PB 161590

^ecUnlcal ^iot& 89

COLLISIONS OF LIQUID DROPS

WITH LIQUIDS

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
THE NATIONAL BUREAU OF STANDARDS

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NATIONAL BUREAU OF STANDARDS

technical ^lote

89

MAY 1961

COLLISJONS OF LiQUID DROPS WITH LIQUIDS

Olive G. Engel

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

listed or referred to in the open literature. They are for sale by the Office of Technical Services, U. S. Department of Commerce, Washington 25, D. C.

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CONTENTS

Page

1. Introduction — 2

1.1 Previous investigations 2

1.2 Description of the collision event 3

a. Major stages 3

b. Effect of velocity 6

c. Effect of properties of the liquids used 7

d. Effect of proximity of container walls 7

2. Correlation with hypervelocity-pellet and meteorite craters- 8

3. Quantitative data on liquid-drop-into-liquid collisions 12

3.1 Cavity depth 13

3.2 Ratio of cavity depth to cavity diameter 19

3.3 Effect of the energy of the impinging drop 19

4. Current research 25

5. Bibliography 30
COLLISIONS OF LIQUID DROPS WITH LIQUIDS*

by

Olive G. Engel

ABSTRACT

The information available on collisions of liquid drops

with liquids is reviewed. Measurements of the depth and

diameter of the cavity produced in the target liquid made on

G. J. Franz's high-speed motion pictures of collisions of

waterdrops with water are reported. The measurement data are considered to be of a preliminary nature. The most important

provisional observations drawn from them are: the cavity

depth-versus-time curve is a parabola or ellipse tangent

to the depth axis at the origin; the depth/diameter ratio

of the cavity varies with time elapsed since the collision

and has a value of 0,5 at only two points in time. An

equation for maximum cavity depth is derived on the basis

of the three assumptions that the impinging drop is spherical,

that the cavity is hemispherical, and that the volume of the

cavity is proportional to the kinetic energy of the drop.

Test of the equation with the measurement data indicates

that the assumptions are reasonably good in the low free-fall

velocity range. Apparatus that is now being assembled at

the National Bureau of Standards to obtain more accurate data

and to carry out an exhaustive study of this type of

collision is described.

"This NBS Technical Note is based on a report entitled

"Collisions of Liquid Drops with Liquids, Part I. A Review

with Some Preliminary Data", dated June I960, covering work

performed as part of a project sponsored by the Nonmetallic

Materials Laboratory, Materials Central, Wright Air Development

Division, Air Research and Development Command, United States

Air Force. Since this technical report was submitted,

A. C. Charters of the National Aeronautics and Space Administration has published in the October I960 issue of Scientific

American a popularized discussion of some of the fluid-impact

phenomena that have been examined quantitatively herein.
1. INTRODUCTION

Collisions between liquid drops and liquids have become important

in the present era of space flight. This is not only because of the

possibility of collisions between space craft and meteors but also

because of the possibility of collisions between themelted leading

surface of a re-entering space vehicle and waterdrops in the atmosphere.

Meteors travel at velocities of from 12 to 72 km/sec (39,000 to

24.0,000 ft/sec). On the basis of the argument that the aerodynamic

(l/2pV^) pressure at the penetration of a meteor into rock is more than

1,000 times the plastic limit of steel, Opik [l]£/ has suggested that

meteor impact, at least in its initial stages, is the case of the impact

of a liquid drop of density 6. into a liquid medium of density £ . It

appears, however, that concomitant vaporization, and possibly explosion,

of the projectile and eventually of the target may complicate the simple

liquid-drop-into-liquid picture [2, 3].

With regard to collisions between waterdrops in the atmosphere and

the melted leading surface of a space vehicle, it has been recommended . [4-] that hypervelocity collisions between waterdrops and melting aerodynamic surfaces should be studied. It is very difficult to study highspeed collisions between a liquid projectile and a liquid target because

of the problems involved in accelerating liquids. It appears, however,

that a study of free-fall-velocity collisions of this kind may produce

information of value for the high-velocity case because the Reynolds

Number at free-fall velocity is high enough that there should be both

dynamical and geometrical similarity.

1.1 Previous Investigations

Even in the free-fall velocity range, only the qualitative aspects

of this type of collision have been studied. No equation appears to

exist that gives crater depth as a function of collision velocity; no measurements of the depth and diameter of craters produced at different

velocities appear to have been made.

Worthington [5, 6, 7] made a study of collisions of liquid drops

with liquids with use of an approximately 3-^ sec spark that could be

timed to illuminate any stage of the collision repeatedly within a

57"

Numbers in brackets refer to literature references at the end of

this paper.
time range of a few millisec (msec). In his first study [5], after a sufficient number of observations of a given stage had been made to secure accuracy, he made a sketch of the configuration of that stage.

In a later study [6], the sketches were replaced by spark photographs.

High-speed pictures of this type of collision have also been given

by Edgerton and Killian [8] in their book on ultra high-speed photography,

and by Franz [9] who took high-speed motion pictures of waterdrops

falling into water.

The specific liquid-projectile liquid-target combinations that have

been studied are: milk drops falling into water [5, 6, 7], milk drops

falling into olive oil [5], waterdrops falling into milk mixed with

water [6, 7], milk drops falling into milk [8], and waterdrops falling

into water [7, 9]. The stages of the collision for all of these liquidprojectile liquid-target combinations are similar; the degree to which

a given stage progresses, however, appears to depend on the properties

of the liquids involved and on the mass and velocity of the colliding

drop.

