AlAA 92-0112 The Behavior of a Liquid Drop Levitated and Drastically Flattened by an Intense Sound Field C. P. Lee, A.V. Anilkumar, and T.G. Wang Center for Microgravity Research and Applications, Vande rbi It U n ive rsi ty Nashville, TN 37235

30th Aerospace Sciences Meeting & Exhibit

January 6-9/1992 / Reno, NV For permission lo copy or republish, contact the American l n s t i e of Aeronautics and Astronautics 370 L'Enfant Rmenade, S.W.. Washington, D.C. 20024

MA-92-0112 THE BEHAVIOR OF A LIQUID DROP LEVITATED AND DRASTICALLY FLATTENED BY AN INTENSE

SOUND FIELD

C.P. Lee.' A.V. Anilkumar." and T.G. Wang"' Center for Microgravity Research and Applications

Vanderbilt University, Nashville, Tennessee 37235

Ab§lrxl

Acoustic levitation is one of the key methods for wntainerless processing of mens in a microgravity environment. In order to explore its fullest potential, we have studied the deformation and break-up of a liquid drop in levitation through the radiation pressure. Using high-speed photography we have observed ripples on the central membrane of the drop, atomization of the membrane by emission of satellite drops from its unstable ripples, and shattering of the drop after upward buckling like an umbrella, or after horizontal expansion like a sheet. These effects have been captured on a video film which will be shown. From our theoretical study, we believe that the ripples are capillary waves generated by the Faraday

W instability excited by the sound vibration Atomization occurs whenever the membrane becomes so thin that the vibration is sufficiently intense. The vibration leads to a destabilizing Bernoulli correction in the static pressure Buckling occurs when an existent equilibrium is unstable to a radial (Le. tangential) motion of the membrane because of the Bernoulli effect. Besides, the radiation stress at the rim of the drop is a suction stress which can make equilibrium impossible, leading to the horizontal expansion and the subsequent break-up.

3 . I n t r o d W

It is known that a small liquid drop in air can be levitated at a pressure node in a standing sound wave, making use of the acoustic radiation pressure.l.2 On earth, this can be accomplished for a millimeter-sized drop in an acoustic levitator, that consists of a transducer at the bottom, driving

* Research Associate Professor ** Research Assistant Professor *** Centennial Professor and Director Copyright 0 1992 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.

I

'e

a wave at a resonant frequency to a reflector at the top.3 It has been observed4 that when such a drop is increasingly and drastically flattened by the radiation pressure, it will suddenly burst.

Our observations with high temporal and spatial resolutions, reveal in great detail some phenomena that can happen during the brief disintegration process. They have been recorded on a video tape, to be shown in the presentation. The accompanying theoretical investigation, described briefly below, provides a coherent basis for the understanding of these phenomena.

2. Theor-

Let us first consider the equilibrium shape of the flattened drop. For simplicity, the flattened drop is approximated to be a flat circular disc with no thickness, levitated horizontally at a pressure node, as far as scattering of sound wave is concerned. The radiation stress is evaluated1 on its surface using the solution of the wave equation.5 Then the drop is allowed to have an equilibrium shape of finite nonuniform thickness. Its central membrane is allowed to vibrate with the acoustic pressure like a drumhead when it is thin, leading to a Bernoulli correction to the internal hydrostatic pressure in an equilibrium situation. Let R, be the spherical radius of the drop, k be the wavenumber of the wave, R be the radius of the disc, I be the radial position on the disc surface, f(r) is the half thickness of the drop at I, p=po/p be the ratio of the density of air to that of the liquid, A be the pressure amplitude of the wave, cg be the speed of sound in air, and 0 be the surface tension of the liquid in air. Then the equilibrium shape is determined using the Young-Laplace equation:

1

L)

there is no solution for a sound intensity above a critical value determined by the peak of B,.

There is also no solution for flattening beyond a critical R*, marked by DE (dead end). The stress from surface tension at the rim, trying to hold the drop together, is inversely proportional to the radius of curvature E there. The suction stress imposed by the oscillating flow of air around the rim, trying to pull the drop apart, is also inversely proportional to E, but is, in addition, directly proportional to the product AZR. Therefore there exists a point DE, beyond which the suction stress is larger than what the surface tension can withhold, such that the drop cannot be further flattened without losing equilibrium.

The parameter a is the drop size relative to the wavelength of the sound wave, and represents the strength of the radiation pressure on the membrane (Eq.(l)). In general, for a given flattening R*, a smaller drop, with a smaller a, is held together more tightly by surface tension, and therefore calls for a larger acoustic intensity represented by B,. So the B,-R* curve tends to be displaced upward as a decreases.

