Fluid Dynamics Research 12 (1993) 61 93 FLUID D Y N A M I C S North-Holland RESEARCH

Phenomena of liquid drop impact on solid and liquid surfaces

Martin Rein

Max-Phmck-hTstitu! ['ur Stri'mmng,s[orschung. Bunsenslrafle 10, 37073 Gi~ttingen. Germany

Received 1 February 1993

Abstract. The fluid dynamic phenomena of liquid drop impact are described and reviewed. These phenomena include bouncing, spreading and splashing on solid surfaces, and bouncing, coalescence and splashing on liquid surfaces. Further, cavitation and the entrainment of gas into an impacted liquid may be observed. In order to distinguish properly between the results of different experiments different impact scenarios are discussed. The specific conditions under which the above phenomena did occur in experiments are analyzed and the character- istics of drop impact phenomena are described in detail.

I. Introduction

The fluid mechanics of drop impact with surfaces is of importance in a variety of different fields. High-speed impacts with solids can cause severe erosion as, for example, in steam turbines. Engineering applications of drop impact are spray cooling and ink-jet printing. Thin coatings can be obtained on surfaces by spray coating and spray painting. In severe reactor accidents the accumulation of molten core debris on the containment walls may be reduced by vigorous splashing. The entrainment of bubbles by drops falling into a superheated liquid can enhance nucleate boiling. In filtration aerosol droplets are only absorbed when adhesion is obtained on contact. Drop impact is also of interest in non-engineering fields. In agriculture the prevention of soil erosion due to rain plays an important role. A knowledge of the spreading behaviour of pesticides of foliage enables a reduction of the quantity of pesticides applied per unit area. Further, splashed pesticide droplets can be blown away by wind and pollute neighbouring places. Splashing raindrops provide mechanisms for the dispersal of fungus spores. Atmospheric and oceanographic sciences investigate phenomena connected with rain formation and the interac- tion of rain with the surface of the ocean. High pressures occurring during meteor impacts can cause a fluidization of the rocky material. The resulting flows may lead to the formation of central peaks in craters that can be found, for example, on the moon. Further, estimates of the size distribution of microtektites were derived by studying these flows. Finally, in criminalistics the study of characteristics of stain patterns of impacted blood drops can be of significance in reconstructing crimes. It is, therefore, not surprising that investigations of drop impact focus on very different aspects of this process.

The fluid flow associated with impinging drops is rather complicated and not understood in detail. Depending on the special circumstances different characteristic features are observed. Consider, for example, the various phenomena that can appear when it is raining. Raindrops falling into a pool can splash, coalesce with or bounce off the water. If the impact energy is high enough, the impact of a drop on a liquid surface leads to the formation of a crater in the liquid. At

Correspondence to: M. Rein, Max-Planck-Institut fiir Str6mungsforschung, BunsenstraBe 10, 37073 G6ttingen, Germany.

0169-5983/93.,$09.25 ~ 1993 The Japan Society of Fluid Mechanics. All rights reserved

62 M. Rein ' Phenomena ~!['liquid drop impact

the crater wall a sheet of liquid is raised above the surface. Often this sheet is unstable and tiny droplets appear at its upper rim. This occurrence is called crown formation (sometimes also coronet formation) because of its resemblance with a real crown. At high impact velocities the sheet of liquid closes above the crater and a bubble is formed. When it rains heavily such bubbles can be observed floating on the surface of pools. Further phenomena are the formation of jets which emerge out of the center of the crater, and the generation of vortex rings which propagate into the liquid. The jets will be called central jets hereafter .-Raindrops colliding with a solid surface splash or simply spread on the surface. They may even bounce. Very high pressures can arise inside a drop when it impacts a solid surface. This results in sideways jetting of the liquid from beneath the drop. The subsequent reduction in pressure caused by strong expansion waves can lead to the occurrence of cavitation inside the drop.

The collision of a drop with a surface is governed by a large number of parameters. Following the H-theorem the number of independent parameters can be reduced by the number of fundamental dimensions that are present. Usually the remaining number of parameters that are important for a single aspect of drop impact, is still relatively large. It is, therefore, difficult to obtain simple relations from dimensional analysis. However, a dimensional analysis provides a complete set of dimensionless groups that governs the process considered. In previous work dimensionless groups were often introduced. However, this was not always the case and, hence, the results reported are sometimes not comparable. When dimensionless groups were considered this was not always a complete set. This explains some of the discrepancfes that can be found between the results of different sources.

This paper focuses on the fluid mechanics of drop impact. Other important aspects such as the separation of electrical charges and the Leidenfrost phenomenon (i.e., the separation of a drop from a hot surface by a layer of vapor) are not considered. Compressibility effects that are typical for high-speed impacts were previously reviewed by Lesser and Field (1983), who were especially concerned with pressure loads and erosion effects. These topics are also of importance in the case of meteor impacts (Melosh, 1989). The generation of sound by drops impinging into liquid layers and related aspects are subject of a recent survey by Prosperetti and O~uz (1993). Abbott (1977) gave an overview of experiments on the interaction of raindrops. The emphasis of the present review is on the different phenomena occurring at low-speed impacts. The different conditions under which experiments were performed and at which phenomena such as splashing occur are examined.

2. Different scenarios of drop impact

Many different cases of drop impact can be distinguished according to the circumstances under which the impact happens. The outcome of a collision depends on the properties of the drop, of the impacted surface, and of the fluid which the drop traverses before impact. Further, the impact velocity and geometric aspects are important. Often single drops of a spherical shape are considered which collide normally with a flat solid surface or with a deep liquid layer. The fluid above the surface is usually assumed to be a gas of negligible influence. However, experimental arrangements that have been used, are not always that simple. In order to compare results reported in different sources, the different initial and boundary conditions must be considered in great detail. This is the subject of the next sections.

2.1. Properties 0[" the drop

Figure 1 provides an overview of different parameters that are of importance during drop impact. It begins with the drop, which is usually assumed to consist of a homogeneous liquid. The

M. Rein / Phenomena qf liquid drop impact 63

. . . . liquid drop: / 0 \

internal spherical deformed oscillating circulation

impact:

surfactants

0 normal impact oblique impact

l boundary wake r//iayer

o o

surface:

liquid

shallow deep

solid

/ / /7/ / / / / / plane curved

"/// / / / / ,

flat wavy smooth rough

target liquid: unyielding yielding

© @ i F _-_ v ~_ v ; ~ Y// / / /x -C_-5-55: I,,, ,,,

same different

Fig. 1. Survey of parameters governing the impact of a liquid drop.

liquid is described by its thermodynamic state, and by its surface tension, viscosity and compressibility. Mixtures of water and gelatine have also been used in experiments by Field et al. (1979) in order to form drops of a well-defined non-spherical shape. The flow properties of such mixtures were shown to be similar to the ones of water in the parameter range applied in the experiments. Further, emulsified liquid droplets may be used, as in the experiments of Avedisian and Fatehi (1988) who, however, studied evaporation characteristics which are of no interest here.

Most of the theoretical and numerical calculations are based on the assumption that the drops are spherical. This is also the case in experimental work, although the shape of drops moving

64 M. Rein / Phenomena ~/ liquid drop impact

through a fluid will always be rendered slightly ellipsoidal by aerodynamic forces. Stow and Hadfield (1981) and Cai (1989) seem to be the only ones who determined experimentally the eccentricity of drops at the moment of impact.

Another cause for deviations from the spherical shape are drop oscillations. An oscillation renders the shape of drops time dependent. Hence, the phase of the oscillation at the moment of impact may be relevant. This was corroborated by experiments of Thompson and Newall (1885) and, later, by Rodriguez and Mesler (1988). The latter authors found that the penetration depth of vortex rings that may be formed after drop impact into liquids, is greatest when the shape of the drop on impact is prolate. With the exception of Engel (1955), who looked briefly at the difference between a prolate and an oblate drop impact on solid surfaces, nobody has yet investigated whether the oscillation of drops also changes the characteristics of impacts with shallow liquids or solid walls. Usually it is argued that oscillations which may be formed by the process of drop formation, are damped off at the moment of impact. However, oscillations can also be excited by the wake of the drop (Winnikow and Chao, 1966).

The oscillation of a drop is associated with a flow within the drop. An internal flow can also be generated by the friction between a moving drop and the fluid surrounding it. This leads to an internal circulation which results in a well-known drag reduction (Rybczyfiski, 1911; Hadamard, 1911). This type of internal flow is normally not considered in connection with drop impacts. The internal circulation depends very much on the cleanliness of the liquid surface. Small amounts of surfactants can shield the interior of the drop so that there is no circulation at all. Another cause for alterations of the surface tension can be the process of aging. The surface tension of solutions, for example, may be changed by diffusion of the solute to a newly formed surface (Ward and Tordai, 1946). However, in the case of pure liquids like water a virgin surface ages very quickly and the final value of the surface tension is assumed within a period much shorter than any of the characteristic fluid dynamic time scales before impact.

The impact of liquid masses of shapes other than spherical was also investigated. In order to gain a deeper insight into the phenomena occurring during high-speed impact, Brunton and Camus (1970), Field et al. (1979), and Dear and Field (1988) investigated the impact with solid walls of disc like shapes and of wedges that were cut out of a mixture of water and gelatine. Using optical methods they followed the propagation of shock waves in the drop, and the formation of sideways jetting. This is thoroughly discussed in the review by Lesser and Field (1983). In erosion research the wheel-and-jet technique is often used to study liquid impact (Springer, 1976). Instead of a drop a liquid jet is laterally impacted by a model surface that is fixed at the rim of a rotating wheel.-Huang et al. (1973) and Glenn (1974) simulated numerically the collision with a solid wall of a liquid cylinder of finite length which moves parallel to its axis. The latter author discusses critically the findings of the former. The emphasis of Glenn's work is on the shock compression of the liquid and on the propagation of lateral expansion waves into the cylinder. The possibility of cavitation is also discussed.

In many experiments the impact of single drops was studied. Sometimes, however, also droplet streams were used (Siscoe and Levin, 1971; Riedel, 1977). These are easily obtained by taking advantage of a forced break up of liquid jets (see, e.g., Hiller and Kowalewski, 1989). Droplet streams were also applied in investigating the impact of single drops (Jayaratne and Mason, 1964; Zhbankova and Kolpakov, 1990; Schotland, 1960). The frequency of the stream then needs to be low enough so that the effect of preceding drops has died away when a new drop collides with the surface. In some cases droplet streams were used without reporting the frequency (Hobbs and Osheroff, 1967; Macklin and Metaxas, 1976). Ford and Furmidge (1967) let droplet streams impact onto a slowly rotating disc, thus avoiding the interference of succeeding drops.

