We observe stable holes in a vertically oscillated 0.5 cm deep aqueous
suspension of cornstarch for accelerations a above 10g. Holes appear only if a
finite perturbation is applied to the layer. Holes are circular and
approximately 0.5 cm wide, and can persist for more than 10^5 cycles. Above a =
17g the rim of the hole becomes unstable producing finger-like protrusions or
hole division. At higher acceleration, the hole delocalizes, growing to cover
the entire surface with erratic undulations. We find similar behavior in an
aqueous suspension of glass microspheres.Comment: 4 pages, 6 figure

Daniel I. Goldman

E. E. Meshkov

Erin C. Rericha

Florian S. Merkt

G. I. Taylor

Harry L. Swinney

J. D. Ferry

M. Faraday

M. S. Longuet-Higgins

R. D. Richtmyer

Robert D. Deegan

English

arXiv

We observe stable holes in a vertically oscillated 0.5 cm deep aqueous suspension of cornstarch. Holes appear only if a finite perturbation is applied to the layer for accelerations a above 10g. Holes are circular and approximately 0.5 cm wide, and can persist for more than 10(6) cycles. Above asimilar or equal to17g the rim of the hole becomes unstable, producing fingerlike protrusions or hole division. At higher acceleration, the hole delocalizes, growing to cover the entire surface with erratic undulations. We find similar behavior in an aqueous suspension of glass microspheres.We observe stable holes in a vertically oscillated 0.5 cm deep aqueous suspension of cornstarch. Holes appear only if a finite perturbation is applied to the layer for accelerations a above 10g. Holes are circular and approximately 0.5 cm wide, and can persist for more than 10(6) cycles. Above asimilar or equal to17g the rim of the hole becomes unstable, producing fingerlike protrusions or hole division. At higher acceleration, the hole delocalizes, growing to cover the entire surface with erratic undulations. We find similar behavior in an aqueous suspension of glass microspheres

Merkt, FS

Deegan, RD

Goldman, DI

Rericha, EC

Swinney, HL

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Persistent holes in a fluid

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Persistent Holes in a Fluid

© 2004 The American Physical SocietyThe electronic version of this article is the complete one and can be found online at: http://link.aps.org/doi/10.1103/PhysRevLett.92.184501DOI: 10.1103/PhysRevLett.92.184501We observe stable holes in a vertically oscillated 0.5 cm deep aqueous suspension of cornstarch. Holes appear only if a finite perturbation is applied to the layer for accelerations α above 10g. Holes are circular and approximately 0.5 cm wide, and can persist for more than 10⁶ cycles. Above α ≃ 17g the rim of the hole becomes unstable, producing fingerlike protrusions or hole division. At higher acceleration, the hole delocalizes, growing to cover the entire surface with erratic undulations. We find similar behavior in an aqueous suspension of glass microspheres

Merkt, Florian S.

Deegan, Robert D.

Goldman, Daniel I.

Rericha, Erin C.

Swinney, Harry L.