1.2 Description of the Collision Event

The collision event, as it occurs under various sets of conditions,

has been described in the literature and some Information on the time

required for the successive stages to take place has been obtained

5, 6, 7, 8, 9]. A general description of the major stages is given

below for a collision of this kind at moderate free-fall velocity. The

effects of impingement velocity, of the properties of the liquids used,

and of the proximity of the container walls are then discussed.

a. Major stages

The major stages in the collision of a 0.2-g waterdrop that fell

from a height of 4-0 cm into a target liquid consisting of milk mixed

with water are given below [6, 7]. For convenience of later reference,

successive stages in the collision are arbitrarily identified by capital

letters.

(A) As the drop collides with the target liquid, a little cup of liquid,

having small protruding spicules of liquid around its mouth, is thrown up

around the trailing end of the drop, which is not appreciably distorted^

This cup of liquid grows wider and higher to form a short bellmouthedE/

cylinder of liquid above the target liquid at the periphery of a (more or less hemispherical) cavity that forms in the target liquid.

The drop that impinged apparently rides down into the target liquid on the receding surface of the floor of this cavity as the cavity grows

in size. As the drop rides down into the target liquid it also flows

out radially so that the interior both of the hemispherical cavity,

E7 The term "bellmouthed cylinder" is used to designate a cylindrical

shape with a flared end.

3
which forms in the target liquid, and the interior of the bellmouthed

cylinder of liquid, which stands at the periphery of this hemispherical

cavity, are lined with liquid that originally belonged to the drop,,

The bellmouthed cylinder of liquid at the periphery of the cavity

rises rapidly and its walls thicken« The spicules of liquid, which

originally sloped outward at the mouth of this cylinder of liquid, neck

off small drops and contract to form lobes of liquido

(B) After the cavity in the target liquid has grown to maximum size,

it begins to collapse a During the collapse process, it increases in

size radially while it simultaneously shallows; the cylinder of liquid

at its periphery recedes toward the surface of the target liquid „ After

a total elapsed time of about 55 msec, all that remains on the surface

of the target liquid is a ring of lobes surrounding a central hollow

(C) When the total elapsed time is about 65 msec, a jet of liquid

begins to rise from the center of the residual hollow in the target

liquido A picture of the jet just emerging from the target liquid is

shown in figure 1» This jet carries at its summit the liquid of the

drop that impingedo It reaches a maximum height after approximately

116 msec, and, due to instability, shows a tendency to neck off a drop«

(D) After reaching maximum height this jet subsides,, As it recedes

toward the surface of the target liquid, a mound of liquid collects at

its base The edge of this mound of liquid spreads outward in an everwidening circular ripple By the end of 217 msec the head of this first

jet sinks into the mound of liquid that collected at its base« The final

subsidence of the jet gives rise to a vortex ring that descends through

the liquid,,

(E) If the first jet does not neck off a drop, its disappearance is

quickly followed by the rise of a second jet of different shape after

a total elapsed time of about 24-0 msec This jet also subsides forming

a mound of liquid on top of the mound that was formed during the subsidence of the first jet, and this mound of liquid forms a ripple that

follows after the first one

If, however, the summit of the first jet necks off a drop (or even

nearly necks off a drop) the impact of this drop forms a crater in the

top of the mound of liquid that formed during the subsidence of the

first jet, and the rim of this crater spreads outward to form a second

ripple, which, in this case, quickly follows the first one The second

jet then forms in the center of this crater, but it forms at a later

time than it did for the case that a drop did not neck off the first

jet
FIGURE I. CENTRAL JET EMERGING FROM FLOOR OF SUBSIDING

CAVITY IN TARGET LIQUID
b. Effect of velocity

In stage (A) , for collisions at very low velocity, the drop that

impinged only moves out radially to the extent of flattening as it rides

down into the target liquid on the receding surface of the floor of the

cavity that is produced by the collision [5]. It eventually sinks into

the target liquid at the bottom of the cavity but later reappears in a flattened lobed form and becomes the summit or head of the jet of liquid

that rises [5]o

In stage (A) , for collisions at relatively high velocity, the drop

flows out radially as it rides down into the target liquid on the receding

surface of the cavity floor [7] The liquid of the drop is spread out

in 'a thin lining over the interior of the hemispherical cavity and over

the inside walls of the bellmouthed cylinder of liquid that rises at

the periphery of the cavity [7]„ See section 1.2„ao

For collisions at even higher velocity the cylinder of liquid that

exists at the periphery of the cavity in stage (A) rises to a considerable

height and its top constricts, as though it were pulled in by a drawstring,

to form a dome of liquid over the cavity [6, 7] „ Whether or not this

surface closure occurs seems to depend on the mass as well as on the

velocity (height of fall) of the drop [7], Closure of the liquid cylinder

is sometimes either incomplete or only temporary, and the liquid dome

reopens at the top to make way for the rising jet of liquid in stage (C)

;

the liquid jet is thick and of low height [7], If the liquid dome does

not reopen the rising jet strikes the dome and bursts through it [7].

Worthington [6, 7] explained the constriction of the top of the

liquid cylinder to form a liquid dome as being the result of surface

tension. He suggested, however, that surface tension may not be the

only cause and that the cylinder may be in part forced in by excess external air pressure because of the lessening of air pressure within

it that is produced by the deepening of the cavity in the target liquid

[6]..

