Persistent holes in a fluid

Persistent holes in a fluid

Florian 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, TX 78712

(Dated: February 12, 2004)

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. Holesare circular and approximately 0.5 cm wide, and can persist for more than 106 cycles. Above a � 17gthe rim of the hole becomes unstable, producing finger-like 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.

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 verticalfinger-like protrusions.

The catalog of interface morphologies in acceleratedfluids is broad and well documented. Sinusoidal accel-eration produces Faraday waves [1], solitons [2, 3], andjets [4]. Impulsive acceleration produces the Richtmyer-Meshkov instability [5, 6], and continuous accelerationthe Rayleigh-Taylor [7] instability, characterized by spiresand bubbles [8]. In contrast, holes and fingers exhibit nei-ther the oscillation about a flat state of Faraday wavesand solitons, the finite lifetime of jets, nor the unboundedgrowth of the Richtmyer-Meshkov or Rayleigh-Taylor in-stabilities.

Figure 1 shows holes in vibrated aqueous suspensionsof cornstarch (a-b) and glass microspheres (d-e). De-pending on the container acceleration and frequency, theinitial fluid surface is either flat or corrugated by Fara-day waves. Above a critical acceleration, a finite localizedperturbation of the fluid surface will grow into a stablecylindrical void that extends from the top to nearly thebottom of the fluid layer. The most noteworthy feature ofholes is their permanence: they do not close despite thehydrostatic pressure of the surrounding fluid, persistingas long as our observations (> 106 cycles). At yet higheraccelerations, the holes lose stability to a temporally andspatially erratic state (see Fig. 1(c) & (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 acceler-ation a up to 27g (controlled to ±0.01g). The containerhad an aluminum baseplate, acrylic sidewall (inner diam-eter 9.4 cm), and acrylic top. The container was sealedto reduce evaporation and was attached to the shakerthrough an insulating rod to avoid heat transfer fromthe shaker to the container. The layer depth was 0.5 cmfor the cornstarch mixture and 0.2 cm for the glass mi-crospheres.

The patterns were recorded with 30 frame/second and

2000 frame/second cameras. Lighting in Fig. 1 was pro-vided by a ring of LEDs 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

FIG. 1: Top view of a vibrated layer of (a-c) cornstarch and of(d-f) glass microspheres in liquids matched in density to theparticles. White corresponds to the highest points, and blackto depressions that reach near the container bottom. Holeswithout Faraday waves: (a) a = 12g, f = 150 Hz; (d) 30g,100 Hz. Holes with Faraday waves: (b) 12g, 60 Hz; (e) 27.3g,92 Hz. Delocalized state: (c) 30g, 120 Hz; (f) 30g, 60 Hz.

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was determined by illuminating a narrow section of thehole with a sheet of laser light projected perpendicular tothe layer surface, and photographing from an angle thedeviations from straightness of the laser line.

We used cornstarch consisting of 27% amylose and 73%amylopectin from Sigma Aldrich. The powder was driedat 50◦C for a week and stored in a desiccator. A mix-ture was prepared daily by combining 23.22 g of corn-starch and 36.78 g of a density-matched aqueous solu-tion of CsCl with a density of 1.68 g/cm3. The quanti-tative 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 µm) fromJaygo (Union, NJ), and 18.36 g of an aqueous solution ofsodium 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 witha puff 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 holemay close within seconds. In the region of the phasediagram (Fig. 2) marked “unstable” all holes decay inthis way. In regions marked “stable” and “metastable”,holes adjust to a size that has a fairly well-defined averagevalue (Fig. 3), but over extended periods of time the sizeslowly wanders within a 30% band. At f = 120 Hz thecharacteristic hole diameter is 0.4 cm; at f = 180 Hz,0.6 cm. Holes have a broad distribution of lifetimes, evenfor fixed f and a; in the metastable phase the maximumlifetime is less than 105 cycles and in the stable phase itis greater than 105 cycles. The value of 105 cycles wasselected as the cutoff for stable holes because holes thatlive that long will almost always live for more than 106

