the neck region where the drop meets the film at times immediately after impact. This phenomenon has been confirmed in the present more accurate numerical study for We=100,1000, corresponding to Fig. 3, Fig. 4 of Weiss and Yarin (1999), but with some differences. Instead of growing progressively, the jet is found to wax and wane, sometimes disappearing completely. Associated with these changes, the
Acknowledgements
The author wishes to thank Murray Rudman (CSIRO Division of Building, Construction and Engineering) for the use of his VOF code. He also grateful to A.L. Yarin and D.A. Weiss for helpful discussions relating to their numerical method.
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