In the higher-velocity collisions, which result in the formation of a dome of liquid over the cavity in the target liquid, a downward-moving

jet is produced „ It seems to have its origin at the apex of the liquid

dome, where closure occurs, and its movement is toward the cavity floor,,

Worthington [7] observed this downward jet in the case of collisions of

solid spheres with liquids but failed to observe it in his study of the

collision of liquid drops with liquids. The existence of this jet is

clearly visible in pictures taken by Franz [9] who was interested in the

sounds produced by these collisions and did not make an analysis of the

collision events themselves.
c. Effect of properties of the liquids used

If the properties of the liquid used for projectile and target are notably different, the stages in the collision are similar but the extent

to which they progress is changed,.

With regard to the effect of a change in surface tension of the

target liquid, Worthington [6] stated that addition of milk to the water

into which a waterdrop fell did not make a noticeable change in the

phenomena if the addition was only to the extent of producing a 1 to 3

ratio of milk to water; if pure milk was used, the liquid cylinder thrown

up (see stage A) was higher and had longer spicules indicating a smaller

efficiency of the surface tension in opposing the rise of the liquid

„

With regard to a change in the viscosity of the target liquid,

Worthington [5] stated that the stages in the collision when a drop of

milk falls into petroleum or olive oil are similar to those that result

when it falls into water, modified, however, by the greater or less

mobility of the liquids in question,,

d Effect of proximity of container walls

The effect on the stages of the collision of the proximity of the

walls of the vessel that contained the target liquid has not been discussed. The observations of Franz [9] were obtained using a 91o4--cm

(3-ft) -square glass-walled tank 152 . 4 cm (5 ft) high that was nearly

filled with water. The target-liquid reservoir used by Edgerton and

Killian [8], who were primarily interested in the qualitative .aspects

of the collision from the standpoint of high-speed photography, appears,

from their pictures, to have been about the size of a drinking glass.

The glass vessel with parallel sides that was used by Worthington [7]

to photograph the stages of the collision below the surface was an inverted clock shade.

May [10] has investigated the effect of proximity of container

walls on the water-entry behavior of solid spheres. He has concluded

that, within the range of his investigation, it appears that wall

effects will be avoided if the width of the target-liquid tank is at

least five times the maximum width of the cavity that is formed on entry.

The effect of this variable in collisions of liquid drops with

liquids should be of considerable interest with regard to cavities

formed by meteor collisions. If liquefaction occurs as a result of

meteor collisions, as has been postulated [l], it will probably be

restricted to the immediate site of the collision,,
2„ CORRELATION WITH HYPERVELOCITY-PELLET AND METEORITE CRATERS

In this section several aspects of cavities produced in collisions

of liquid drops with liquids, which have been reported in the preceding

section, are correlated with aspects of craters produced in hypervelocity

and meteor collisions, to which they may bear some relation*

Worthington [7] observed that, in high-velocity collisions of

liquid drops with liquids, the cavity formed in the target liquid was

hemispherical and that the liquid belonging to the drop that impinged

flowed out radially and lined both the hemispherical cavity and the

bellmouthed cylinder of liquid that formed at the periphery of this

cavity. Similarly, it has been observed that craters produced by hypervelocity collisions of metal pellets with metal targets are, in general,

approximately hemispherical [2] Evidence that the material of the

pellet may flow out radially to form a lining in the hemispherical

cavity [ll] or "be plated on the inside of the crater [2] also exists

The bellmouthed cylinder of liquid that rises at the periphery of

the cavity in the target liquid provides another basis for /comparison,,

This bellmouthed cylinder is shown in panels a, b , and c- of figure 2

for three types of collision, namely, for a 182-mg waterdrop colliding

with water at 700 cm/sec, for a 2 54--cm (l-in„) steel sphere colliding

with water at 1,100 cm/sec, and for a 0,,3175-cm (l/8-in„) 302 stainless

steel sphere colliding with a 2024--T4- aluminum target plate at hypervelocity, respectively,, In panels a and b the cavity below the surface

can be seen through the transparent target liquid; in panel c only the

bellmouthed cylinder of metal above the surface of the opaque aluminum

target plate can be seen There is a strong similarity in the appearance

of the structure above the surface of the target for the three types of

collision; differences correlate with the velocity at which the collisions occurred

Because the steel sphere shown in panel b of figure 2 did not flow

out as a result of the collision, the bellmouthed cylinder of liquid

seen in panel b is composed entirely of the target liquid which was watero In panel a, on the other hand, the bellmouthed cylinder of

liquid that was thrown up by a waterdrop colliding with water is lined

with water that belonged to the drop because the drop flows out radially

during the collision,, In panel c, the bellmouthed cylinder of metal

thrown up in the hypervelocity collision may or may not be lined with

metal that belonged to the colliding sphere depending on the extent to

which liquefaction of the 302 stainless steel sphere and the 2024--T4-

aluminum target has progressed

„

-/ These pictures were supplied by Dr Albert May, U S„ Naval Ordnance

Laboratory, White Oak, Maryland

„

-* These pictures were supplied by Lt. Thomas E Sheldon, Headquarters

Air Proving Ground Center, Eglin Air Force Base, Florida

„
"<££»

m

m

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FIGURE 2. INITIAL STAGES OF THREE TYPES OF COLLISION
To account for the curling-over of the liquid thrown up around the

cavity, Worthington [6] stated that at first the flow of liquid appears

to be very much along the surface whereas at a later time it is much

more perpendicular to the surface, and that in the early stages each

particle continues to move for some distance in the straight line along

which it was first projected from the surface.

