Hole dynamics in vertically vibrated liquids and suspensions
aa r X i v : . [ c ond - m a t . s o f t ] D ec Hole dynamics in vertically vibrated liquids and suspensions
Stefan von Kann, Matthias van de Raa, and Devaraj van der Meer
Physics of Fluids group, University of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands (Dated: July 11, 2018)We study the dynamics of holes created in vertically vibrated dense suspensions and viscousNewtonian liquids. We find that all holes oscillate with the driving frequency, with a phase shiftof π/
2. In Newtonian liquids holes always close, while in suspensions holes may grow in time. Wepresent a lubrication model for the closure of holes which is in good agreement with the experimentsin Newtonian liquids. The growth rate of growing holes in suspensions is found to scale with theparticle diameter over the suspending liquid viscosity. Comparing closing holes in Newtonian liquidsto growing holes in dense suspensions we find a sinusoidal, linear response in the first, and a highlynon-linear one in the latter. Moreover, the symmetry of the oscillation is broken and is shown toprovide an explanation for the observation that holes in dense suspensions can grow.
PACS numbers: 82.70.Kj, 45.70.Vn, 47.50.-d
I. INTRODUCTION
When a hole is created in a horizontal layer of (viscous)liquid at rest, the hydrostatic pressure will cause the holeto close. And, in spite of its more complicated rheology,the same thing is expected to happen in a non-Newtonianliquid. Recently however, the reverse has been shown tooccur in experiments where layers of various particulatesuspensions and emulsions were subjected to vertical vi-brations: Holes created in these vibrated liquids do notnecessarily close, but may stabilize [1, 2], grow [3], or leadto chaotic dynamics [4, 5]. Although phenomenologicalmodels are suggested in the literature [3, 6] our under-standing of this behavior is far from complete. In this pa-per we will shed light onto its dynamics by investigatingthe analogies and differences between vertically vibratedviscous Newtonian fluids and a suspension of monodis-perse particles in liquids with the viscosity of water andhigher.A concentrated particulate suspension consists of amixture of a homogeneous liquid and particles that arelarge enough ( > µm ) such that their Brownian motionis negligible. They can be found in many places, rangingfrom quicksand, through freshly mixed cement and paintsto the inside of flexible armor suits. Their flow is impor-tant in nature, industry and even health care [7]. In spiteof their common presence and significance, many aspectsof the flow of these dense suspensions remain poorly un-derstood. In order to study these materials people haveused methods inspired by classical rheology, and typi-cally characterized them in terms of a constitutive rela-tion of stress versus shear rate [8–13]. A general resultis that, when increasing the shear rate, dense suspen-sions first tend to become less viscous (shear thinning)and subsequently shear thicken. In recent experimentspeople found mesoscopic length scales [12, 14], fractur-ing [15], and a dynamic jamming point [10] to be im-portant in such suspensions. Connected to the above,normal stress divergence in the approach to a wall [16]and non-monotonic settling [17] have been reported forobjects moving through dense cornstarch suspensions. Turning to vertically vibrated suspensions, Merkt etal. [1] observed in a vertically shaken, thin layer ofcornstarch suspension that –amongst other quite exoticphenomena– stable oscillating holes can be formed forcertain values of the shaking parameters [1]. These stableholes were subsequently described using a phenomenolog-ical model based on a hysteretic constitutive equation [6].In other particulate suspensions, Ebata et al. foundgrowing and splitting holes and a separated state [3, 4],where the latter is attributed to a convective flow in therim and the first are still not understood. Stable holesand kinks (which appear to be similar to or even identicalto the separated state mentioned above) have also beenreported in emulsions [2]. At present we are still far froma detailed understanding of dense suspensions, and whydifferent suspension behave differently.Here, we will investigate the dynamics of opening holesin a layer of vibrated suspension of monodisperse parti-cles of various sizes suspended in a glycerol-water mix-ture. We will investigate how this dynamics depends onparticle size and viscosity and will compare it to the dy-namics of closing holes in a layer of vertically shaken vis-cous Newtonian liquids, for which we will present a modelwithin the lubrication approximation. We will then shedlight upon how the differences arise and in what mannerthese can explain the observation that the holes in thesuspension do not close as a result of hydrostatic pres-sure.The paper is organized as follows: We will start with ashort description of our setup in Section II. After this wewill present experiments for the dynamics of closing holesin a (vibrated) layer of a viscous Newtonian liquid (Sec-tion III A), followed by the introduction and discussion ofa lubrication model for this system (Section III B). Sub-sequently, in Section IV we turn to the dynamics of open-ing holes in vibrated particle suspensions and discuss thesimilarities and differences with the closing holes. Thepaper will be concluded in Section V. f = 20 - 200 HzΓ = 0 - 60 D = 11 cm H = 8 cm h = 0.4 - 2 cm FIG. 1: A schematic view of the used setup. At the lower endwe have the shaker, on top of which with the container withthe suspension is mounted, which is subsequently vibratedvertically. Above that is the high speed camera, recordingthe suspension from above.