M, Rein / Phenomena of liquid drop impact 65

2.2. Fluid between the drop and the impacted surface

In the following the fluid through which the drop passes before it reaches a surface is considered to be a gas. It is described by its thermodynamic state, and by its viscosity and compressibility. Normally the influence of this gas is neglected. It was already mentioned, however, that the interaction between the surrounding fluid and the drop can give rise to an internal circulation and to oscillations of the drop. Further, in the case of a solid sphere entering into water the density and pressure of the surrounding gas are important parameters which determine whether the cavity in the water produced by the impact is closed in a deep or surface seal (Birkhoff, 1960). Very high pressures are present when a fuel is injected into a combustion chamber. The fuel droplets that are formed, impinge on the walls where they can splash. The splashing characteristics may be different at high pressures.

The influence of the pressure in the surrounding gas needs a greater attention. Worthington (1877) studied the impact of drops of oil and mercury at low pressures. In a first quantitative approach, the influence of the density of the gas was investigated by Schotland (1960). He performed experiments in a depressurized chamber and found that the density of the gas may be of some importance. In experiments by Engel (1966) the pressure was reduced to obtain greater impact velocities. A possible influence of the low pressure on the results was not considered. In the case of high-speed impact Brunton (1966) observed a light flash during impact. The flash is probably due to luminescence of air entrapped between the drop and the wall. A photographic study of the breakup of the air film between an impacting drop and a !iquid surface was conducted by Sigler and Mesler (1990) for the case of low-speed impacts. It is interesting that the rupture takes place at a time when already a small crater has formed in the target liquid.

2.3. Normal and oblique impact, and impact velocity

It is necessary to distinguish between normal and oblique impacts. In the case of normal impacts there may be a difference between a drop impacting a stationary surface, and a surface impacting a stationary drop. In the former case the wake of the drop can excite oscillations and a circulation may be generated inside the drop. More important is that the drop will assume a slighly ellipsoidal shape. This causes a delay in the commencement of sideways jetting that was investigated by Perel'man (1975). When the surface moves towards a stationary drop the air blast ahead of the impacting surface may distort the drop. In experiments with moving sliders Field et al. (1989) shielded the drop against the blast by a cowling. In order to minimize disturbing effects of the gas, erosion experiments by the wheel-and-jet technique are normally performed in chambers with very low pressures.-Similarly, an oblique impact with a stationary surface does not necessarily produce the same results as a normal impact onto a surface moving perpendicu- larly to the direction of impact. Parker (1970) and Reske (1987) observed a lift-off of drops after these impinged normally onto a moving wall. They related the lift-off to the influence of the boundary layer at the wall. A reflection of drops colliding with a moving wall was also observed by Povarov et al. (1976a, b) when the boundary layer is turbulent. In the case of laminar boundary layers they obtained spreading. Due to aerodynamic forces acting on the distorted drop reflected drops acquire a rotational velocity. At high velocities of the wall the drop does no longer touch the surface. In a general diagram Povarov et al. (1976b) show regions of spreading, partial reflection and total reflection as a function of the squared impact velocity scaled with the wall velocity, and of the boundary layer thickness in terms of the drop diameter.

In the case of supersonic impacts (with respect to the gas between the surface and the drop) shocks will be present. When the surface approaches the drop supersonically, the drop is hit by a shock wave prior to impact. After the passage of shocks drops usually shatter. However, if the

66 M. Rein / Phenomena ~?] liquid drop impact

distance between the surface and the shock is small, the time is not sufficient for shattering and the drop is merely deformed (Hsiang and Faeth, 1992). Photographs of this event can be found in Poddubenko and Yablonik (1990). Further, behind oblique shocks the droplet trajectory will be changed (Serafini, 1954). The impact of a liquid mass at supersonic speed with a solid wall was investigated, e.g., by Bowden and Brunton (1961). A detached shock ahead of a high-speed liquid jet was observed by Field and Lesser (1977).

At low impact velocities it may make a difference whether the drop is in a steady or accelerated motion. The latter is usually the case if the drop falls in a gravitational field and the drop height is small, i.e., the drop does not reach its terminal velocity before impact.

From a theoretical viewpoint the impact velocity is an elementary parameter that can easily be changed, In experiments, however, the impact velocity is determined in a variety of different ways. In some cases it is directly measured. In other cases the fall height of the drop is cited. Often the impact velocity is assumed to equal the terminal velocity of the drops. The terminal velocity is then determined from correlations of various sources. It is difficult, however, to take into account the influence of surfactants. Wallis (1974) gave an overview of different correlations and showed that these are not always compatible. Further, he presented a general calculation procedure for the terminal velocity which is based on dimensional analysis and on many previous studies. Even using this general procedure, the impact velocities obtained sometimes deviate from actual values by more than 10%. This leads to an error in the kinetic energy of the drop of more than 20%. One should keep this in mind when results of different experiments are compared.

2,4. Liquid sur/aces

The surface impacted can be either solid or liquid. In the liquid case the surface of a liquid layer will be considered since the interaction of different drops is not examined at this place. In most cases the layer is assumed to be deep, i.e., the bottom does not affect the process of drop impact. In some papers the depth is not explicitly mentioned, but it is clear from the context, or from photographs that a deep layer was present. Some authors have investigated the influence of the depth of shallow liquid layers on the splash produced by impacting drops. Levin and Hobbs (1971) report that they observed little difference between the splashing characteristics of drops impinging on thin liquid films and solid surfaces, the only exception being the time scale which is much shorter in the latter case. Several numerical simulations of drops splashing on shallow and deep liquid layers were performed by Harlow and Shannon (1967a,b).

In virtually all cases the impact of drops with plane liquid surfaces was investigated. In some experiments, however, a wavy surface was present. When the surface is impacted by droplet streams, waves excited by preceding drops are normally considered to be minor disturbances (see, e.g., Schotland, 1960; Jayaratne and Mason, 1964). Zhbankova and Kolpakov (1990) assumed that the surface of the liquid layer recovers between two succeeding collisions. This may be true when the impact frequency is small. At high frequencies the surface will be deformed by preceding drops. Siscoe and Levin (1971), therefore, looked at the interaction of surface waves with impinging drops. In their experiment they used an axisymmetric geometry. Drops fell into the center of a jar filled with water. They found that drops colliding with the crest of a wave are absorbed and amplify the surface waves, whereas drops impinging into a trough produce an unusually high central jet.

The target liquid needs not necessarily be the same as that of the drop. When the liquids are different it needs to be further distinguished between miscible and immiscible liquids. Experi- ments with different liquids are often reported in early work. When a drop is placed on a target liquid of lower density it will sink and a vortex may be produced. Tomlinson (1864) studied differences in the patterns of such vortex rings, and also of those produced when a drop impacts

M. Rein / Phenomena of liquid drop impact 67

Fig, 2. Impact of drops of distilled water into a shallow layer of a mixture of water and glycerol (radius R = 1.56 mm, fall height: 0.1 m, depth of liquid layer: h ~, R). The flow patterns after the 12th drop impact are shown. The target liquid is: (a) a 2% solution of acetic acid (surface tension smaller than that of drop liquid); (bl a pure mixture of water and glycerol (surface tension about that of drop liquid); (ct a 20% solution of ammonium chloride (surface tension greater than that of

drop liquid). (from Smith, 1975)

with a small velocity. He suggested that the differences can be useful, e.g., in identifying adulterations of oils. Further investigations of the influence of different drop liquids on the formation of vortices due to drop impact can be found in Thompson and Newall (1885). Worthington (1908) added milk to the target liquid (water) and observed "certain differences in detail", but only "little difference in the resulting splash". The age of the surface of the target liquid was reported to be of some influence in experiments by Esmailizadeh and Mesler (1986) and by Hsiao et al. (1988), respectively. In these cases the surface tension appears to have been altered by contamination and oxidation.

The effect of a difference in the surface tensions of the drop and the target liquid was investigated more closely by Smith (1975) in connection with the so-called drop picture method. When drops fall into shallow liquid layers the resulting flow patterns depend on the ratio of the surface tensions. Different patterns ("drop pictures") corresponding to different ratios of the surface tensions were proposed to be useful in determining the quality of liquids. Usually a series of water drops falls into a mixture of water and glycerol. Then Schlieren pictures are taken. Three examples from Smith (1975) are shown in fig. 2. The flow patterns after the 12th impact of a water drop into the target liquid are shown. In fig. 2a the surface tension of the target liquid is rendered smaller than the one of the drop by addition of acetic acid. In fig. 2b the surface tensions are about the same, and in fig. 2c the surface tension of the target liquid is greater than the one of the drop. The viscosities are about the same in figs. 2a-c. However, the influence of different viscosities was also addressed by Smith (1975).

In more recent investigations the drop and target liquid are usually the same in both, experiments and theoretical considerations. Exceptions are flow visualization by ink and the immersion method for determining droplet-size distributions. In the latter method droplets are collected in an immersion liquid that is immiscible with the drop liquid (Rupe, 1950).

2,5. Solid surfaces

Now the impact of drops on solid surfaces is considered. Usually the surface is formed by a plane wall. In some cases the wall is curved as in experiments of Levin and Hobbs (1971) who used a copper hemisphere. However, the curvature of the hemisphere was much smaller than that of the drop. The advantage of curved surfaces is twofold. First, the optical observation of the

68 M. Rein / Phenomena ~?f liquid drop impact

process of impact is more easily performed. Second, the liquid runs off the point of impact. This is useful when droplet streams are used in the experiments.-Dear and Field (1988) investigated by Schlieren optics the effect of different surface shapes on the propagation of shock waves in both, the liquid and the impacted medium. Depressions, as well as projections in the solid surface were examined. The background of this work was the investigation of erosion processes due to repeated impact.