In figure 2, the extent of curling-over of the cylinder of liquid

around the cavity is greatest for the highest-velocity collision (panel c)

and is least for the lowest-velocity collision (panel a) ; the height of

the bellmouthed cylinder of liquid with respect to the diameter of its

base is in the same order.

It has been suggested that the craters on the moon may have been

caused by impinging meteors. Lunar craters l) are cuplike depressions

with steep inside slopes that decrease gradually to a nearly level floor

depressed below ground level; 2) the volume of the rim is equal to the

volume of tke subsurface part of the crater; 3) some of the craters

contain a central peak whereas others do not; and 40 some craters have

a system of rays about them [12] Official U, S> Navy photographs of some craters on the moon are shown in figure 3 , Baldwin [12] has con- cluded that lunar craters are explosion pits. However, the four

attributes of lunar craters cited above may correlate with stages in

collisions of liquid drops with target liquids.

With regard to the first two attributes, at the time that the first

liquid jet begins to emerge (see section 1,2, a and fig, l) , the hemispherical cavity that originally formed in the target liquid has become

wide and shallow. Compression in the target liquid has been relieved

and it is reasonable to suppose that the volume of the residual crater

is roughly equivalent to the volume of the rim that surrounds it.

With regard to the third attribute, if meteor collisions are a kind

of liquid-drop liquid-target collision, and if the liquid melt caused by

the high-speed collision should freeze just as the jet began to protrude

from the bottom of the cavity (see fig, l) , the frozen structure would

be a crater containing a central protuberance; if the liquid melt froze

just before the head of the jet emerged from the bottom of the cavity,

the frozen structure would be a crater without a central protuberance.

The liquid crater shown in figure 1 may be compared with lunar craters

that contain a central protuberance. Such craters are marked with

arrows in figure 3„

~n

' These photographs were supplied by Mr, J, L, Gossner of the U, S,

Naval Observatory in Washington, D, C,

10
FIGURE 3. CRATERS ON THE MOON

11
With regard to the fourth attribute, Worthington [5] remarked that

symmetrical rays form in the cavity that develops in the target liquid

when a liquid drop collides with a liquid,, As seen in his sketches,

these rays terminate in projecting points around the rim of the cavity,,

3. QUANTITATIVE DATA ON LIQUID-DROP-INTO-LIQUID COLLISIONS

Worthington [6] made a first step toward obtaining quantitative

data* He followed the time required for accomplishment of the various

stages of the collision of a a 2-g waterdrop that fell 4.0 cm into a target liquid of milk mixed with water He found that the cylinder of

liquid at the periphery of the cavity rose to its maximum height in

20 msec and remained poised with little change for the next 10 msec

It then widened out and subsided until nothing but a lobed rim surrounding

a central hollow was left above the surface; this required 25 msec The

subsidence of the liquid cylinder was followed by the rise of the central

jet, which required 50 msec, and by its subsidence, which required 70 msec*

The total time accounted for was 175 msec These stages of the collision,

therefore, took place in less than one fifth of a second,,

For a theoretical analysis of this type of collision, more quantitative data are needed* The very limited quantitative study of the collision incident that is presented in this section has been made with the

use of the high-speed motion pictures of waterdrops falling into water

that were taken by Franz [9] in connection with his study of the sounds

that are produced,, His kind loan of these motion picture films, for

whatever other data could be derived from them, is gratefully acknowledged*

The films that were made available comprised collisions of three

sizes of drops for two combinations of drop mass with collision velocity:

11-mg drop colliding at velocities of 4-00 and 650 cm/sec, 56-mg drop

colliding at velocities of 4-00 and 700 cm/sec, and 182-mg drop colliding

at velocities of 4-00 and 700 cm/sec „ The stages in the collisions are similar to those described in section l 2 a Pictures of the stages of

similar collisions have been given by Franz [9]„ The collisions of the

56-mg and 182-mg drops at a velocity of 700 cm/sec resulted in surface

closure of the cavity in the target liquid in a way similar to that

described in section l a 2,b <,

For the purpose for which the films used in this study were made,

there was no need to photograph a reference standard of length or of

optical distortion in the system,, That this was not done, however,

reduced the accuracy with which the cavity depth and diameter measurements reported below could be determined „ Nevertheless,' the data

obtained from these films are valuable in giving the broad outlines

of some quantitative aspects of this type of collision, and in indicating points that should be studied in future experiments

„

12
)

3.1 Cavity Depth

Measurements of the depth of the cavity in the target liquid,

at known intervals of time after the collision incident, were made on magnified prints of the motion picture films,, These prints were obtained

with use of a microfilm reader-printer. The measurements were made using

a stereomicroscope set for low magnification and a steel rule graduated

to l/2 mm. Each recorded measurement is the average of two readings.

The meniscus appeared as an indistinct line in the prints. The

mid-point of this indistinct line was arbitrarily taken to be the

undisturbed surface of the target liquid. The depth of the cavity was measured with respect to it. The fact that the bottom of the cavity

and the periphery of the mouth of the cavity at the surface of the

liquid were also indistinct reduced the accuracy with which the measurements could be made.

The time that, had elapsed from the first instant of the collision

was found from the reported speed at which the pictures were made. This

was 2,800 frames/sec or 357 jxsec per frame

To convert measured values of cavity depth to real values, the magnifying factor was determined for the prints made from each film. For

the case that the freely falling drop was spherical, the magnifying

factor was taken to be the ratio of the real diameter of the drop to

the measured diameter. The real diameter was calculated from the

volume of a sphere and the reported mass of the drop, taking the

density of the water used to be one. For the case that the freely

falling drop was an oblate spheroid, the magnifying factor was taken

to be the ratio of the real major axis of the drop, calculated from its

mass, to the measured major axis. The real major axis was calculated

from the volume of an oblate spheroid, the ratio of the measured semimajor to the measured semi-minor axis of the drop, and the reported mass of the drop, taking the density of the water used to be one.