Scholarly Materials And Research @ Georgia Tech

P H Y S I C A L R E V I E W L E T T E R S week ending7 MAY 2004VOLUME 92, NUMBER 18Persistent Holes in a FluidFlorian S. Merkt, Robert D. Deegan, Daniel I. Goldman, Erin C. Rericha, and Harry L. SwinneyCenter for Nonlinear Dynamics, The University of Texas at Austin, Austin, Texas 78712, USA(Received 10 November 2003; published 5 May 2004)184501-1We observe stable holes in a vertically oscillated 0.5 cm deep aqueous suspension of cornstarch. Holesappear only if a finite perturbation is applied to the layer for accelerations a above 10g. Holes arecircular and approximately 0.5 cm wide, and can persist for more than 106 cycles. Above a ’ 17g therim of the hole becomes unstable, producing fingerlike protrusions or hole division. At higheracceleration, the hole delocalizes, growing to cover the entire surface with erratic undulations. Wefind similar behavior in an aqueous suspension of glass microspheres.DOI: 10.1103/PhysRevLett.92.184501 PACS numbers: 47.50.+d, 47.20.–k, 47.54.+r, 47.55.KfFIG. 1. Top view of a vibrated layer of an aqueous suspensionof (a)–(c) cornstarch and of (d)–(f) glass microspheres (imagediameter, 9.4 cm). White corresponds to the highest points, andblack to depressions that reach near the container bottom.Holes without Faraday waves: (a) a12g, f150Hz; (d) 30g,through an insulating rod to avoid heat transfer fromthe shaker to the container. The layer depth was 0.5 cm100 Hz. Holes with Faraday waves: (b) 12g, 60 Hz; (e) 27:3g,92 Hz. Delocalized state: (c) 30g, 120 Hz; (f) 30g, 60 Hz.The free surface of a fluid at rest in a container is flat.Departures from flatness induce a restoring flow whetherthe fluid is Newtonian, viscoelastic, or liquid crystalline:poke the surface, and the resulting indentation will befilled by the ensuing flow. In contrast, we have discoveredthat a vibrated aqueous suspension of cornstarch or glassmicrospheres can permanently support holes and verticalfingerlike protrusions.The catalog of interface morphologies in acceleratedfluids is broad and well documented. Sinusoidal accelera-tion produces Faraday waves [1], solitons [2,3], and jets[4]. Impulsive acceleration produces the Richtmyer-Meshkov instability [5,6], and continuous accelerationthe Rayleigh-Taylor [7] instability, characterized byspires and bubbles [8]. In contrast, holes and fingers donot exhibit the oscillation about a flat state of Faradaywaves and solitons, the finite lifetime of jets, or theunbounded growth of the Richtmyer-Meshkov orRayleigh-Taylor instabilities.Figure 1 shows holes in vibrated aqueous suspensions ofcornstarch 1(a) and 1(b) and glass microspheres 1(d) and1(e). Depending on the container acceleration and fre-quency, the initial fluid surface is either flat or corrugatedby Faraday waves. Above a critical acceleration, a finitelocalized perturbation of the fluid surface grows into astable cylindrical void that extends from the top to nearlythe bottom of the fluid layer. The most noteworthy featureof holes is their permanence: they do not close despite thehydrostatic pressure of the surrounding fluid, persisting aslong as our observations (>106 cycles). At yet higheraccelerations, the holes lose stability to a temporallyand spatially erratic state [see Figs. 1(c) and 1(f)].Experiment.—A layer of cornstarch or glass micro-spheres in liquid was vertically oscillated sinusoidallywith a frequency f from 50 to 180 Hz and a peak accel-eration a up to 27g (controlled to 0:01g). The containerhad an aluminum base plate, acrylic sidewall (inner di-ameter 9.4 cm), and acrylic top. The container was sealedto reduce evaporation and was attached to the shaker0031-9007=04=92(18)=184501(4)$22.50 for the cornstarch mixture and 0.2 cm for the glassmicrospheres.The patterns were recorded with 30 and 2000 frames/second cameras. Lighting in Fig. 1 was provided by a ring2004 The American Physical Society 184501-1P H Y S I C A L R E V I E W L E T T E R S week ending7 MAY 2004VOLUME 92, NUMBER 18of light-emitting diodes strobed at f or f=2. The illumi-nation was such that peaks on the surface appear brightand valleys dark. The layer was lit from above for themeasurements of the size of the holes. The hole shape wasdetermined by illuminating