The surface of the wall can be smooth or rough. Engel (1955) reports that splashing was reduced when highly polished surfaces were used. In accordance with this Levin and Hobbs (1971) note that they employed a rough surface in order to promote splashing. Stow and Hadfield (1981) investigated the influence of the surface roughness on the splashing threshold in a rigorous manner. At small surface roughnesses the splashing threshold depends strongly on the roughness, whereas the threshold is little influenced by the surface roughness when it is large. These results are in qualitative agreement with Parker (1970). It is interesting that a qualitatively similar dependence on the surface roughness of the number of impacts required to produce erosion pits was reported by Hancox and Brunton (1966). In a few cases the surfaces were carefully washed with cleaning liquids and distilled water (Stow and Hadfield, 1981; Loehr, 1990). In order to simulate the impact on foliage, Ford and Furmidge (1967) prepared 'artificial' surfaces of beeswax and cellulose acetate. The impact with different types of paper was examined by Wallace and Yoshida (1978), who were concerned with determining droplet-size distributions in pesticide sprays, and by Oliver (1984), who investigated image development and ink drying in the process of ink-jet printing. A1-Durrah and Bradford (1982) studied the amount of soil detachment due to single raindrop impact onto different soil surfaces. Porous surfaces that can be of technical relevance were used in experiments by Avedisian and Koplik (1987) in the contex of the Leidenfrost phenomenon. Often, however, the state of the surface is not described in detail.

In many problems the elastic response of the surface is insignificant. However, the elasticity of the wall can no longer be neglected when high-speed drops collide with the wall. The elasticity of the wall was taken into account in an analytical calculation by Lesser (1981). His results show that the onset of sideways jetting of liquid from beneath the drop will be delayed by an elastic response of the wall. This was also observed in experiments by Field et al. (1989), who let drops impinge on compliant walls. The response of surfaces to liquid impact is of primary importance in erosion research but will not be further considered here.

3. Discussion of phenomena

3.1. hnpact of drops onto solid surfaces

The preceding sections dealt with the different conditions under which drops may impact. Now the phenomena of drop impact are more closely examined. First, the impact on solid surfaces is considered. In this case the result of a drop impact can be bouncing, spreading or splashing. This is sketched in fig. 3. In some cases cavitation was observed inside the drop. In the next section a theoretical description of the initial phase is given. Then, the spreading and splashing behaviour is considered. Finally, mechanisms leading to cavitation are discussed.

3.1.1. Initial phase The impact of liquid drops on solid surfaces results in bouncing, spreading or splashing

(cf., fig. 3). In all cases the compressibility of the liquid plays a major role during the initial phase

M. Rein ,' Phenomena q[liquid drop impact 69

©

~ \ ~ \ \ \ \ \ \ \ \ \

t 0

~ \ \ \ \ ~ ~ \ \ x bouncing

o o

o o

o o

~ \ \ \ \ \ \ \ \ \ ~

spreading splash in g

Fig. 3. Impact of a drop on a solid surface: bouncing, spreading, and splashing.

of the impact. In the acoustic limit the pressure increase equals the waterhammer pressure pcui where p and c are the density of the liquid and the sound speed in the liquid, and ui is the impact velocity. Even at low-speed impacts the pressure can rise appreciably. Since the events occurring right after impact are important for the subsequent processes the initial phase is considered in the following.

Bowden and Field (1964) first pointed out the significance of the shock wave that is formed in a drop when it collides with a solid wall. This shock and other important parameters are shown in fig. 4. In the ideal case, the first contact between the base of the drop and the wall is point like. A contact zone of radius re then develops. The contact edge moves outward. Initially the velocity uc of the contact edge is supersonic with respect to both, the sound speed in the liquid, and the speeds of the dilatational and shear waves in the solid wall. This is called the tri-supersonic case (Lesser and Field, 1983). Here, elastic properties of the wall will not be considered. As the drop further approaches the wall the contact angle /~ between the drop and the wall increases and, hence, the contact edge velocity decreases. The relation between the impact velocity ui, the contact edge velocity ue, and the contact angle/~ is (cf., fig. 4)

ue = ui/tan ft.

Inside the drop the shock propagates with a velocity cs that is of the same order of magnitude as the sound speed. As long as the impact velocity is greater than cs sin fl the shock remains attached to the contact edge. The liquid ahead of the shock is not yet disturbed by the impact. When the contact angle becomes larger than the critical angle tic,

/~c = sin- 1 (ui/Cs),

the shock separates from the contact edge and moves up the undisturbed surface of the drop. Then, the compressed liquid is no longer enclosed by the shock and the wall, and sideways jetting becomes possible (cf., fig. 5). At the same time expansion waves propagate into the drop possibly causing cavitation. This picture of the initial phase of a drop impact was developed for high-speed impacts. In the limit of low-speed impacts the critical contact radius (i.e., that radius where the shock overtakes the contact edge) becomes very small. However, a jetting motion that originates under the drop right after impact can still be observed (Stow and Hadfield, 1981; Loehr, 1990). In the low-speed case jetting begins at a contact radius that is larger than the critical contact radius for jetting. Further, the jet velocity is smaller than that observed in the case of high-speed impacts. The jetting flow either is the initial phase of a purely spreading motion of the liquid on the surface, or it ends in splashing. In the former case the outcome of the impact is a liquid lamella that

70 M. Rein / Phenomena ~?] liquid drop impact

Z

ha

Cs

re I Ue

Ui

r

Fig. 4. Impact of a drop on a solid surface: the initial stage.

Fig. 5. Propagation of a shock wave into the drop after it impacted a solid surface, formation of an expansion wave, and the onset of sideways jetting.

shrinks and eventually assumes its equilibrium shape, whereas the drop disintegrates into many smaller ones in the latter case.

During the initial stage of impact the drop is merely deformed and compressed at its base. Hence, surface tension forces and the viscosity of the liquid do not enter the scenario at that early stage. The influence of the surrounding gas will usually be of minor importance causing a slightly ellipsoidal shape of the drop at the most. To simplify the problem the wall is often assumed to be rigid. This is a good approximation as long as the impact velocity is not too high. The important parameters then are the density and compressibility of the liquid, and the impact velocity and

M. Rein Phenomena o]l iquid drop impact 71

radius of the drop. All physical quantities of interest are functions of these variables. In non-dimensional form the dependencies can be reduced to functions only of a single variable. This variable is normally chosen to be the impact Mach number Mi = ui/c. Note that Mi is formed with the sound speed of the liquid. Typical physical quantities of interest are the radius re and time t~ of jetting commencement, and the pressure in the compressed liquid volume. There exist different characteristic times t* and pressures p* and one characteristic length l*, namely, the radius R of the drop. Here, three characteristic times are formed:

t* = R/c, t* ~ - - R/Hi, t~ = (ui/c)(R/c).

The meaning of t*,2 is obvious. The time t* can also be useful as will be seen below. The characteristic pressures are the waterhammer pressure, the stagnation pressure and the inverse of the compressibility K:

p* = Uix/p/K = pCUi, p'~ = pu 2, p~' = l/to.

Returning to the Bowden Field (1964) picture (cf., fig. 4) the angle between the drop surface and the wall equals tic = sin- 1 M~ at the moment of shock detachment. The corresponding contact radius is simply given by

G/R = Mi.

Some straightforward analysis yields for the time between impact and shock detachment

t¢ c/R = ~-1, [1 - cos (sin- 1 Mi)] ~ Mi/2.

The relation on the r.h.s, is correct to O(M3). Using the third characteristic time for scaling simply yields t¢/t~ = 1/2.

In the contact zone between the drop and the wall the pressure is not uniform. It is highest at the contact edge where it exceeds the waterhammer pressure, and lowest at the center. This is due to the spherical geometry of the drop. At the contact edge the shock is oblique with respect to the wall, and its slope (and thus the nonuniformity of the pressure) increases with the contact radius, i.e., with time. In the acoustic limit Lesser (1981) determined the wave front using the Huygens principle. He also obtained a solution for the pressure in the compressed region, This solution includes the edge pressure given in terms of the edge Mach numberMe = ue/c. However, the edge pressure becomes singular when Me ---, 1.The problem can no longer be treated linearly. For the case of one-dimensional liquid impact Heymann (1968) proposed an empirical relation between the impact and the shock velocity, cs = c + k u i , where k is some constant whose value depends on the liquid. The initial pressure rise is then given by Ap = pcui(l 4- kMi). Using this relation which implicitly defines an equation of state, Heymann (1969) calculated the edge pressure of an impacting drop. In this he used an approximated two~dimensional geometry and then adapted the approach of Walsh et al. (1953), who considered the formation of jets in the case of high-speed collisions of two solids from a hydrodynamic viewpoint. The equations obtained by Heymann could only be solved iteratively.

Lesser (1981) also used nonlinear shock relations in conjunction with an explicit state equation to derive a relation between the impact Mach number, the edge Mach number and the nondimensional edge pressure, Pe = p~/pCU~, obtaining:

E 2 M ~ F ( M ~ E ) - ( E - M i ) 2 - M~ z = O,

where the function F(x) = [1 - (1 + 7x)-1/;']/x contains information on the state equation used (e.g., for water ? = 7). Lesser summarized the initial phase of drop impact in an M e - Mi

72 M. Rein Phenomena ~fi liquid drop impact

parameter plane. At shock detachment Me¢ = cos(sin-lMi), hence, the corresponding edge pressure scaled with the waterhammer pressure, Pec, is again a function of Mi only. At impact Mach numbers up to about M~ = 1/2 the edge pressure is close to p~c ~ 3. This theory was extended to oblique impacts by Faddayev et al. (1988) and by Surov and Ageyev (1989). Then the pressure is nonuniform on the circumference of the contact zone. Further, the critical angle for shock detachment depends on the azimuthal angle. In a previous paper Surov and Ageyev (1988) also determined the shock front in the drop via the Huygens principle, thereby calculating the shock velocity from an equation of state. A more detailed description of the first stage of drop impact can be found in papers by Heymann (1969), Field et al. (1979) and Lesser (1981).

As soon as the shock moves up the free surface of the drop a lateral motion of the liquid becomes possible. Typically jets with velocities much greater than the impact velocity are observed. Bowden and Brunton (1961) noticed that the mechanism leading to the formation of high-speed jets is of the same kind as the one producing powerful jets in the case of shaped explosive charges (Birkhoff et al., 1948). This was corroborated by Bowden (1966), who examined experimentally the formation of such jets in different geometries. In the case of high-speed impacts the jet velocity exceeds the impact speed by about an order of magnitude. At low-speed impacts when the impact velocity is of the order of the terminal velocity, the jet velocity observed initially is about twice the impact velocity (Stow and Hadfield, 1981; Loehr, 1990).