The cavity-depth data are plotted in figure 4-° Curves A and A' are for the 11-mg drop colliding at velocities of 4-00 and 650 cm/sec,

respectively; curves B and B' are for the 56-mg drop colliding at

velocities of 4-00 and 700 cm/sec, respectively; curves C and C are for the 182-mg drop colliding at velocities of 4-00 and 700 cm/sec,

respectively.

From figure 4- it can be seen that for each of the collisions the

cavity that forms in the target liquid undergoes a relatively smooth

increase in depth until maximum depth is approached. At the time of

maximum depth, the cavity appears to vibrate. Each curve of figure 4- has been drawn arbitrarily through the vibration fine structure so that

the maximum of the curve is the average depth of the cavity at the time

of maximum depth. The maximum of the curve of cavity depth versus time

is hereafter referred to as the maximum depth of cavity produced.

13
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CURVE A II-•mg DROP COLLIDING AT 400 cm /sec

-

.4 CURVE A'll-mg DROP COLLIDING AT 650 cm /sec

— CURVE 6 56-mg DROP COLLIDING AT 400 cm /sec

CURVE B* 56-mg DROP COLLIDING AT 700 cm /sec

.2 CURVE C 182 -mg DROP COLLIDING AT 400 cm /sec —

CURVE C'l82 -mg DROP COLLIDING AT 700 cm /sec

' J l 1

l 20 30 40 50

TIME , msec

FIGURE 4. DEPTH OF CAVITY PRODUCED IN WATER BY AN IMPINGING

WATERDR0P PLOTTED AGAINST ELAPSED TIME

60

14
By comparing curve A with A', curve B with B', and curve C with C

it can be seen that increasing the collision velocity with drop mass constant in each case increases both the maximum depth of the cavity

produced in the target liquid and the time that was required to produce

maximum depth. Similarly, by comparing curves A, B, and C, or curves A',

B', and C, it can be seen that increasing the mass of the drop with

collision velocity constant also results in an increase both in the maximum depth of the cavity produced in the target liquid and in the time

required for the cavity to grow to its maximum depth. By comparing

curve B with curve B

1 , and then comparing curve B' with curve C, it

appears that an increase in the mass of the drop may be less important

than a proportional increase in collision velocity in producing the

effect of increase in maximum cavity depth and of increase in the time

required to produce maximum depth.

The maximum depth of cavity produced and the time required to

reach this maximum cavity depth for each of the six collisions that

have been studied are listed in table 1 and plotted in figure 5.

The plots of cavity depth versus time shown in figure 4- appear

to be parts of a family of parabolas or ellipses tangent to the cavitydepth axis at the originf/. The general equation of a family of ellipses

is

2| + ^_b]f = , (1) a b

where D is the cavity depth, t is the time, a is the semi-major

axis, and b is the semi-minor axis. When the cavity depth D is

zero, t = and t = 2b. The semi-major axis is the maximum cavity

depth produced, D , and the semi-minor axis is the time required to

produce the maximum cavity depth, t . Solving eq (l) for cavity depth D

produces the expression,

+ 1/2

D = a (2 - p (t/b) . (2)

The cavity-depth-versus-time data for the 56-mg waterdrop colliding

with water at a velocity of 700 cm/sec is plotted in figure 6 with the curve calculated by use of eq (2) . In calculating points by eq (2) , the

17A theoretical analysis of this aspect of collisions of liquid drops

with liquids is in progress.

15
Table 1

Properties of the Cavity at Maximum Size

Proje ctile Drop Cavity

Depth, D

mm

Elapsed

Time, t

sea

Cavity Depth

divided by

Cavity Diameter Mass

mg

Velocity

cm/sec

11 400 7.25 0.0139 0.64

11 650 9.70 0.0168 0.58

56 400 11.9 0.0161 0.56

56* 700* 14.4* 0.0207* 0.55*

182 4.00 16.8 0.0246 0.52

182* 700* 21.9* 0.0286* 0.54*

Surface closure of the cavity occurred.

16
c.c

1 i o

2.0 —

1.8

o —

1.6

^ 1.4 o

m^m X

Q.Ui

2

X

o —

s |.0 mm— o

*—

.8 — —

.6 — -

.4 —

.2 — -

, s

Fl

o®8 ,02 .03

TIME TO PRODUCE MAXIMUM DEPTH, S6C

IMUM DEPTH OF CAVITY PRODUCED IN WATER

BY IMPINGING WATERDROPS PLOTTED AGAINST

THE TIME REQUIRED TO REACH IT

17
10 20 30

TIME ,msec

FIGURE 6. DEPTH OF CAVITY PRODUCED IN WATER BY COLLISION OF

56-mg WATERDROP AT 700 cm/sec PLOTTED AGAINST

ELAPSED TIME

40

18
empirical values of Dmmand t for this collision were used. The

divergence of the observed values from the calculated curve after

32 msec may be the result of energy dissipation (hysteresis)

,

3.2 Ratio of Cavity Depth to Cavity Diameter

Measurements of the diameter of the cavity produced in the target

liquid were made at intervals of time after the collision for which

depth measurements had been made. The diameter that was measured was

the maximum diameter at the surface of the target liquid , Both the

cavity depth and the ratio of cavity depth divided by cavity diameter

are plotted in figure 7 against the time elapsed since the collision

for the collision of the 182-mg waterdrop with water at a velocity of

4.00 cm/sec ,

From the plots of figure 7 it can be seen that the depth/diameter

ratio for the collision of a liquid drop with a target liquid is not

constant with time. It goes through a maximum and it reaches its maximum sooner than the depth curve does. From the depth/diameter curve plotted in figure 7 it would appear that this ratio is equal to 0.5

only at two points in time. For very short times after the collision,

the cavity is oblate; just before and also during the time that it

reaches maximum depth, it is prolate; during the stages of its subsidence, it is again oblate.