The parameter p takes into consideration the vibration forced by the acoustic pressure. It is assumed to be important in the membrane but not in the ring-shaped periphery of the flattened drop. But if the liquid is viscous, then viscous diffusion spreads the vibration even to the periphery. In such a situation. the whole drop vibrates almost uniformly, and the Bernoulli correction (not to be confused with the Bernoulli stress due to air flow around the rim) to the internal pressure is essentially nullified. So p is given a finite value only for a low-viscosity liquid like water, but should be set to zero for a viscous liquid like glycerine. The Bernoulli correction tends to lower the internal pressure of the membrane relative to that in the periphery. But since the internal pressure in the flat membrane remains close to the acoustic radiation pressure, i.e. unchanged, that in the periphery has to be elevated in equilibrium. Thus the Ba-R* curve meets its DE earlier, with the additional pressure in the periphery tipping the balance between the much stronger surface tension and suction stress at the rim, in favor of the latter. Therefore the maximum flattening attained by a water drop, before losing equilibrium, is less than that by a glycerine drop.

Another effect of the Bernoulli correction to the -

where a = k R , , s = r / R , q = f I R , R*=R/R,, &=R,A~/(opoco2), B.*=B,R*/+', Q is a constant proportional to the internal pressure, and E is the smaller radius of curvature at the rim scaled by R. Equation (1) balances surface tension represented by the left side, and the pressure part of the acoustic radiation stress which compresses the drop into a disc-like form, the suction part of the stress at its rim due to potential flow of air around a sharp edges, internal hydrostatic pressure, and Bernoulli correction due to its membrane vibration, represented in that order by the right side. The dimensionless number B. may be called the acoustic Bond number. The boundary conditions are dq I ds =-at the rim where s=l , and d q l ds = 0 at the center where s=O. An additional condition is that the volume of the drop is unchanged by deformation. The detailed derivation, numerical procedure and solutions can be found in ref.7.

In Fig.1 the bold curve is the schematic drawing of a typical B. versus R* curve for given a and p. It is noted that flattening first increases, then decreases, with the sound intensity. Initially, the drop becomes more flattened when the radiation pressure on its top and bottom surfaces increases with sound intensity. When the drop is thin enough, the suction stress at its rim is also significant such that flattening is easier. Then there comes a point beyond which the suction stress becomes dominant, such that flattening can continue with decreasing sound intensity. So

t

. R'

Fig.1. Bold curve: a typical equilibrium curve. Path 1 : transition from one equilibrium point to another. Path 2: transition fails by missing the curve. Path 3: A simple loss of equilibrium in the absence of resonant frequency shift.

2

u internal pressure is the buckling instability described below. If the flattened drop in equilibrium is disturbed such that a part of its membrane becomes thinner, then that part is subjected to more vibration forced by the acoustic pressure. The resulting Bernoulli pressure drives the liquid away to the neighboring region, causing the membrane there to thicken. Surface tension then brings about a build-up of internal pressure there to reverse the flow. Overshooting leads to an oscillation. But if the membrane is thin enough, the Bernoulli pressure generated by the sound vibration can become too strong for surface tension to resist, and the motion is unstable. Since the drop is levitated against gravity, its membrane at equilibrium is slightly buckled upward to start with. The instability causes the membrane to buckle upward violently, as observed experimentally and described in the next section. According to our estimation,' the instability criterion is r>36, where rSPB,*/h3, in which h is the membrane thickness, assumed uniform, scaled by R. From our numerical work, B,* is close to 1 in the limit where the drop is drastically

W flattened. So for a millimeter-sized drop, the membrane thickness is of the order of 100pm when buckling instability occurs. Since the vibration of a thin membrane can be very strong, capillary ripples should also appear on the membrane due to parametric instability, before the drop buckles, unless the liquid is very viscous.

For a smaller drop, with the same flattening R*, the membrane can be so thin that another effect comes into the picture. With a lighter inertia, the forced vibration can be so violent that the capillary ripples might emit satellite drops vertically in both directions, in what we call 'atomization.' In this case, the membrane is still thick enough such that it remains intact while the satellite drops break off from the wave crests of its ripples.

For a still smaller drop, the membrane is even thinner that holes might form amid the ripples through van der Waals fluctuations, leading to another form of break-up. We shall call this break- up process 'fragmentation.'

This instability can occur to a drop expanding horizontally such that its membrane is stretched excessively, due to loss of equilibrium. The loss happens because the drop becomes so flattened that it moves beyond the limit DE, or the sound intensity is so high that the drop moves above the W

peak of B., referring to Fig.1. The bold curve in Fig.1 shows that for each value

of B. there are two values of R*. The higher Value of R*, corresponding to the downward slope of the curve, can exist experimentally because there is an interaction between the drop and the levitator field. The presence of a drop causes a resonant frequency shift* in the levitator, resulting in a power loss if the latter is operated at a fixed frequency, as it is in our case. The shift, and therefore the power loss, increases when the drop becomes increasingly flattened, making possible a downturn in B, in the experimental situation.

of FxDer-

The acoustic levitator is operated at a fixed frequency of around fo=21.76 kHz. A millimeter- sized drop is levitated and flattened with a slowly increasing input sound intensity over a period of about 20 secs. from about 162 to 166 dB (measured without the drop). The development was recorded with a high-speed video. In this section we shall give a qualitative description and interpretation of what is shown on the video tape. For more details, the readers are referred to the work that we have recently completed (ref.7), and a further work that we shall submit for publication.