II. EXPERIMENTAL SETUP
The experimental setup is shown in Fig. 1. Thecore consists of a cylindrical container with a diameter D = 11 . H of 8 . f between 20 and 200 Hz and a dimensionlessacceleration Γ from 0 up to 60. Here, Γ = a (2 πf ) /g ,where a is the shaking amplitude and g the gravita-tional acceleration. The container is filled up to a height h = 6 . ± . ,
000 frames persecond (fps), and is imaged from the top. The bottomof the container was covered with an adhesive sheet forimproved contrast between liquid and container bottom.When using transparent liquids, a small amount of pow-dered milk was added to whiten the liquid. Of course itwas checked that adding adhesive sheet or milk powderdid not influence the dynamics of the system.
III. VISCOUS NEWTONIAN LIQUIDS
Before turning to the –anomalous– opening holes indense suspensions consisting of monodisperse particlesin a mixture of glycerine and water, we will first studythe regular case of holes closing in a viscous Newtonianliquid. We will both discuss the case where the holes closepurely due to the hydrostatic pressure in the liquid andthe case in which a periodic forcing is added by vibratingthe system vertically. In the second subsection we willsubsequently present a model to describe both cases. d ( m ) t (s) ExperimentModel
FIG. 2: The diameter of a closing hole in a layer of honey( µ = 6 . · s) with a thickness of h = 6 ± f = 50 Hz,Γ = 30 and recorded at a framerate of 250 fps (blue line).The black line is the result of a calculation using the lubrica-tion model. The inset is from an experiment using the sameshaking parameters, but twice the recording speed (500 fps). A. Experiment
We prepare a layer with a thickness of h = 6 . ± . µ = 6 . · s,and several glycerine-water mixtures with viscosities of µ = 1 .
30, 1 .
10, 0 .
45, and 0 .