Depth/diameter curves for the other five collisions for which data

were available were of a similar shape. The maximum value of the curves ranged from 0,55 to 0.65. Small differences in these curves may or may

not be significant; more data are needed to evaluate them. The value of

depth/diameter at the time of maximum cavity depth is listed in table 1

for each of the six collisions.

3.3 Effect of the Energy of the Impinging Drop

The total energy of each drop that collided with the target liquid

is the sum of its kinetic energy and surface energy. The kinetic energy

of each drop was calculated from the mass and velocity reported for it.

The surface energy of each drop was found from the calculated surface

of the drop taking the surface tension of the water used to be 72.75 dyne/cm,,

For the case that the freely falling drop appeared to be a sphere, the

surface area of the drop was taken to be 773 where d is the drop

diameter; for the case that the freely falling drop appeared to be an oblate spheroid, the surface area of the drop was taken to be

2 2

27Ta + T7b /e [in (l + e)/(l - e)] where a is the semi -major axis,

1/2

! The values of d, a, and b were calculated. See section 3,1..

2 2

b is the semi-minor axis, and the eccentricity e is (a - b ) /a.

19
20 30

TIME, msec

FIGURE 7. DEPTH AND DEPTH /DIAMETER RATIO OF THE CAVITY

PRODUCED IN WATER BY COLLISION OF A 182-mg

WATERDROP AT 400 cm/sec

20
The kinetic energy, surface energy, and momentum of the impinging

drop for each of the six collisions for which data were available are given in table 2 a The total energy is not tabulated,, The values of the

kinetic energy and of the surface energy are both good to two significant

figures, but the magnitude of the values is so different that the surface

energy becomes insignificant if it is added to the kinetic energy.

The observed value of maximum depth of the cavity produced in the

target liquid is plotted against the kinetic energy of the colliding

drop in figure 8 and against the momentum of the drop in figure 9,

Points for the two collisions for which surface closure of the cavity

occurred are marked with a cross in each graph because maximum cavity

depth may be affected when surface closure occurs* For these collisions

the mouth of the cylinder of water thrown up around the cavity in the

target liquid constricted and eventually closed to form a dome of liquid

over the cavity* See section 1.2„b.

From a practical standpoint it would be informative if the relation

between the energy of the impinging drop and the maximum depth of the

cavity produced in the target liquid as a result of the collision could

be ascertained* It has been found [2] that, in the fluid region of

hypervelocity collisions of metal pellets against metal targets, the

volume of the crater formed in the target is proportional to the kinetic

energy of the impinging pellet; the craters formed are also nearly

hemispherical in shape « If it is assumed that, at the time of maximum

depth, the volume of the cavity in the target liquid is proportional

to the kinetic energy of the impinging drop, then, for a spherical drop

that produces a hemispherical cavity,

P 1/3

D

m

= Kr(p V*) (3)

where D is the maximum depth of the cavity, r is the radius of the

original drop, p is the density of the drop liquid, V is the

impingement velocity, and K is a constant having the dimensions

2 1/3

/cm, sec /g) which are those of the cube root of reciprocal pressure

„

For the case that p

- 1,

D = KrV2' 3 U) m

For a drop whose shape is an oblate spheroid

1/3 2/3

D = K (a

2

!)) V (5) m

where a and b are the semi-major and semi-minor axes, respectively.

21
Table 2

Energy and Momentum of the Impinging Waterdrops

Drop Mass Drop Velocity Momentum Kinetic Energy Surface Energy

g cm/sec g. cm/sec ergs ergs

0.011 4.00 4-. 4- 880 17.4

0.011 650 7.2 2,300

A

17.4

0.056 400 22. 4,500

A

51.5

0.056 700 39. 14,000 51.£

0.182 400 72.8 14,600 11 £

0.182 700 127. 44,600

A

113.

22
wo Hld30 A1IAV3

23
o

12

o

CM

OO

O

£ -

Q_O

08

o

II 1

o UJo

o

o

cvi

10 OJ 00

• • • wo'Hidaa A1IAV0

24
The equation can be tested with the available data by determining

whether or not K remains constant , In the experiment in which the

moving pictures were obtained [9] the waterdrops were allowed to fall

through air. The three drops having lowest kinetic energy were spherical

in shape at the time that they collided with the target liquid; the

three drops having highest kinetic energy were oblate spheroids. The

values of the constant K, calculated with use of eqs (4.) and (5) and

the empirical values of D are listed in table 3.

The data for the two collisions that resulted in surface closure

of the cavity in the target liquid are marked with asterisks in table 3,

Because maximum cavity depth may be affected for the case that surface

closure occurs, the values of K that are marked with asterisks in

table 3 may be disregarded In evaluating the constancy of K. However,

even if the two values of K that are maiked with asterisks are ignored,

the tabulated values suggest that K may not be 'constant. It appears

to be decreasing as the kinetic energy of the drop is increased but more data are needed to determine whether or not the drift is real.