It is well-known that the critical jetting angle //~ = sin- 1 Mi obtained theoretically, is much smaller than observed in experiments. A review on this topic is given in Lesser and Field (1983). Since the review was written these authors have advanced a new model leading to a delay in jetting commencement. This and several other effects that are also responsible for a delay of jetting are briefly discussed. A well-defined delay is due to compliant walls. This was explicitly included in the calculations by Lesser (1981). This delay, however, is not large enough to explain the discrepancy between experiment and theory. A further source for the delay are aerodynamic forces rendering the drop oblate. With oblate drops the critical angle will be reached at greater contact radii. Perel'man (1975) estimated the effect of drop oblateness on jetting commencement and reported that these estimates compare well with experimental results (actual- ly, jets impacted laterally were considered). Field et al. (1985) proposed a somewhat different picture of jetting commencement. As soon as the shock detaches from the edge point the liquid behind the shock is accelerated normally to the surface of the drop. Hence, the velocity of the ejected liquid has components both, perpendicular and tangential to the wall. The former component increases the effective impact velocity and is thus responsible for a delay in jetting. Based on this picture Field et al. (1989) derived a new critical angle for jetting commencement which agrees well with experimental results. Finally, it should be mentioned that the jetting motion may not be observable in its initial stage. This can either be due to the jet velocity being less than the contact edge velocity (cf., Lesser and Field, 1983), or to problems in resolving the initial phase experimentally. Following the incompressible theory of Birkhoff et al. (1948) the jet velocity increases strongly with decreasing jetting angle. At the same time, however, the mass of the jet tends to zero.

3.1.2. Spreading After a contact is formed between a drop and a solid surface, the liquid normally starts

spreading out. Different scenarios are possible. In the limiting case when the drop is carefully placed onto the surface, i.e., when the kinetic energy of the drop is extremely small, the process of spreading is dominated by intermolecular forces. The dependence on time of both, the radius of the wetted spot and the contact angle, can be described by universal scaling laws (de Gennes, 1985). Usually a precursor film, also called primary film, having a thickness of a few tens to several

M. Rein / Phenomena of'liquid drop impact 73

thousand ~,ngstr6ms, precedes the spreading liquid (Cazabat, 1987). In models for describing the advancing contact line the assumption of a no-slip boundary condition leads to the appearance of infinite forces (Dussan, 1979). The presence of the thin primary film removes this singularity at the macroscopic contact line. In theoretical work the singularity can be circumvented by allowing for slip near the contact line.

If the drop strikes the surface with a finite velocity, spreading is greatly influenced by the kinetic energy of the drop. Further, its surface energy is important. Spreading starts at the moment of jetting commencement. A smooth motion of the liquid that results in the formation of a thin liquid disc is characteristic of low-speed impacts. The expanding liquid disk is also called a lamella. In a more limited sense the notion "spreading" is often used for this type of fluid flow. This is also the case hereafter. At higher impact velocities the jetting motion leads to a disintegra- tion of the liquid and splashing occurs. The transition of spreading to splashing is the topic of the section on the splashing threshold.

Spreading- in the sense of the last paragraph proceeds as follows. First, the lamella expands very quickly and reaches a maximum radius within a short time. The kinetic and surface energy of the drop are dissipated by viscous processes in the thin sheet of liquid, and are transformed to additional surface energy. During a second stage the lamella shrinks to a small size. Ford and Furmidge (1967) distinguished a second expansion stage that is again followed by a contraction. In a few cases they observed even further spreading phases. Eventually the drop assumes its equilibrium shape on the surface. In some cases the recoil of the lamella may cause the drop to separate from the surface. This picture, proposed by Ford and Furmidge (1967), is a possible mechanism for drops to bounce off after impact. An oscillation in the size of the contact area between the liquid and the wall was also present in the experiments by Gillespie and Rideal (1955). At a given impact velocity they observed some drops to rebound while some did not.

Initially the spreading motion of the liquid is supercritical, i.e., the flow velocity is faster than the wave speed of capillary waves. This can well be seen on photographs by Loehr (1990) that show a Mach cone developing behind a small disturbance. (Similarly, such Mach cones can usually be observed behind small disturbances when a water jet impacts a kitchen sink.) After the lamella has reached its maximum radius Loehr observed capillary waves running from the rim of the lamella back to the center. During this time the radius of the lamella stays approximately constant for a short while. The transition from a supercritical to a subcritical flow probably starts at the center where the supply of liquid decreases. This information propagates radially, along characteristics, toward the rim. The velocity is further reduced by viscous dissipation. Calcu- lations of Bechtel et al. (1981) also yielded a contraction phase in the inviscid case, thus showing that viscosity is not necessary for slowing down the spreading liquid. In contrast to this, numerical computations of Harlow and Shannon (1967a) where both viscosity and surface tension were neglected, show that the lamella does not stop spreading. As soon as the velocity of the rim drops below the wave speed of capillary waves, disturbances can propagate inward. By this time the spreading rate is already very small and, finally, the stage of contraction begins. This stage lasts for a time that can be about two orders of magnitude longer than the duration of the expansion phase (Loehr, 1990). The contraction stage depends very much on the properties of the surface, such as its cleanliness and roughness. Chandra and Avedisian (1991) mention that the Weber number needs to be large for the drop to contract. They do not, however, report a critical Weber number.

The thickness of the liquid lamella is not constant. This was already observed by Worthington (1876). Chandra and Avedisian (1991) suggest that the variation in film thickness may be caused by hydraulic jumps. The rim of the lamella appears to be thicker than the rest of the liquid sheet. This renders an observation of the film thickness difficult.

The radius of the liquid tamella scaled with the drop radius is called the spread factor/~. The maximum spread factor, /~m, that is reached at the end of the first expansion phase is a quantity

74 M. Rein Phenomena of liquid drop impact

often considered in experimental work. Further, the spreading rate, i.e., the velocity of the rim of the lamella, and the initial velocity of the expanding lamella are sometimes reported. In a few cases the time to reach the maximum spread radius, and the thickness of the liquid disk have been briefly discussed. Besides the radius, the density and the impact velocity of the drop, these quantities will also depend on the viscosity,/u, of the liquid and on the interracial tensions between both liquid and gas alg, and liquid and surface ais. Effects due to gravity are negligible since the height of the lamella is very small. Normally the Weber number We = pu~ D/rrlg and the Reynolds number Re = pui D/la are introduced, using the diameter of the drop, D = 2R, as the characteristic length. The Weber number is a measure for the ratio of the kinetic to the surface energy of the drop. When the tension between the liquid and the solid surface is taken into account a third dimensionless group needs to be formed to complete the set of dimensionless groups. The ratio of the two interracial tensions is an obvious choice.

An estimate of the spread factor fl as a function of time can be obtained from an energy balance. This was performed by several authors the earliest being Engel (1955). The sum of the kinetic, potential and surface energy, Ek, Ep and Es, of the impacting drop is equated with the sum of the kinetic, potential, surface and dissipated energy, E[,, E'p, E~ and E~, of the liquid lamella:

Ek + Ep + E, = E;, + E'p + E~ + E~.

In order to obtain an expression for the kinetic energy of the expanding lamella assumptions need to be introduced concerning the velocity profile in the liquid sheet and the height of this sheet. The uncertainties involved in such assumptions are limited to the dissipation if one is only interested in the maximum spread factor tim, since the contribution of the kinetic energy of the lamella then vanishes. The contribution of the potential energy is very small and is usually neglected. An expression for the surface energy E~ of the lamella that includes the contact angle 0 between liquid and surface, can be found in Ford and Furmidge (1967). Except for the dissipated energy E~ all energy terms are then well defined.

The dissipation can be calculated from the energy balance when tim is known. Stow and Hadfield (1981) give an expression for the surface energy of the liquid lamella (at its maximum radius) scaled with the impact energy. This expression shows that the dissipated energy is large as long as We is not too small. Ford and Furmidge (1967) plotted the ratio E'a/E'~ of the dissipated to the surface energy versus the dimensional impact energy. They considered surfaces of different materials yielding different contact angles, and found a strong dependence of the ratio E'd/E's on the contact angle, i.e., on the ratio aig/al~. However, they also report that the impact energy is the more important parameter for tim. Numerical calculations of Trapaga and Szekely (1991) show that most of the energy is dissipated in the early stages of spreading.

In order to calculate tim Chandra and Avedisian (1991) approximated the dissipated energy in a straightforward manner. The energy balance then yields:

2 We ----fl4m + (1 -- COS0)fl 2 -- (~We + 4) ~ 0. 3 Re

This correlation that is compared with their own experiments, slightly overestimates tim. When Re is large, tim is approximately given by fl2m = (½We + 4)/(1 -- cos 0). Loehr (1990) also fitted his experimental data to a correlation of this form, the coefficients on the r.h.s., however, being different.

Numerical results of Trapaga and Szekely (1991) suggest that tim ~ Re1/5 when the Weber number is kept constant (at We = 1290). This is similar to the correlation of Chandra and Avedisian (1991), but it again overestimates experimental results, e.g., those of Loehr (1990), and of Stow and Hadfield (1981).

M. Rein / Phenomena of liquid drop impact 75

Trapaga and Szekely (1991) also correlated the non-dimensional time rm = Zm/t~ = tm 'u i /R required to reach t im with the Reynolds number. Since their numerical results do not reproduce the contraction phase /3m is approached asymptotically. They therefore actually consider the

4 ~1/5 time ('~'m)90 that is needed to reach 90% of /3m. They obtain (rm)90 = ~R~ . This relation is at variance with experimental results of Shi and Chen (1983), who found that 17 m decreases with increasing impact velocity when the drop diameter is kept constant. On the other hand, Stow and Hadfield (1981) measured ~'m ~ 6 when Re = 3500 and We = 100, which is not too far from the corresponding (rm)9o = 6.8. In addition to being a function of Re, "~m will also depend on We.

Measurements of the dependence on time of the spread factor /~ have been reported by several authors. The data were usually obtained by optical methods. In one case an electrical resistance probe was used. An insulating gap in a conducting surface is bridged by the impacting drop, the current through the bridge depends on the spread radius (Shi and Chen, 1983).

In the experiments of Loehr (1990) We ranged from about 200 and 1500 and Re was varied between 4200 and 19700. All of his results are well represented by the empirical formula /3(t) = 1 - exp( - t/T) where t is the dimensional time and T = 7.77 x 10 -2 s is a damping parameter that is constant in Loehr's experiments. Loehr used water and changed the viscosity by adding glycerol. Hence T will not be a function of the viscosity. It is interesting that in all cases T ~ D/ui. On average D/ui = 0.83 x 10 -3 s. This points to a nondimensional form of Loehr's correlation in the form

fl(r) = 1 - exp( - c~),

where r = tu~/R is the normalized time and c is a nondimensional parameter that may still depend on, e.g., the surface tension.