A relatively large water drop (kR-1) changes from convex to concave when the input intensity increases. Its liquid is squeezed to the rim that becomes a donut-shaped ring, leaving a thin membrane at the center. Short-wavelength ripples then appear on the membrane. The membrane suddenly bulges upward while being flattened, with the thicker periphery contracting inward soon after, turning the drop into a close shell. The violent closing leads to the formation of two jets moving vertically in both directions, the upward one piercing and shattering the shell. This phenomenon has been referred to as a 'buckling' instability. A glycerine drop in a similar situation behaves similarly, but displays no capillary ripples because of suppression by the high viscosity.

Smaller drops can stay stable while the ripples become unstable and emit satellite drops in both directions. The satellite formation may be referred to as 'atomization.' It appears that buckling instability and atomization are exclusive of each other, because in the latter too much (surface and kinetic) energy is carried away by the satellite drops

3

such that the former cannot occur. Viscous drops such as a glycerine one can buckle but cannot atomize because there are no ripples.

A still smaller water drop can suddenly expand horizontally to a larger radius and stop. Ripples appear on the stretched membrane, become unstable and emit satellite drops. When the drop size is significantly reduced by the mass loss, the drop stops atomizing and goes through flattening anew. The expansion to a larger radius corresponds to the path 1 in Fig.1, which means the following. When the sound intensity is increased beyond the peak, somehow it has to decrease in order that the drop can stay in equilibrium. With a larger drop, as we have just described, this is achieved easily because of the power loss due to the resonant frequency shift, such that the drop stays close to equilibrium until the end. When the drop is small, it might take a while for it to expand sufficiently to induce the necessary frequency shift. Before that is done the drop is in a transient state along path 1 away from equilibrium. When that is done it returns to the equilibrium curve. After that its membrane might end up being so stretched that it atomizes. It is also possible that the drop misses the equilibrium curve by passing beyond DE when it tries to return, like path 2 in Fig.1, such that the drop continues to expand horizontally and atomize until its final disintegration. For a glycerine drop in the same situation, buckling instability occurs instead.

An even smaller water drop can expand horizontally suddenly to disintegration after some moderate flattening, because it can hardly induce a frequency shift. So after reaching the peak of the &-R* curve, the drop flies off almost parallel to the R*-axis along path 3. At the beginning of the expansion, it has been observed that a knife-edge first appears briefly along the rim. This provides strong evidence that the suction part of the acoustic radiation stress at the rim is responsible for tearing the drop apart, after overcoming surface tension there. A glycerine drop also undergoes sudden horizontal expansion, but displays no knife-edge structure because of the high viscosity.

Figures 2 and 3 show two of the possible modes of the phenomenon of the sudden horizontal expansion. In Fig.2 we show a sequence of the top views of the expansion of such a small water drop. It is noted that a regular wave pattern first appears on the membrane immediately behind the

ring-shaped periphery, before spreading inward throughout the membrane. Then the outer part of the drop, including the ring, breaks up into droplets, leaving the central part of the membrane contracting into a smaller disc.

For a relatively larger drop, as in the sequence in Fig.3, waves might appear throughout the membrane causing atomization, but the membrane is so thin and further deprived by the mass loss, that holes appear in an annular region between the ring-shaped periphery and the center of the membrane. The holes grow and merge into a gap separating the two portions of the drop, both of which then collapse inward.

The work described in this paper was carried out at the Center for Microgravity Research and Applications at Vanderbilt University, under contract with the National Aeronautics and Space Administration.

\J References

1. L.V. King, Proc. R . SOC. London Ser. A 147, 212 (1934). 2. T.G. Wang, Ultrasonics Symposium Proceedings (IEEE, New York, 1979), pp.471- 475. 3. E.H. Trinh, Rev. Sci. instruments 56, 2059 (1985). 4. E.G. Lierke, E.W. Leung, and D. Luhmann, Drops and Bubbles: Third International Colloquium, Monterey, CA 1988. AIP Conference Proceedings 197, ed. T.G. Wang, 1989, pp.71- 80. 5. C.Flammer, Sphefoidal Wave Functions (Stanford University Press, Stanford, Californie, 1957), p.1-55. 6. G.K. Batchelor, An lntroducfion to Fluid Dynamics (Cambridge University Press, 1967), p.411-412. 7. C.P. Lee, A.V. Anilkumar, and T.G. Wang, 'Static Shape and instability of an acoustically levitated liquid drop,' Phys. Fluids (in press). 8. E. Leung, C.P. Lee, N. Jacobi, andT.G. Wang, J. Acoust. SOC. Am. 72, 615 (1982).

4

d’

F i g 2 Sequence of top views of a small water drop Fig 3. A small water drop (kR,=0.32) undergoing (kR,=0.21) undergoing sudden horizontal horizontal expansion. Ripples appear in (c) and expansion. A regular wave pattern first appears in atomization takes place. Holes appear in (d). (b), and develops into violent ripples in (e), behind the ring-shaped periphery. The ring breaks up leaving the membrane in contraction in (1).

5