The average value of K for the four collisions for which surface

closure did not occur is 0.093. Points calculated by use of eqs (4-) and (5) and with K = 0,093 are connected with a line in figure 8,

k. CURRENT RESEARCH

Although 80 years have passed since Worthington [5] started his

investigation of collisions of liquid drops with liquids, it can be

seen from the preceding sections that even the simplest example of this

type of collision (waterdrops falling into water) is very poorly understood. High-speed photography is the tool needed to obtain further

information. More high-speed motion pictures, taken under conditions

and with controls that will make it possible to evaluate the accuracy

of measurements made on them, are required.

Apparatus with which an exhaustive study of collisions of liquid

drops with liquids can be made is now being assembled at the National

Bureau of Standards," It consists essentially of a 15»2-cm (6-in,)-diam

pipe that is 20 m (66 ft) long attached to a sealed tank that contains

the target liquid. The purpose of the pipe is to provide gravity

acceleration of the liquid drops under reduced pressure. The system

is being assembled in a stairwell. The target-liquid tank is located

at the basement level? the pipe extends through four stories of the

building and terminates in the penthouse,

4*1 Evacuated System

To increase the kinetic energy of the projectile drop either its

velocity, or mass, or both must be increased. The velocity acquired

by a drop for any given height of fall will be greater in an evacuated

25
Table 3

Values of the Proportionality Constant K

Drop Mass

g

Drop Velocity

cm/sec

Constant K

f 2 / ^/3 (cm. sec /g)

0.011 400 0.097

0.011 650 0.094-

0.056 4-00 0.092

0.056* 700* 0,077*

0,182 4.00 0.088

0.182* 700* 0.079*

* Surface closure of the cavity occurred.

26
system than in air because of the reduction in drag. The size of the

drop can also be made larger in an evacuated system than in air because

the tendency for breakup due to air currents encountered during fall

will be reduced. The lowest feasible pressure for the simple system

in which both the drop and target are of the same liquid is the room- temperature vapor pressure of the liquid selected for study. If the

pressure in the system were to be made lower than this, both the target

liquid and the projectile drop would boil.

The drop-accelerator pipe and the target-liquid tank have been

designed so that both the volume of the pipe and the free space above

the target liquid in the tank can be exhausted to the vapor pressure of

the liquid under study. The evacuation will be accomplished by means

of an air-cooled simplex-type vacuum pump having a free air displacement

of 36.8 l/min (13 cu ft/min) . Because the vapor pressures of the liquids

to be used will be in the millimeter range, pressure in the system will

be checked by means of a mercury manometer.

The end of the pipe is connected to the target-liquid tank through

a 15,2-cm (6-in.) aluminum ball valve. With the ball valve closed, the

pipe can be evacuated without concomitant evaporation of the target liquid

and the consequent need to trap out large quantities of vapor „ Vapor

must be trapped out when the free space above the target liquid is

evacuated, but the volume involved is not large. After the pipe and

tank have been evacuated separately, the valve between them will be

opened and the target liquid will be allowed to evaporate until

equilibrium is reached.

The throat of the connecting ball valve is large enough so that

drops, falling through the pipe, can pass through it and strike the

target liquid in the tank below. The valve is equipped with natural

rubber seats and seals to insure long service life in the atmosphere

saturated with water vapor that will exist within the system,

4.. 2 Materials

The drop-accelerator pipe and target-liquid tank are both made

of 5052 aluminum alloy to minimize corrosion and changes in the properties of the liquid under study that corrosion might produce. This

aluminum alloy has been used for distilled water lines. Welding for

the system was done with 5154 aluminum rod. Small model tanks made

of 5052 aluminum sheet and welded with either 5154- or 5356 aluminum

rod were each found to be vacuum tight on the first trial when tested

with a helium leak detector. Both of these aluminum welding rods were considered satisfactory for corrosion resistance.

The inside surface of the target-liquid tank was coated with black

pigmented Epon 820 epoxy resin to reduce light reflection when highspeed shadowgraph motion pictures are taken of the liquid-drop collisions , The tank was then baked at 70°C (158°F) to cure the coating.

The epoxy coating will not contaminate the liquids to be investigated.

27
Vacuum seals at flanged joints in the pipe and tank were made with

O-rings of Buna-N synthetic rubber of Durometer 70 hardness . The 0-rings

were cleaned with ligroine and were used without application of grease*

U» 3 Design of the Drop-Accelerator Pipe

The drop-accelerator pipe is of seamless circular tubing; it has

a 0.64.-cm (0.25-in.) wall thickness, an inside diameter of 15.2 cm

(6 in.), and a total length of 20 m (66 ft). It consists of four flanged

sections with a port between each section and at the top of the pipe;

liquid drops can be introduced into the pipe at the four ports. The

lengths of the pipe sections are such that the drop-admittance ports

occur at about 1.5 m (5 ft) above the floor at each floor level.

a„ Drop-admittance ports

The four drop-admittance ports make it possible to study the collision of a fixed drop mass at four widely different velocities „ > They are 45.7-cm (l8-in.) sections of pipe that have a short side arm welded in.

The open end of the side arm is flanged, and, when not in use, each port

is closed vacuum tight with a flat circular cover that bears against an

0-ring.