Chandra and Avedisian (1991) compared their results for /3(t) with numerical (Harlow and Shannon, 1967a) and analytical (Savic and Boult, 1955) calculations. In the early period of impact the agreement is good, although in both calculations viscosity and surface tension are neglected. Chandra and Avedisian (1991) provide an estimate of the ratio • of the stagnation pressure force that drives the radial flow, to the viscous force: • ~ Re//~ 5. Similarly they derived an expression for the ratio A between the stagnation pressure force and the surface tension force: A ~ We//32. In this estimate the stagnation pressure is approximated by pu2/2 which is rather conservative since the initial spreading motion is initiated by the waterhammer pressure. Hence, as long as/3 is small the spreading process is independent of viscosity and surface tension effects.

Both, Bechtel et al. (1981), and Shi and Chen (1983) obtained/3(t) by variational methods. The former authors compared their theoretical results with experiments by Wachters and Westerling (1966) although those were performed for drops assuming the spheroidal state after impact (Leidenfrost effect). Not surprisingly the comparison is not satisfactory. The theory of Shi and Chen (1983) agrees well with their own data after a free parameter was fitted, so that a good correlation with other data by Toda (1972) was obtained.

Stow and Hadfield (1981) examined the influence on spreading of the surface roughness. The roughness was varied by two orders of magnitude, starting with a buffed surface. They found that there is no discernible difference in the time dependence of the spread factor. Further, the initial velocity of the lamella appears to be independent of the surface roughness.

The initial velocity Vo of the lamella is often approximated by extrapolating the curve of /~ versus time to r = 0 although jetting commences only at times greater than r = 0. Typical values reported in the literature are Vo/Ui = 2 . . . . . 10. Stow and Hadfield (1981) pointed out that a detailed study of the initial development of the liquid sheet is necessary. It is difficult, however,to observe this stage because it is hidden under the impacting drop.

76 M. Rein Phenomena ql'liquid drop impact

Stebnovskii 11979) resolved the first stage (r _< 8/3 x 10 3) of the development of the wetted spot by high-speed photography. The rate at which the spot increases its radius starts at 1500ul and then quickly drops by one order of magnitude. Stebnovskii suggested that the quick wetting is caused by a very thin liquid film. However, it may well be that at this early time the wetted spot consists mainly of the contact zone between the impacting drop and the wall. The rate of increase of this spot then equals the contact edge velocity ue that is extremly high right after impact.

In the case of oblique impacts the spreading behaviour depends on the azimuthal angle. Stow and Hadfield (1981) examined a typical case and found that initially the difference in the spread factor measured both, down the slope and perpendicular to the slope, is approximately 4%. Later, for r > 3, the lamella becomes elongated down the slope and the difference between the spread factors in the two directions becomes more pronounced.

The apex height h,(cf., fig. 4) of the impacting drop is often considered in connection with spreading. Typically the apex approaches the wall with the impact velocity as long as the time after impact is less than r = 1 (Stow and Hadfield, 1981; Chandra and Avedisian, 1991). Only after this time the apex slows down. These experimental results are in good agreement with analytical calculations by Savic and Boult (1955). It is interesting to note that the shock wave propagating within the drop does not noticeably change the shape and velocity of the apex when it is reflected at the upper surface of the drop.

3.1.3. Instabilio' of the rim of the liquid lamella In the case of low-speed drop impacts onto solid surfaces disturbances can normally be

observed on the rim of the expanding lamella. These disturbances that appear to be of a regular form, are present right after spreading commencement. They have been called rays, radial arms, and fingers. A typical and well-known pattern that was obtained by letting an ink drop fall onto a sheet of paper, is shown in fig. 6a.

The development with time of these appearances was first investigated by Worthington (1876, 1877) by means of short time exposure using electric sparks. He distinguished between rays and arms. The former are formed first. At a later stage of spreading the rays are overtaken by the expanding liquid. A sketch of this stage, as observed by Worthington (1877), is reproduced in fig. 6b. After this stage Worthington reports that arms are formed by merger of two rays thus reducing the number of disturbances. Although the instability of the rim of the expanding lamella is well-known it did not receive much attention after Worthington. A chemically mapped water blot with radial arms is shown in a paper by Engel (1955). However, one hundred years passed before a first quantitative explanation was given in terms of that instability which is normally referred to when an interface between different fluids is accelerated toward the denser one, namely, the Rayleigh-Taylor instability.

Normally all modes are unstable in the case of the Rayleigh-Taylor instability. Surface tension, however, renders modes corresponding to short wave lengths stable. Moreover, there exists a mode of maximum instability when surface tension is present. Allen (1975) suggested that it is this mode that is observed in the case of the spreading lamella. Following Allen, the acceleration of the gas-liquid interface toward the liquid is given by the retardation of the expanding lamella. Using gross approximations for the retardation and other quantities such as the impact velocity, he obtained an estimate of the number of disturbances that is of the same order of magnitude as observed on the circumference of a typical final blot.

Recently, Loehr (1990) investigated the dependence on the Reynolds number of the instability by image processing photographs of different drop impacts. One of his photographs is displayed in fig. 6c. He determined the spectrum of the disturbances by spatial Fourier analysis and found that there exists a distinct maximum in the spectrum. This maximum corresponds to the number

M. Rein / Phenomena ~/'liquid drop impact

(a)

77

Fig. 6. Instability of the rim of the spreading lamella after the impact of a drop on a solid surface: (a) blob produced by an ink drop that fell onto a sheet of paper; (b) rays at the circumference of a spreading lamella (from Worthington, 1877); (c) spreading lamella at t = 0.9 ms after the impact of a drop of bidistilled water onto a plane glass plate (R = 1.8 mm,

U~ = 5.3 m/s). View is from below. (from Loehr, 1990)

N of waves on the circumference of the lamella. Loehr represented his results by the following correlation between N and the Reynolds number:

N = 3 x 10 - 3 Re + 24.

This is in agreement with Worthington (1876, 1877), who observed that N increases with the radius and the height from which the drops fall. Loehr and Lasek (1990), who did not find ray-like patterns but always obtained wavy structures on the rim, suggested that the observation of fingers may be due to an illumination in a stroboscopic manner.

The mechanism that produces the instability of the rim is not clear. Loehr (1990) emphasized that the number of waves is already fixed right in the beginning of spreading. In this context he critically discussed the approach of Allen (1975). It appears that several of the approximations involved are too rough. Further, it is not clear whether the Rayleigh Taylor instability applies to this axisymmetric geometry (cf., Birkhoff, 1954; Plesset and Mitchell, 1956). Loehr (1990) also briefly discussed several other mechanisms based on vortex dynamics such as the G6rtler instability. These ideas, however, appear not to be applicable.

78 M. Rein / Phenomena ~[ liquid drop impact

In the final stage of spreading Worthington (1876) observed the radial arms to break up into droplets. This is the first stage in the transition from spreading to splashing. At lower impact velocities no secondary droplets are formed (Walzel, 1980).

3.1.4. Splashing and splashing threshoht A drop is said to splash whenever it disintegrates into two or more secondary droplets

after colliding with a solid surface. This definition includes the separation of tiny droplets from the wave crests of the unstable rim of the lamella in the last stages of spreading, as well as vigorous splashing events at high-speed impacts. In the latter case the lamella is often observed to lift off the surface before secondary droplets break apart. Similar to drop impacts into liquid layers a crown like structure is formed and tiny droplets are ejected from the rim of the crown. The presence of splashing then is obvious. A sequence of high-speed photo- graphs of a splash showing this behaviour can be found in the book by Edgerton and Killian (1954).

The splashing threshold is determined by the conditions at which the transition from spreading to splashing, and vice versa, takes place. It was already mentioned that high impact velocities yield splashing whereas spreading is obtained at low-speed impacts. This points to the import- ance of the kinetic energy of the drop. Further, the surface tension plays an important role. Scaling the kinetic energy by the surface energy yields an expression that eqttals the impact Weber number, We = pu2D/a, except for a constant factor. A critical Weber number, Wee, is often reported where splashing begins. However, the threshold also depends on other properties of the drop not present in the Weber number, and on properties of the surface. Hence, a critical Weber number alone does not supply a splashing threshold. This explains the widely differing critical values Wec reported in the literature.

Stow and Hadfield (1981) examined the splashing threshold of water drops. The impact velocity and the radius of the drops were changed. They found that drops spread without splashing as long as

R u 1"69 < Sc,

where Sc is a dimensional function of the propefiies of the liquid and the surface. They rewrote this relation as Re We 2 < ~ and suggested that this relation may be appropriate for any liquid,

being a function only of the surface roughness. However, a correlation of their data by Ru( ~ Wec < S'c also appears possible. S'c, which is non-dimensional, still depends on the properties of the surface and the liquid. In a similar manner Baker et al. (1988) correlated data of various sources, describing different impact scenarios which include collisions with liquid sur- faces. This latter correlation should be considered with care since there will be great differences, e.g., in the splashing threshold on dry and liquid surfaces, respectively.

The influence of the surface roughness on splashing was already mentioned in section 2. The results of Stow and Hadfield (1981) reveal a strong dependence of the threshold on the roughness at very small values of the surface roughness parameter only. At greater values of this parameter, as long as the surface is smooth, its influence on the splashing threshold is small.

Schmidt and Knauss (1976) suggested including effects due to the viscosity of the liquid. This was later accomplished by Walzel (1980). From experimental data he derived the following correlation between the critical Weber number and the Ohnesorge number Oh = l~/(Dap)l/2:

Wec = 7.9 × 101°Oh 2'8.

Walzel states that a similar relation also holds when drops impact with shallow liquid layers. The correlation by Walzel shows that an increase of the viscosity should delay splashing. This is in agreement with an observation by Geguzin (1978), who considered the impact with a thin liquid

M. Rein / Phenomena t~[' liquid drop impact 79

film. But it contradicts the results of Loehr (1990) concerning the impact with solid surfaces. Further, a comparison with experimental data of Stow and Hadfield (1981) shows that Walzel's correlation predicts a critical Weber number that is one order of magnitude too large.

Splashing may be enhanced when a drop collides obliquely with a surface. On impact the drop experiences a sudden force tangential to the surface. The resulting deformations have a destabiliz- ing effect. This mechanism is also present when a drop impacts normally on a moving wall. The boundary layer at the wall then increases the deforming forces. This was investigated by Yao and Cai (1988) for the case of drops impacting a rotating disk. They obtained a noticeable reduction in the critical Weber number for splashing as the rotating speed was increased. In their experiments the disk was above the critical temperature at which the Leidenfrost phenomenon can be observed.