a narrow section of the holewith a sheet of laser light projected perpendicular to thelayer surface, and photographing from an angle the devi-ations from straightness of the laser line.We used cornstarch consisting of 27% amylose and73% amylopectin from Sigma Aldrich. The powder wasdried at 50 C for a week and stored in a desiccator. Amixture was prepared daily by combining 23.22 g ofcornstarch and 36.78 g of a density-matched aqueoussolution of CsCl with a density of 1:68 g=cm3. The quan-titative results we present are for cornstarch, but we alsoused glass microspheres in an aqueous solution for thequalitative comparison in Fig. 1. The glass mixture wasprepared with 29.37 g of glass balls (1–20 m) fromJaygo (Union, NJ), and 18.36 g of an aqueous solutionof sodium polytungstate with a density of 2:55 g=cm3.Hole dynamics and size selection.—The interface of avibrated fluid is flat at low accelerations and rippled byFaraday waves above a critical acceleration [1]. Holespersist in the cornstarch mixture only if an indentationdeeper than about 50% the layer depth and 0.4–2.0 cmwide is applied to the surface. Holes were generated with apuff of air through a nozzle pointed at the surface. Thesubsequent evolution of a hole depends on the parametersa and f. For all parameter values explored, the hole may60 80 100 120 140 160 1800510152025Faraday wave onset ↑↓ Faraday wave extinction Frequency (Hz)Acceleration (g)DelocalizedMetastableStableUnstableUnstableFIG. 2. Phase diagram for a vibrated aqueous cornstarchsuspension as a function of acceleration and frequency. In eitherthe metastable or stable region, a surface perturbation forms apersistent hole. In the metastable regime these holes alwayscollapse within 105 container oscillations; in the stable regimesome holes last for more than 105 oscillations. In the areasmarked unstable, holes collapse in less than 104 cycles. In thedelocalized regime the surface is highly irregular, as shown inFig. 1(c). Faraday waves appear with increasing acceleration at, and disappear with decreasing acceleration at .184501-2close within seconds. In the region of the phase diagram(Fig. 2) marked ‘‘unstable’’ all holes decay in this way. Inregions marked ‘‘stable’’ and ‘‘metastable,’’ holes adjustto a size that has a fairly well-defined average value(Fig. 3), but over extended periods of time the size slowlywanders within a 30% band. At f  120 Hz the charac-teristic hole diameter is 0.4 cm, and at f  180 Hz,0.6 cm. Holes have a broad distribution of lifetimes,even for fixed f and a; in the metastable phase themaximum lifetime is less than 105 cycles, and in thestable phase it is greater than 105 cycles. The value of105 cycles was selected as the cutoff for stable holesbecause holes that live that long will almost always livefor more than 106 cycles.Short-lived holes collapse within a few seconds by auniform contraction. Holes lasting longer than about 104oscillations develop a hump on their rims, which thenfalls onto the hole, either covering it or rendering it sosmall that it collapses rapidly by uniform contraction. Onoccasion this latter mechanism causes the hole to dividerather than collapse.Holes typically have a nearly circular horizontal crosssection [see Fig. 1(a)].Vertical cross sections through holecenters are shown in Fig. 4(a). At the base of the hole,there is usually a ribbon of material about 0.1 mm highand 2 mm wide that bisects the shaft, as seen in the solidline in Fig. 4(a). This feature is not visible in all of theradial profiles because it is not azimuthally symmetricand may be missed in a single profile of the hole. Ataccelerations immediately above the stability line thematerial surrounding a hole remains level with the restof the layer, while at higher acceleration, the hole’s rimrises about 0.5 mm above the surrounding fluid.0 20 40 60 80 1000123456t (s)0 10 20 301.82.02.22.42.6t (ms)r (mm)r (mm) FIG. 3. Time evolution of the radii of six holes formed with a2 s long puff of air. Holes grow during the forcing and thenrapidly shrink after the air is cut off. Inset: hole diameteroscillates synchronously with the forcing; note time scale (a 15g, f  150 Hz).184501-2FIG. 5. Side view of the first steps toward the delocalizedstate in