At the level of the side arm in the drop-admittance port that is

being used, the lower section of the pipe is sealed off with a removable

circular plate and ring that are supported by a fixed ring welded into

the pipe. Vacuum tightness is insured with 0-ring seals. In the center

of the removable plate there is a tapered hole into which a 19/38 ground

glass joint, which is fitted with an 0-ring, may be inserted. The ground

glass joint supports a dripper for admitting drops of a given size into

the evacuated system.

b. Dripper design

The dripper consists of a pipet nozzle, two liquid reservoirs, and

two stopcocks. The inner liquid reservoir, which is sandwiched between

the stopcocks, is at first empty and the stopcock that separates this

reservoir from the evacuated pipe is left operu The outer liquid

reservoir is separated from the inner reservoir by the stopcock furthest

from the evacuated system; this reservoir is open to atmospheric pressure

and is full of the liquid that is to be used. When the system has come

to equilibrium, the inner stopcock is closed and liquid is admitted into

the inner reservoir from the outer reservoir at the pressure that prevails within the system. Drops are admitted into the system from the

inner reservoir.

The stopcocks have been given a special treatment that permits easy

turning without the use of grease which might contaminate the drop liquid.

Vacuum tightness of the stopcocks is insured either by spring compression

or by liquid seals. By using pipets of different nozzle size the mass

of the drop that is admitted can be changed.

28
4.4 Design of the Target-Liquid Tank

The target-liquid tank is a cylinder 35<»6 cm (14 in,) in diameter

and 55,9 cm (22 in.) high that was rolled from 1,27-cm (0,50-in,) sheet

and welded. Two 30,5~cm (12-in.) side arms, which were also rolled and

seamed, are welded into it,

a# Criteria for size

The main criterion used in choosing the diameter of the tank was

that it should be large enough to bypass the effect of the container

walls on the cavities formed in the target liquid by the liquid-drop

collisions. On the basis of available information it appeared that a diameter of 25*4 to 30,5 cm (10 to 12 in,) would be adequate for this

purpose. Excessive size was avoided both to reduce problems of procurement or fabrication and to minimize the volume of target liquid that

would be required for the experiment. The second consideration will be

important when liquids other than water are investigated.

b. Optical windows

The open ends of the side arms in the tank terminate in 1,9-cm

(0,75-in,) flanges. They are closed vacuum tight with 2.54-cm (l-in,)-

thick covers that contain glass windows through which the liquid-drop

collisions will be photographed. The windows are of No, 7913 optical

grade Vycor glass of shadowgraph quality. The glass blanks, which were furnished with a commercial polish, were made sufficiently flat, symmetrical, and parallel to insure minimum distortion in the pictures,

4.. 5 Scope of the Experiment

According to present plans, the apparatus will first be used to

carry out an exhaustive study of collisions of waterdrops with water

By use of the four drop-admittance ports in the pipe, it is planned

to study the collision of a drop of given mass at four different

velocities. By changing the nozzle of the pipet in the dripper, it

is planned to study the effect of varying the drop mass. By maximizing

both the mass and the velocity of the impinging drop, it is planned to

explore further the dependence of cavity depth on the kinetic energy

of the drop.

By introducing movable glass barriers into the target liquid, it

is planned to study the effect of the proximity of walls. By varying

the degree of evacuation of the system, it is planned to study the

effect of a change in pressure.

If it is possible, the investigation will then be extended to a consideration of the effect of changing the surface tension, viscosity,

and density of the liquids used for drop and target.

29
5. BIBLIOGRAPHY

1. Opik, Ernst,. Researches on the physical theory of meteor phenomena I,

Toimetused Acta et Commentationes, Universitatis Tartuensis, XXX,

[A] 4 (1936) (Tartu, Estonia).

2. Proceedings of the third symposium on hypervelocity impact, Vol. 1,

Armour Research Foundation of Illinois Institute of Technology,

Chicago, 111. (Feb. 1959).

3. Cook, Melvin A., The science of high explosives (Reinhold Publishing

Corporation, New York, N. Y. , 1958).

4.. Ungar, E. W., Investigation of high-speed particle erosion of melting

aerodynamic surfaces, Final Report on Contract No. DA-33-019-0RD-2774,

Battelle Memorial Institute, Columbus, Ohio (1959).

5. Worthington, A. M. , On impact with a liquid surface, Proc Roy. Soc. 34- , 217 (1882).

6. Worthington, A. M. , and Cole, R. S., Impact with a liquid surface,

studied by the aid of instantaneous photography, Phil. Trans. Roy. Soc,

182A, 137 (1897).

7. Worthington, A. M. , A study of splashes (Longmans, Green, and Company,

New York, N. Y., 1908).

8. Edgerton, Harold E., and Killian, James R», Jr., Flash! Seeing the

unseen by ultra high-speed photography (Charles T«, Branford Company,

Boston, 1954).

9. Franz, G. J., Splashes as sources of sound in liquids, J Acous* Soc.

Am. 11, 1080 (1959).

10. May, Albert, The influence of the proximity of tank walls on the

water-entry behavior of models, NAVORD Report 2240, U. S. Naval

Ordnance Laboratory, White Oak, Maryland, October 1951.

11. Partridge, William S. , VanFleet, Howard B., and Whited, Charles R.

,

An investigation of craters formed by high-velocity pellets,

Technical Report No. OSR-9* Air Force Office of Scientific Research,

May 1957.

12. Baldwin, Ralph B«, The face of the moon (University of Chicago Press,

Chicago, 111., 1948).

USCOMM-NBS-DC

30
U. S. DEPARTMENT OF COMMERCE

Luther H. Hodges, Secretary

NATIONAL BUREAU OF STANDARDS

A. V. Astin, Director THE NATIONAL BUREAU OF STANDARDS

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