A splash results in the formation of many secondary droplets. The dependence on different parameters of the number and size distribution of these splash droplets was investigated by Stow and Stainer (1977). Besides the radius, surface tension and impact velocity of the drop, the surface roughness and curvature were varied. In practically all cases the radius of the splash droplets could be approximated by a log-normal distribution. This was also reported by Levin and Hobbs (1971). The size distributions were determined by arresting the splash droplets on a so-called collecting plane parallel to and at a small distance below the target surface. A collecting plane perpendicular to the impacted surface was used by Allen (1988) in order to determine the direction of splash droplets. Stainer and Stow (1976) described and compared different methods for detecting droplets on collecting planes. A common feature of these methods is that the droplets are arrested on the plane. Hence, their spreading behaviour becomes important and the calibration may be difficult. The calibration can be avoided by collecting the droplets in an immersion liquid that is immiscible with the drop liquid. In the immersion method droplets penetrate the surface of the immersion liquid, fall slowly to the bottom of the cell that contains the liquid, and can then be photographed and counted. In order to yield good results the immersion liquid needs to satisfy certain requirements (Hiroyasu and Kadota, 1976). A different method for determining droplet-size distributions that does not require mechanical contact with the droplets, is to analyze high-speed photographs of splash droplets taken before they collide with the surface. A great difficulty in this approach is to determine whether or not the photographic images are in focus. This question is addressed by Fantini et al. (1990), who were concerned with obtaining size distributions of spray droplets by image processing.

A theoretical approach for the determination of a splashing threshold was given by Cohen (1991). It is based on the assumption that there exists a minimum radius Rml, for secondary droplets. (For details such as the definition of Rmin, see the original paper.) The maximum number N m of secondary droplets that can be formed is then given by

Ek + E~(R) N n l - -

E s ( R m i n ) '

where Ek and E~(R) are the kinetic and surface energy of the impacting drop, and Es(Rmi,) is the surface energy of a drop of radius Rmi,. To obtain splashing N m needs to be greater or equal to two. However, the critical Weber number obtained in this manner appears to be too small.

There are other more general theoretical approaches in the literature that aim at obtaining fragment size distributions. Grady (1982) treated the fragmentation of arbitrary bodies of material due to an impulsive process by an energy balance between the kinetic and surface energy. Melosh and Vickery (1991) applied this method to determine the size distribution of microtektites. T h e microtektites were assumed to be melt droplets formed on impact of meteors. Englman (1991) reviewed the maximum-entropy method. This method yields fragment size distributions and may lead to physical conclusions concerning, e.g., the mechanism of fragmentation.

80 M. Rein / Phenomena ~)f liquid drop impact

Here the mechanism causing splashing will be some kind of instability of the spreading lamella. In the case of droplet streams impacting a thin liquid film that is formed by preceding drops, an experimental investigation by Weiss (1993) has shown that an instability of capillary waves on the spreading lamella is of great importance. But other mechanisms may also contribute to splashing. In the first stage of impact the compressibility of the liquid cannot be neglected. Although it has never been considered in connection with splashing it appears likely that the compressibility also plays a role. Another dimensionless group, e.g., the Mach number, then needs to be considered in determining the splashing threshold. Gonor and Yakovlev (1977) suggested that the occurrence of cavitation within the drop initiates splashing. Further, effects such as the solidification of a liquid on cold walls can enhance splashing (Savic and Boult, 1955).

3.1.5. Cavitation Already Engel (1955) observed that cavitation bubbles can be present in a drop right after its

collision with a solid surface. This was investigated in more detail by Brunton and Camus (1970). In order to obtain undistorted photographs of the events occurring inside the liquid they placed a disc-shaped drop made by a mixture of gelatine and water, between two parallel transparent plates. A third plate was projected against the drop. They found that there exist two different zones where bubble fields are formed. One lies just above the contact area between the drop and the wall, and the other is under the apex of the drop. A single bubble that originates at the point of impact was observed by Chandra and Avedisian (1991). In a sequence of photographs this bubble can be seen to rise into the drop.

The occurrence of bubbles inside an impacted drop can be due to air entrainment or to cavitation. In the last stage of the drop's approach to the surface its base can be deformed so that a bubble can be entrapped at impact. This mechanism may be present in the case described by Chandra and Avedisian (1991). These authors report, however, that it was not possible to decide whether air entrainment or cavitation caused the bubble in their experiments. Another mechan- ism leading to the entrainment of a gas bubble is due to the spreading and retracting motion of the lamella. Elliot and Ford (1972) observed that the central area of the lamella may rupture and form a drained area in the last stage of spreading. In the succeeding stage of contraction the receding liquid may close over the drained area trapping a gas bubble. When bubbles are formed within the volume of liquid it appears likely that cavitation is present. Here, in the case of drop impact, there are two mechanisms leading to cavitation. These were already proposed by Engel (1955). In both cases the superposition of pressure waves causes tensions. Since ordinary liquids like water are usually contaminated by nuclei this leads to a sudden rupture of the liquid. The dynamics of this type of cavitation inception were investigated by Rein and Meier (1990a, b). Both mechanisms appear to be present in the case of Brunton and Camus (1970) thus leading to the formation of two bubble fields.

The first mechanism is responsible for the cavitation bubble field above the interface between the wall and the liquid. As soon as the shock that is formed in the drop after impact, lifts off the contact edge an expansion wave propagates into the drop. The amplitude of this wave is determined by the excess pressure at the contact edge. It was mentioned earlier that the pressure is nonuniform in the high pressure region, the highest pressure prevailing at the contact edge. Therefore, tensions will eventually be created when the expansion wave propagates inward. This is enhanced by the cylindrical symmetry of the geometry that leads to an increase in the amplitude of the expansion wave when it approaches the center.

The formation of the other bubble field below the apex is similar to the formation of cavitation zones in the case of underwater explosions (Kedrinskii, 1976). When a shock pulse is reflected at an interface of low impedance a tension wave is formed. The curvature of the surface causes a focussing of the reflected tension wave. Again, a bubble field is formed.

M. Rein / Phenomena of liquid drop impact 81

Transient cavitation is well-known to cause severe erosion. When a cavitation bubble collapses near a wall a high-speed jet is formed penetrating the cavity in the direction toward the wall (Plesset and Prosperetti, 1977). Further, shock waves can be emitted by cavity collapse. Hansson and Morch (1980) considered the collapse of bubble fields on the whole. They found that the collapse proceeds from the outer boundary of the field toward its center. This 'concerted' collapse yields very high pressures at the center and can have a strong damaging effect. Hence, besides high impact pressures and shear stresses occurring during jetting, cavitation inside the drop may contribute to erosion due to drop impact.

3.2. Impact q[" drops into liquid layers

3.2.1. Regimes oJ bounc&g, coalescence, and splashing The collision of a drop with a liquid surface may result in bouncing coalescence of splashing

(cf., fig. 7). Sometimes, after impact, a drop floats on the surface for several seconds and then disappears. This was already observed by Reynolds (1881), who pointed out that the cleanliness of the liquid surfaces is most important for producing floating drops. The result of a collision that does not lead to floating or bouncing can be classified as coalescence or splash, respectively. In the former case the drop disappears quickly in the target liquid. A small carter is formed but otherwise the impacted surface is hardly disturbed. Practically no secondary droplets are produced. This event is usually connected with the formation of a vortex ring below the surface. In the case of splashing the liquid surface is greatly disturbed. The formation of a liquid column that rises out of the center of the crater formed after impact, above the original surface of the target liquid is characteristic of splashing. The appearance of this central jet was used as a criterion for splashing by Rodriguez and Mesler (1985). In contrast to splashing on solid surfaces this definition does not necessarily require the formation of secondary droplets. In the splashing case the target surface is greatly deformed. A hemispherical cavity with a radius that can be an order of magnitude greater than the drop radius, is formed. At the circumference of the

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. . . . . . 0 _ . . . . . . © _;-_~_-_;--_--_~_-_-_:~_-_-_= ~ _ - _ - _ - - - ; -_-..-_-~

f l o a t i n g b o u n c i n g

o o

o o

o o

c o a l e s c e n c e -~-==-- splashing

i e

- _ 2 = 2 2 2 2 2 2 2 2 2 2 2 2 - . . . . . . . . . . . . . . . . - - - - - _ - - - _ - _ - _ - _ ~ _ - - _ _

vo r tex . . . . . . . . . . . . . . . r ng

Fig. 7. Impact of a drop on a liquid surface: floating, bouncing, coalescence, and splashing.

82 M. Rein Phenomena fff liquid drop impact

cavity a liquid sheet, the so-called crown, rises above the original level of the target liquid. This sheet of liquid may close above the cavity forming a bubble. When the cavity collapses, the central jet rises out of the center of the cavity. Usually, secondary droplets are produced in splash. These are shed from the rim of the crown and from the tip of the jet. Splashing is sometimes further divided into crown formation, jet formation, and closure of the bubble. In the following the conditions under which the different phenomena occur are discussed.

When drops rebound from a liquid surface a contact can temporarily be formed between the liquids. The reflected drop may then be smaller than the incident one. This is also called partial coalescence. The bouncing behaviour of single drops was investigated by Ching et al. (1984). In the parameter range examined they never observed single drops to rebound. Bouncing was only obtained with droplet streams. They suggest that disturbances on the liquid surface produced by preceding drops are important for bouncing to occur, and that with single drops most of the impact energy is lost in forming a crater; the remaining energy is not sufficient for rebound. In contrast to this Rodriguez and Mesler (1985) observed single drops to bounce, provided their radius, impact velocity, and, hence, the Weber number, are small enough. Rayleigh (1882), who was concerned with the scattering of nearly vertical jets due to the rebound of colliding drops found that a small contamination of the liquid with soap or milk will cause drops to coalesce on collision.