cornstarch. These photographs were taken every 0.9 s;time increases from left to right and top to bottom. An initialhump on the rim begins growing upward, reaches a maximumheight, and then topples outward, enlarging the area of fluidmotion. This process repeats until the entire surface of theliquid is active in the creation and destruction of verticalstructures and voids [see Fig. 1(c)] (a  25g, f  80 Hz).40 60 80 100 120 140-8-4048f (Hz)tan(θ)(b)160depth (mm)r (mm)-5.0 -2.5 0 2.5 5.0432105r(a) rFIG. 4. (a) Vertical cross sections of holes at a  14:5g(dashed and dotted lines) and a  17g (solid and dot-dashedlines) for f  150 Hz. The two profiles for each accelerationshow the extremes in a hole’s shape, separated by one-halfcycle. (b) The tangent of the phase lag versus frequency at a 15g. The solid line is a fit to =	1 f=f02, where  and f0are fitting parameters.P H Y S I C A L R E V I E W L E T T E R S week ending7 MAY 2004VOLUME 92, NUMBER 18In addition to a slow variation over hundreds of oscil-lations, the radius rt of a hole oscillates synchronouslywith the driving frequency (inset of Fig. 3): rt  rr sin2ft , where r is the mean radius, r is theamplitude of the oscillation, and  is the phase relative tothe container motion zt  z0 sin2ft, where z0 a=2f2 and z > 0 when the container is above itsrest point. r is typically 0:15r. The phase lag is shown asa function of frequency in Fig. 4(b).Holes do not interact when their centers are separatedby more than about two diameters. Therefore, as shown inFig. 1(a), holes do not form regular patterns. Further, theycan be located anywhere in the container. Holes occa-sionally come sufficiently close to interact, and then theymerge or annihilate.Faraday waves.—Though Faraday waves are excited incornstarch suspensions, the formation of holes is unre-lated to these waves, as can be seen from the existence ofholes when no Faraday waves are present [Figs. 1(a) and1(d)]. Further, the phase boundaries for holes and Faradaywaves are distinct. The Faraday transition is hysteretic,and, moreover, the flat and surface wave state can coexist.As a is increased at a fixed f, small patches of surfacewaves appear at the boundary ‘‘Faraday wave onset’’ inFig. 2. As a is raised further, the patches grow and184501-3ultimately engulf the entire surface at a values typically50% higher than onset. With decreasing a, the surfacewaves break up into patches that finally disappear at theboundary ‘‘Faraday wave extinction’’ in Fig. 2.Delocalization.—In the ‘‘delocalized’’ region in Fig. 2,a perturbation generates a hole that immediately grows aprotrusion from its rim, as Fig. 5 illustrates. The protru-sions can rise as high as 2 cm and remain upright forthousands of oscillations before falling and nucleating anew hole. The process continues until the entire surfacewrithes with fingers and holes, yielding the spatially andtemporally erratic state shown in Figs. 1(c) and 1(f). Thetransition from the metastable hole region to the delocal-ized state is not hysteretic. If a is decreased through thevalue for delocalization onset, the erratic motion ceasesand the irregular pattern imprinted by the delocalizedstate evolves according to hole dynamics.Shear thickening.—We attribute the stability of holesto shear thickening, i.e., an increase of viscosity withshear rate. We measured the viscosity of the cornstarchsuspension (Fig. 6) and found a behavior typical of shearthickening fluids [9]; i.e., an initial decrease of the vis-cosity for increasing shear rate is followed by a rapidincrease at a critical shear rate, which in this case is _c 8 s1. Our conclusion is predicated on four observations:the critical shear rate is similar in magnitude to the shearrate at the throat of a hole; the phase lag of the hole’sradius relative to the driving force indicates the materialresponse is primarily viscous; holes are unstable in anonshear thickening fluid; and holes are stable in shearthickening fluids other than cornstarch suspensions.184501-310−1100101102103104105Maximum shear rate (Hz)Viscosity (centipoise)shear thickeningshear thinning .