cycles.Short-lived holes collapse within a few seconds by a

uniform contraction. Holes lasting longer than about 104

oscillations develop a hump on their rims, which then fallsonto the hole, either covering it or rendering it so smallthat it collapses rapidly by uniform contraction. On oc-casion 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 throughhole centers are shown in Fig. 4(a). At the base of thehole, there is usually a ribbon of material about 0.1 mmhigh and 2 mm wide that bisects the shaft, as seen in thesolid line in Fig. 4(a). This feature is not visible in allof the radial profiles because it is not azimuthally sym-metric and may be missed in a single profile of the hole.At accelerations immediately above the stability line thematerial surrounding a hole remains level with the rest

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Faraday wave onset ↑↓ Faraday wave extinction

Frequency (Hz)

Acc

eler

atio

n (g

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Delocalized

Metastable

Stable

Unstable

Uns

tabl

e

FIG. 2: Phase diagram for a vibrated aqueous cornstarch sus-pension as a function of acceleration and frequency. In eitherthe metastable or stable region, a surface perturbation willform a persistent hole. In the metastable regime these holesalways collapse within 105 container oscillations; in the stableregime some holes last for more than 105 oscillations. In theareas marked unstable, holes collapse in less than 104 cycles.In the delocalized regime the surface is highly irregular, asshown in Fig. 1(c). Faraday waves appear with increasingacceleration at �, and disappear with decreasing accelerationat �.

of the layer, while at higher acceleration, the hole’s rimrises about 0.5 mm above the surrounding fluid.

In 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 + δr sin (2πft + θ), where r is the mean radius, δr is

0 20 40 60 80 1000

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t (s)

0 10 20 301.8

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2.2

2.4

2.6

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r (m

m)

r (m

m)

FIG. 3: Time evolution of the radii of six holes formed witha 2 s long puff of air. Holes grow during the forcing andthen rapidly shrink after the air is cut off. Inset: hole diame-ter oscillates synchronously with the forcing; note time scale.(a = 15g, f = 150 Hz.)

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the amplitude of the oscillation, and θ is the phase rela-tive to the container motion z(t) = zo sin (2πft), wherezo = −a/(2πf)2 and z > 0 when the container is aboveits rest point. δr is typically 0.15r. The phase lag isshown as a function of frequency in Fig. 4(b).

Holes do not interact when their centers are separatedby more than about two diameters. Therefore, as shownin Fig. 1(a), holes do not form regular patterns. Further,they can be located anywhere in the container. Holesoccasionally come sufficiently close to interact, and thenthey merge 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 existenceof holes when no Faraday waves are present (Fig.1(a)and (d)). Further, the phase boundaries for holes andFaraday waves are distinct. The Faraday transition ishysteretic and, moreover, the flat and surface wave statecan coexist. As a is increased at a fixed f , small patchesof surface waves appear at the boundary “Faraday waveonset” in Fig. 2. As a is raised further, the patches growand ultimately engulf the entire surface at a values typi-cally 50% higher than onset. With decreasing a, the sur-face waves breakup into patches which finally disappearat the boundary “Faraday wave extinction” in Fig. 2.

Delocalization. In the “delocalized” region in Fig. 2,a perturbation will generate a hole which immediately

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FIG. 4: (a) Vertical cross sections of holes at a=14.5 g (dashedand dotted lines) and a=17 g (solid and dotted-dashed lines).The two profiles for each acceleration show the extremes in ahole’s shape, separated by one-half cycle. (b) The tangent ofthe phase lag versus frequency at a = 15g. The solid line is afit to α/(1− (f/fo)

2), where α and fo are fitting parameters.(f=150 Hz)