Bouncing was investigated more closely with droplet streams. Under certain circumstances streams colliding obliquely with liquid surfaces rebound. When the conditions are slightly changed a transition to coalescence occurs. This transition usually begins with partial coales- cence. A parameter important for the transition is the impact angle. When only this angle is varied Jayaratne and Mason (1964) found that alternating zones exist in which drops bounce off or coalesce with the target liquid. These zones were only observed at small drop radii (R < 0.12 × l0 3 m). With larger drops they found only one transition from bouncing to coales- cence with increasing impact angle. The impact velocity also plays a role. When it is increased the zone of coalescence becomes smaller and eventually disappears. Zhbankova and Kolpakov (1990), who performed similar experiments, emphasized that there are no sharp boundaries between bouncing and coalescence, but rather regions where one outcome is more likely than the other. They generalized their experimental data by plotting these regions in the Weber number versus impact angle plane sketched in fig. 8. At higher Weber numbers, indicated by the upper solid line in fig. 8, another transition from bouncing to coalescence takes place, for splashing is eventually reached via coalescence (but not from bouncing) when the Weber number is even more increased.

In order to obtain coalescence the layer of gas between the drop and the surface needs to be ruptured. It can therefore be expected that the state of the gas is of some importance for the threshold of coalescence. This was examined by Schlotland (1960), who varied the density of the surrounding gas by about one order of magnitude. By changing the height from which the drop fell he determined the critical height at which coalescence is first obtained. The results show that the critical height increases with the ratio of the density of the gas to that of the liquid. Since Schotland does not report alternating zones of bouncing and coalescence, the corresponding critical Weber number appears to provide the upper limit for bouncing. Further, Schotland found that the transition from bouncing to partial coalescence depends mainly on the normal compon- ent of the impact velocity. Hence, the Weber number formed with this component of the velocity will determine the threshold. The critical Weber number for coalescence, Weco, is quite small (e.g., Weco ~ 3 at some constant but not explicitly mentioned value of the density ratio). Schotland finally suggested that besides the Weber number and the ratio of the densities there may be additional parameters determining the threshold.

Bouncing and coalescence are phenomena that occur only at low Weber numbers. When the dimensionless impact energy increases drops will splash. The conditions at which the transition

M. Rein/Phenomena of liquid drop impact 83

We

alesce~ :+--

s bouncing I

0 ~')'4 ~)2 ~ Fig. 8. Oblique impact drops on liquid surfaces: regions where bouncing and coalescence, respectively, is likely.

from coalescence to splashing takes place were investigated by Rodriguez and Mesler (1985). They considered single water drops falling normally onto a water surface. They found that the boundary between coalescence and splashing is not sharp but slightly uncertain. An increase in either the impact velocity or the radius of the drop leads to the splashing transition. Within the uncertainties involved, the threshold did not depend on the oscillation phase on impact when the drops oscillate. Zones of splashing and coalescence were displayed in a Froude number versus Reynolds number plot where the Froude number Fr was defined as Fr = ul/(gD) vz (g is the acceleration of gravity). Single splashing limits of other authors that were included in this plot agree with these results. In a preceding paper Carroll and Mesler (1981a) pointed out that numerical computations by Harlow and Shannon (1967a, b) yielded splashing within the zone of coalescence. Carroll and Mesler (1981a) suggested that this contradiction may be due to neglecting the surface tension in the numerical work, and further, that the Weber number could likely enter the criterion for splashing.

Hsiao et al. (1988) reconsidered the splashing limit in terms of the Weber number. They defined the Weber number as the ratio of two time scales: a time scale characteristic of surface tension effects, T 1 = (pD3/a) 1/2, and a convective time scale re = D/ui. Normally We is taken to be the square of this ratio: We = ( r l / T 2 ) 2. Following Hsiao et al. (1988), if T 1 <~ T2, the convective time scale is so large that the drop can easily be deformed into a vortex ring during impact. Conversely, when z2 '~ zl, the time is too short for the surface tension to deform the drop. This results in splashing. These considerations suggest that the splashing limit is determined by a critical Weber number alone, at least as long as the viscosity is low. In order to validate their hypothesis Hsiao et al. (1988) performed experiments with mercury whose surface tension is almost an order of magnitude greater than that of water which was used in other experiments. Evaluating their own experiments, as well as the data given by Rodriguez and Mesler (1985), they found that the critical Weber number for splashing, Weep, is nearly independent of the Froude number (Wesp ~ 60). Above Weep splashing will be obtained. However, the influence on the splashing limit of the viscosity was not discussed.

Thompson and Newall (1885) performed experiments with liquids of different kinematic viscosities. This time the influence of the surface tension was not considered. According to their

84 M. Rein Phemmzena ¢~l liquid drop impact

results splashing is obtained at low, and coalescence at high values of the kinematic viscosity. The drop radius and the impact velocity appear to be approximately constant in these experiments. Hence, the Reynolds number formed with the drop radius and the impact velocity, is also an important parameter for the splashing limit and the critical Weber number will be a function of the Reynolds number.

3.2.2. Coah, scence and cortex rhlgs

It is known since long that a vortex ring can be formed when a drop coalesces with a liquid. The vortex propagates into the liquid. Thereby the vortex ring becomes wider and it slows down. Eventually it will stop or become unstable and break up. The first scientific contribution on this topic can be attributed to Rogers (1858), who observed and discussed the analogy of 'rotating rings' in air and liquids. Reynolds (1875) ascribed a calming action on sea surfaces to vortex rings generated by raindrops. He proposed that vortex rings penetrating into the water cause a momentum transfer between the surface and deep liquid layers which are at rest. A detailed experimental investigation on the formation of vortex rings by drop impact was then carried out by Thompson and Newall (1885). They examined the influence of various liquids on the formation of vortex rings. A vortex was only generated when the drop and the target liquid are miscible. Further, they found that the penetration depth of a vortex ring varies about sinusoidally with the height from which the drop falls onto the liquid. If the fall height is expressed in terms of the time the drop needs to reach the surface, a good correlation between the penetration depth and the eigenfrequency of the drop oscillation is obtained. Several forms of the development of vortex rings as observed by Thompson and Newall (1885) are reproduced in fig. 9.

The influence of drop oscillations on the impact can be used to determine the eigenfrequency of the drop. Kutter (1916) applied this method to determine the surface tension of liquids dynamically. He measured the eigenfrequency of a drop by varying the fall height, so that different maxima and minima in the penetration depth of the vortex rings were obtained. The difference between two succeeding fall heights that yield maximal penetration depths, corresponds to the eigenfrequency. The surface tension can then easily be calculated from the well-known relation for the eigenfrequency of a drop (cf., e.g., Lamb, 1932). Chapman and Chritchlow (1967) studied the phase of the oscillation at the moment of impact more closely. They state that the penetration depth of a vortex ring is greatest when the impact happens while the drop changes its shape from oblate to prolate. Similar observations were also reported by Keedy (1967) (cf., Rodriguez and Mesler, 1985). Later, however, an investigation by high-speed photography by Rodriguez and Mesler (1988) revealed that most penetrating vortex rings are obtained on prolate, and least penetrating ones on oblate impacts. These authors traced this behaviour to a complex interaction between the process of vortex formation and the crater dynamics. An often cited photographic study of the formation of vortex rings by drop impact by Okabe and Inoue (1961) does not exhibit these details. It is interesting, however, to note that the photographs of the latter authors reveal a similarity in large-scale structure between a descending vortex ring and a mushroom cloud as was pointed out by Sigurdson (1991). The development in time and space of several properties of the vortex ring was measured by Hallett and Christensen (1984). They also reproduced the sinusoidal dependence of the penetration depth on the distance the drop falls prior to impact. Further they found that the penetration depth of the vortices decreases With the impact Weber number. In the limit of vanishing impact velocities Anilkumar et al. (1991) derived a power law for the penetration depth L. Assuming that the surface tension energy of the drop is fully transformed into kinetic energy of the vortex, and using an estimate for the dissipation, they obtained L ~ D 5/4. This power law correlates well with their experimental results.

M. Rein / Phenomena of liquid drop impact 85

Fig. 9. Normal impact of a drop into a liquid: coalescence. Different forms of the formation and propagation of a vortex ring. (from Thompson and Newall, 18851.

Coalescing drops do not always succeed in forming fully developed vortex rings. Thompson and Newall (1885) observed that a drop remains close to the surface without changing its shape significantly if the Reynolds number is small. A similar phenomenon was reported by Cai (1989). Cai who was not aware of earlier work on the dependence on the fall height of the penetration depth of vortex rings, found that certain fall heights exist at which the drop submerges, producing practically no vortex ring, but lasts for several seconds below the surface before it diffuses. These fall heights obviously coincide with those yielding least penetrating vortex rings. This becomes also apparent by comparing the photographs of Cai (1989) with those of Rodriguez and Mesler (1988) for the case of oblate drop impacts. Actually, in Cai's fig. lb a vortex ring can be seen before it deforms into a blob of dye. This is much better resolved in high-speed photographs of an oblate drop impact by Rodriguez and Mesler (1988). However, the latter authors also report that the appearance of the vortex ring was sometimes similar to a dyed blob.

In order to form a vortex ring a source of vorticity is needed. However, a satisfactory explanation for the generation of vorticity has not been presented in the past as was pointed out by Morton and Cresswell (1992). In a subsequent paper Cresswell and Morton (1992) proposed a model for vorticity generation based on "free surface stress relaxation", i.e., on the condition of zero tangential stress at free surfaces. On impact, at the junction between the drop and the target liquid the curvature of the free surface is very large. Hence, in the presence of surface tension the surface is accelerated normally to itself and the cusp-like junction is smoothed out. This causes a curvature in the streamlines and a finite rate of strain. However, at the free surface the rate of

86 M. Re in ,' P h e n o m e n a o f l iquid drop impact

strain needs to vanish. Viscous forces accelerate the surface parallel to itself and vorticity is generated. The model is finally shown to be consistent with experimental observations.

3 . 2 . 3 . S p l a s h i n g c h a r a c t e r i s t i c s

The main features of splashing are the formation of a crown and the rise of a liquid column out of the middle of the crater. The column is also called a central jet. The central jet often becomes unstable and drops separate from its tip. Apparently, because of this instability the jet is sometimes called Rayleigh jet. Shin and M c M a h o n (1990) used the term Wor th ing ton jet which seems quite appropriate, since Wor th ing ton (1908) performed the first extensive study of splashes. Fascinating early photographs of different stages of splashing drops can be found in the book of Wor th ing ton (1908).