γc FIG. 6. Apparent viscosity as a function of the maximumshear rate _ for two cornstarch suspensions prepared in thesame manner as in Fig. 1. Measurements were made with aplate-plate rheometer (Paar Physica TEK 150) with radius R 2:5 cm and gap 0.1 cm. The apparent viscosity was calculated asfor a Newtonian fluid:   2= _R3 where  is the torque.Since the shear rate increases with the radius, our measurementof the viscosity is a convolution of the geometry and theresponse of the fluid. The dashed extension of the curverepresents measured values that are not reliable due to aninstability in the flow (temperature, 27:6 0:2 C).P H Y S I C A L R E V I E W L E T T E R S week ending7 MAY 2004VOLUME 92, NUMBER 18We can estimate the shear rate for an oscillating hole(inset of Fig. 3). The root-mean-square shear rate at throatof the hole is approximately the interface speed,2=2prf, divided by a length scale of order the depthof the layer h. Using r=r  0:15, f  100 Hz, andr=h  0:4, we obtain _  27 s1. The similarity of thisvalue to the critical shear rate strongly suggests that shearthickening is an essential ingredient for holes.The frequency dependence of the phase lag  indicatesa combined viscous and elastic response of the fluid.Modeling the hole motion as a spring in parallel with adashpot (i.e., a Voigt element; see [10], for example)yields tan  =	1 f=f02, where  is a constantproportional to the dissipation and f0 is the resonantfrequency. As shown in Fig. 4(b), the data are wellmodeled by this equation; the resonance frequency isaround 58 Hz. Since the Voigt element is primarily vis-cous at high frequency, the phase data indicate that atfrequencies above 60 Hz the material response becomesdominantly viscous; it is noteworthy that stable holesform only in this high frequency region.We also tried to form persistent holes in a Newtonianfluid (silicone oil with a viscosity of 3.4 P) and viscoelas-184501-4tic fluids (polybutadienes with viscosities of 8.3, 28, and72 P). In all cases, however, an initial hole was backfilledwithin a few hundred container oscillations.Barnes [9] states that any sufficiently concentratedsuspension of solids in a fluid will shear thicken. If shearthickening is crucial for hole formation, then any suffi-ciently dense suspension of solids ought to supportpersistent holes. Indeed, we conducted experiments on adense suspension of glass microspheres and found the holeand delocalized states, as shown in Figs. 1(d) and 1(e).Conclusions.—We have shown that a vertically vi-brated aqueous suspension of cornstarch displays anumber of unexpected patterns — holes, fingers, and adelocalized state — that can be attained only by theapplication of a finite perturbation. Holes vibrate syn-chronously with the container and are unrelated to thewell-known Faraday waves. At higher acceleration thehole becomes unstable to the creation of fingerlike struc-tures, which at even higher accelerations are respon-sible for the transition to an irregular delocalized state.Outstanding questions include the following: How doholes stay open? What sets their size? How can the finger-like protrusions grow and remain upright? What drivesthe transition to the delocalized state?We thank M. Schröter, W. D. McCormick, P. Heil,A. Lee, W. Zhang for valuable suggestions, and P. Greenand A. Jamieson for assistance with the viscosity mea-surements. We acknowledge support by the EngineeringResearch Program of the Office of Basic Energy Sciencesof the U.S. Department of Energy under Grant No. DE-FG03-93ER14312. F. S. M. acknowledges support by theFriedrich Ebert Stiftung.[1] M. Faraday, Philos. Trans. R. Soc. London 121, 299(1831).[2] O. Lioubashevski, H. Arbell, and J. Fineberg, Phys. Rev.Lett. 76, 3959 (1996).[3] O. Lioubashevski, Y. Hamiel, A. Agnon, Z. Reches, andJ. Fineberg, Phys. Rev. Lett. 83, 3190 (1999).[4] M. S. Longuet-Higgins, J. Fluid Mech. 127, 103 (1983).[5] R. D. Richtmyer, Commun. Pure Appl. Math. 13, 297(1960).[6] E. E. Meshkov, Sov. Fluid Dyn. 4, 101 (1969).[7] G. I. Taylor, Proc. R. Soc. London A 201, 192 (1950).[8] U. Alon, J. Hecht, D. Ofer, and D. Shvarts, Phys. Rev.Lett. 74, 534 (1995).[9] H. A. Barnes, J. Rheol. 33, 329 (1989).[10] J. D. Ferry, Viscoelastic Properties of Polymers (Wiley,New York, 1961).184501-4

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Persistent holes in a fluid

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