FIG. 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 ini-tial hump on the rim begins growing upward, reaches a maxi-mum height, and then topples outward, enlarging the area offluid motion. This process repeats until the entire surface ofthe liquid is active in the creation and destruction of verticalstructures and voids (see Fig. 1(c)). (a =25g, f = 80 Hz)

grows a protrusion from its rim, as Fig. 5 illustrates. Theprotrusions can rise as high as 2 cm and remain uprightfor thousands of oscillations before falling and nucleatinga new hole. The process continues until the entire surfacewrithes with fingers and holes, yielding the spatially andtemporally erratic state shown in Fig. 1(c) and (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 [10], i.e., an initial decrease of the viscos-ity for increasing shear rate is followed by a rapid increaseat a critical shear rate, which in this case is γc = 8 s−1.Our conclusion is predicated on four observations: thecritical shear rate is similar in magnitude to the shearrate at the throat of a hole; the phase lag of the holesradius relative to the driving force indicates the mate-rial response is primarily viscous; holes are unstable in anon-shear thickening fluid; and holes are stable in shearthickening fluids other than cornstarch suspensions.

We can estimate the shear rate for an oscillating hole(inset of Fig 3). The root-mean-square shear rate atthroat of the hole is approximately the interface speed,(2π/

√2)δrf , divided by a length scale of order the depth

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101

100

101

10210

3

104

105

Maximum shear rate (Hz)

Vis

cosi

ty (

cent

ipoi

se)

shear thickening

shear thinning .γc

FIG. 6: Apparent viscosity as a function of the maximumshear rate γ for two cornstarch suspensions prepared in thesame manner as in Fig. 1. Measurements were made with aplate-plate rheometer (Paar Physica TEK 150) with radiusR = 2.5 cm and gap 0.1 cm. The apparent viscosity wascalculated as for a Newtonian fluid: η = 2τ/(πγR3). Sincethe shear rate increases with the radius, our measurement ofthe viscosity is a convolution of the geometry and the responseof the fluid. The dotted line represents measured values thatare not reliable due to an instability in the flow. (temperature,27.6 ± 0.2 ◦C.)

of the layer h. Using δr/r � 0.15, f = 100 Hz, andr/h � 0.4, we obtain γ � 27 s−1. The similarity ofthis value to the critical shear rate strongly suggests thatshear thickening is an essential ingredient for holes.

The frequency dependence of the phase lag θ indi-cates a combined viscous and elastic response of the fluid.Modelling the hole motion as a spring in parallel with adashpot (i.e., a Voigt element, see [9] for example) yieldstan θ = α/(1 − (f/fo)2), where α is a constant propor-tional to the dissipation and fo is the resonant frequency.As shown in Fig. 4(b), the data are well modelled by thisequation; the resonance frequency is around 58 Hz. Sincethe Voigt element is primarily viscous at high frequency,the phase data indicates that at frequencies above 60 Hzthe material response becomes dominantly viscous; it isnoteworthy that stable holes only form in this high fre-quency region.

We also tried to form persistent holes in a Newtonianfluid (silicone oil with a viscosity of 3.4 Poise) and vis-coelastic fluids (polybutadienes with viscosities of 8.3, 28,and 72 Poise). In all cases, however, an initial hole wasbackfilled within a few hundred container oscillations.

Barnes [10] 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 support persis-tent holes. Indeed, we conducted experiments on a densesuspension of glass microspheres and found the hole anddelocalized states, as shown in Fig. 1(d) & (e).

Conclusions. We have shown that a vertically vi-brated aqueous suspension of cornstarch displays a num-ber of unexpected patterns–holes, fingers, and a delocal-ized state–that can only be attained by the application ofa finite perturbation. Holes vibrate synchronously withthe container and are unrelated to the well-known Fara-day waves. At higher acceleration the hole becomes un-stable to the creation of finger-like structures, which ateven higher accelerations are responsible for the transi-tion to a irregular delocalized state. Outstanding ques-tions include how do holes stay open, what sets their size,how can the finger like protrusions grow and remain up-right, and what drives the transition to the delocalizedstate?

We thank M. Schroter, 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 numberDE-FG03-93ER14312. FSM acknowledges support bythe Friedrich Ebert Stiftung.

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