The main stages of the collision of a drop with a liquid surface leading to a splash are depicted in fig. 10. Right after impact a thin film of target liquid is ejected upward at the periphery of the drop. In the target liquid a cavity is formed. The cavity is initially of a cylindrical shape. The cavity then quickly expands to form a crater of an almost hemispherical shape. The drop deforms and lines the walls of the crater. At the periphery of the crater a sheet of liquid is raised above the original surface and forms the crown. This sheet normally becomes unstable and small droplets are shed from its rim. The crown is mainly composed of target liquid, but usually also contains

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; ~;/%1!%11111111 iii i~ i~ii ;; !i iii i lilj,,,ilbll i ii i!~ i,i; ~ i il ii i~i i~ iii iii i: .... : ?v :::::::iil iilil i iiiiiiii ill i iiii iiii!Vi ''

Fig. 10. The splash of a water drop (colored with ink) that impacts on a deep pool of water (radius R = 1.9 mm, fall height: 0.6 m). Different stages of splashing are shown: ejection of a liquid film, formation of a crown, a crater and a central jet, and separation of a drop from the tip of the jet. The photographs were taken 0.1 ms, 1.9 ms, 11.4 ms, 25.6 ms, 74.0 ms and

113.8 ms after the first contact between drop and target liquid. (from Wilkens, 1987).

M. Rein / Phenomena ~?/" liquid drop impact 87

some of the drop liquid that lines the cavity (Engel, 1966). In the later stages the cavity and the crown start to subside. The crown thickens and the crater begins to collapse. This leads to the formation of the central jet that carries drop liquid at its top. From the tip of the jet drops may pinch off. If the impact energy is large enough the crown rises appreciably, starts to bend inward, and may eventually close above the crater. On closure two jets are formed, one is directed upward and the other downward into the cavity where it interacts with the central jet. In this case the central jet does not penetrate the closed bubble.

In addition to those parameters already mentioned in connection with other impact phe- nomena the events described in the last paragraph can also depend on the acceleration of gravity. This is typically taken into acccount by considering the Froude number Fr = ul/(gD) 1/2. Norm- ally, however, in experiments the impact energy is the important parameter. Therefore, the dependence on the Weber number of the splash will be discussed in the following.

If We is only slightly greater than the critical Weber number for splashing, Wesp, no drops detach from the tip of the central jet. In this case Worthington (1908) observed that a second jet can be formed after the first one has disappeared. This oscillating behaviour of the jet is reminiscent of the alternating phases of spreading and retraction in the case of the spreading lamella after drop impacts on solid surfaces.

As soon as the Weber number exceeds a certain threshold one or more drops pinch off the tip of the central jet. When these drops subsequently reenter the target liquid they bounce or coalesce, respectively, for their impact velocity is small. In the latter case vortex rings are produced that propagate into the bulk liquid. Hence, vortex rings can also be present in the splashing case. The threshold for a drop to detach at the top of the jet was reported to be We ~ 84 by Hallett and Christensen (1984) 1, who performed experiments with water. These authors further state that a crown is formed only if We > 180. This, however, is at variance with Worthington (1908), who obtained a crown at Weber numbers where the central jet is still stable. The crown closes above the crater at high impact energies only. Evaluating the Weber number for different cases where a bubble was formed yields Weber numbers well above We = 1000. A lower limit was given by Hallett and Christensen (1984), who report that they never obtained closure of the crown if We < 370.

An estimate of the maximum radius of the crater was derived from an energy balance by Engel (1966, 1967) and other authors. The kinetic and surface energy of the approaching drop are equated to the potential energy of the crater at its maximum size. At the moment when the crater reaches its maximum extension the kinetic energy of the liquid is assumed to practically vanish, and the shape of the crater is approximated by a hemisphere. Depending on other approxima- tions, e.g., for the shape of the wave swell formed by the crown, different expressions may be obtained. The dissipation is of minor importance (Macklin and Metaxas, 1976) and the sound energy is negligible (Franz, 1959). Prosperetti and O~uz (1993)further neglected the contribution to the potential energy of the wave swell and obtained an expression that approximates the maximum cavity radius well as long as the impact speed is low.

The maximum height to which the central jet rises is a function of the impact energy. In experiments performed with a water drop of constant radius Hallett and Christensen (1984) observed that there is a small range of impact velocities where this height assumes a pronounced maximum. Further, in this range the jet is narrow, whereas a wide jet is formed at higher impact velocities. These authors ascribed this change in the behaviour of the jet to a transition from laminar to turbulent flow into the base of the jet. However, more recent work on bubble entrainment by drop impact shows that there exists a small range of so-called regular entrainment

1 Hallen and Christensen (1984) introduced the ratio L of the kinetic to the surface energy of the drop and note that L = ~We. However, explicitly calculating L yields L = 1½We. This relation was used to transform L into We.

88 M. Rein ~ Phenomena oJ'liquid drop impact

where a drop impact leads to the entrainment of a single bubble at the bottom of the cavity (cf., Prosperetti and O~uz, 1993). A comparison of the two phenomena reveals that the range of impact velocities that yields high narrow jets coincides exactly with that range where a single bubble is sealed off. Since both, the jet and the bubble have their origin at the bottom of the crater, it is not surprising that these two phenomena are closely related. At impact velocities beyond the range of regular bubble entrainment the maximum height reached by the wide central jet increases slowly with the impact velocity. Similarly, the number of splash products was observed to increase with the impact energy (Hobbs and Kezeweeny, 1967).

Related phenomena are present in splashing on shallow liquids. It was first noted by Hobbs and Osheroff (1967) that the depth of a shallow liquid layer has a great influence on the outcome of splashing. There is a certain depth at which the central jet reaches a maximum height. The number of droplets shed from the jet correlates with this height. Later, Macklin and Hobbs (1969) observed that the maximum height reached by the central jet increases as soon as the bottom of the layer begins to affect the shape of the cavity. At depths where the expanding cavity begins to touch the bottom the height of the central jet decreases abruptly. This was investigated in more detail by Shin and McMahon (1990). They found that during the collapse of the crater a small cylindrical stem is formed at its bottom when the depth of the liquid layer is such that the cavity at its maximum size just touches the bottom of the liquid layer. This stem is delayed in retracting upward. Shin and McMahon (1990) pointed out the similarity between the formatiom of this stem and the regular entrainment of a gas bubble beneath the crater in deep liqukts, and proposed that the stem is responsible for the formation of high jets in shallow-liquid splashes. This is strongly corroborated by the findings discussed in the last paragraph that the formation of high central jets is related to regular entrainment m deep-water splashes. - The height to which the central jet rises is very small at depths of the liquid layer below that which yields maximum jet heights. In the limit of splashing on liquid films a central jet is no longer observed and the events become in many respects similar to splashing on solid walls.

3.2.4. Entrainment o f gas bubbles Closely associated with the impact of drops on liquid surfaces is the entrainment of gas bubbles

into the target liquid (Blanchard and Woodcock, 1957). When the impact results in coalescence a large number of small bubbles can be entrained. These are often carried deep into the target liquid by the vortex ring that is formed after impact (Esmailizadeh and Mesler, 1986). A mechan- ism based on surface-tension effects that leads to the entrainment of many bubbles was proposed by O~guz and Prosperetti (1989), who also discussed other possibilities such as a Rayleigh Taylor instability of the approaching liquid surfaces. Under special conditions when very salty water drops fall into stratified salty water, Borgas and Hibberd (1992) observed the entrainment of spherical shells of air that enclose the impacting drop. No vortex is formed but the drops move into the target liquid until they reach an equilibrium position. The air film ruptures eventually and small gas bubbles rise to the surface.

When the impact of the drop results in a splash Pumphrey and Elmore (1990) distinguished between irregular, regular and large bubble entrainment. Under certain conditions a single bubble is sealed off at the bottom of the crater that is formed after impact. This is the case of regular entrainment. It causes the emission of sound that originates from oscillations of the bubble (Pumphrey and Walton, 1988). The process was calculated numerically by O~uz and Prosperetti (1990). Using a boundary-integral method they obtained good agreement with experimental results. In particular, the conditions at which regular entrainment is obtained are reproduced. More details can be found in the comprehensive review by Prosperetti and O~uz (1993). In special cases Pumphrey and Elmore (1990) observed that a large bubble having a volume comparable to that of the crater, is trapped below the surface. A different cause for the

M. Rein ; Phenomena ~J'liquid drop impact 89

formation of large bubbles is the closure of the crown above the cavity. 1here are many conditions under which splashes do not always lead to bubble entrainment. This was termed irregular entrainment by Pumphrey and Elmore (1990). Bubbles may then be entrained by the impact of splash droplets instead of the impact of the primary drop. In particular, the impact of drops that pinch offfrom the tip of the central jet can lead to an entrainment of the same type as is observed in the case of coalescing drops (Carroll and Mesler, 1981b),

4. Concluding remarks

The impact of drops on both, solid and liquid surfaces results in a variety of phenomena. When a drop impacts a solid wall or the surface of a pool the outcome can essentially be classified as bouncing, spreading or coalescence, and as splashing. In the literature some of the phenomena were given close scrutiny while others were only briefly examined. The main purpose of this review was to discuss the different cases and the conditions under which they occur. Usually, there appears to be only little general agreement on the thresholds for the onset of a certain phenomenon. An exception may be the critical Weber number Wesp for splashing of drops falling into a liquid. The difficulty in obtaining reliable thresholds is that the exact conditions of the impact are often not well known. Minor disturbances such as the surface roughness of walls, or the contamination of a liquid surface by surfactants may cause large changes in the critical dimensionless numbers for the onset of a certain phenomenon. Further, the impact of drops is governed by a large number of parameters. Normally only some of these are considered in a single investigation.

Most of the papers on phenomena of drop impact report on experimental work. Except for numerical computations and relations obtained from an energy balance, theoretical work is usually limited to the initial stage of drop impact. The process of drop impact on the whole is an unsteady, free surface flow. In the initial stage of the impact the compressibility of the liquid cannot be neglected. At a later time surface-tension effects and the viscosity become important. The spreading motion of a lamella on a solid wall is a boundary layer flow with a free surface. Some yet unknown instability of this flow appears to cause the transition from spreading to splashing. The process of fragmentation of a splashing drop is a basic problem that has also not been solved. When the collision of a drop with a liquid surface results in splashing gravitation enters the process as an external force. In the splashing case with its many splash droplets, and, in addition, when cavitation occurs inside the drop, a two phase flow is present. Finally, the mechanisms that cause the transition between different results of an impact are not well under- stood. Thus, many questions both, from an experimental and theoretical viewpoint, are still open.

Acknowledgements

The author would like to thank Dr. D. Auerbach, Prof. W. Fiszdon, Dr. F. Ohle and Dipl. Phys. D. Weiss for their valuable comments on a preliminary version of this review.

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