15 Pa · s. Viscosities belowthe last value lead to holes that close extremely fast; inparticular they were found to close within a single cycleof the lowest driving frequency we have used in our study( f = 20 Hz). Moreover, for these low viscosities inertialeffects will start to become important and therefore suchfluids were not considered here.Fig. 2 provides a typical experimental result for a h = 6mm thick layer of honey, vibrated at f = 50 Hz, Γ = 30.After creating a circular hole in the layer, we follow thedynamics of its closing and plot the hole diameter as afunction of time. Over the course of several seconds –i.e.,on time scales larger than the period of the driving– thehole is observed to close in an almost linear manner. Ontop of this the hole oscillates at the same frequency asthat of the driving, which is shown in the inset wherepart of the signal has been magnified in time.When changing the shaking parameters f and Γ, itbecomes clear that the closing time is to a large extentindependent on f and Γ, as is shown in Fig. 3 where weshow results obtained in glycerine. In particular, whenwe do not shake at all and just create a hole in the con-tainer at rest and observe its closing, we find that itstime evolution follows the very same trend. The ampli-tude of the oscillation increases more or less linearly withthe shaking acceleration Γ and is in fact of the same or-der as the shaking amplitude a = Γ g/ (2 πf ) . The latter −1 −0.5 000.010.020.030.040.05 t (s) d ( m ) −1 −0.5 000.010.020.030.040.05 t (s ) d ( m ) no shaking50Hz, Γ =1050Hz, Γ =2050Hz, Γ =30 no shaking50Hz, Γ =10200Hz, Γ =4020Hz, Γ =5 (a) (b) FIG. 3: Time evolution of the hole diameter in glycerine ( µ =1 . · s): (a) For a constant shaking frequency f = 50 Hzand different values of the shaking acceleration Γ = 0 (noshaking, blue line), Γ = 10 (red line), Γ = 20 (magenta line),and Γ = 30 (black line). (b) For varying frequency: f = 0Hz, Γ = 0 [no shaking, blue line, as in (a)]; f = 20 Hz, Γ = 5(black line); f = 50 Hz, Γ = 10 [red line, as in (a)]; and f = 200 Hz, Γ = 40 (magenta line). t (s) d ( m ) µ =6.4 Pa s µ =1.3 Pa s µ =1.1 Pa s µ =0.45 Pa s µ = 0.15 Pa s FIG. 4: Time evolution of the hole diameter in a h = 6 mmthick layer of liquid of varying viscosity µ , vibrated at f =50 Hz and Γ = 10. The solid black lines denote the timeevolution according to the model discussed in Section III B. observation also explains why the amplitude of the oscil-lation decreases so much when the frequency is raised to f = 200 Hz, which causes the shaking amplitude to godown by a factor 16. Moreover, the amplitude appearsto be independent of the hole size, i.e., the amplituderemains largely constant while the hole diameter shrinksdown to zero.In Fig. 4 we compare results for the different liquidviscosities, shaken at f = 50 Hz and Γ = 10. For thelowest viscosity ( µ = 0 .
15 Pa · s ) we observe that theholes closes in less than a tenth of a second, i.e., within afew cycles of the driving. When we increase the viscositythe closing time increases rapidly, and for the highestviscosity (that of honey, µ = 6 . · s) the closing time isover six seconds.In the same Figure we observe that there is a signifi-cant span of time in which the average closure velocityappears to be linear. This allows us to correct the signalby subtracting this linear behavior and afterwards com- −3 t (s) d ( m ) ∆ ψ container positionhole size FIG. 5: Comparison of a sine fit of the trajectory of the ver-tical position of the container (vibrated at f = 20 Hz andΓ = 5), and the trajectory of diameter of a closing hole in a h = 6 mm layer of glycerine ( µ = 1 . · s), corrected linearlyfor the average closing velocity of the hole (see text). Theinset shows the phase difference ∆ φ for every period shownin the main Figure. pare it to the vertical position of the container. This isdone in Fig. 5, where we zoom in on a few cycles only.There is a clear phase shift between the driving and thehole, which is measured to be approximately a quarter ofa period, as shown in the inset of Fig. 5. The fact that thehorizontal oscillation of the hole lags behind ∆ φ = π/ B. Modeling
To model the dynamics of closing holes in a viscousNewtonian liquid we use axisymmetric lubrication the- −2−10123x 10 −3 t (s) d ( m ) ˙ d ( m / s ) t (s) FIG. 6: (a) Superposition of several cycles of the correctedtrajectory of the diameter d ( t ) of a closing hole shifted overan integer number of periods of the driving. The (almostcompletely overlapping) black and yellow curves indicate theaverage position and a sinusoidal fit to the average respec-tively. (b) The instantaneous velocity ˙ d ( t ) of the closing holeaveraged over all cycles (red + symbols). The solid (blue)line shows the derivative of the sine fit of (a). Taken from anexperiment with glycerine ( µ = 1 . · s, h = 6mm, f = 50Hz, Γ = 20). ory. In absence of the driving, the equation of motion forthe liquid profile h ( r, t ) can be derived from continuityand a lubrication ansatz for the velocity profile withinthe layer (See Appendix A [19]). ∂h∂t = ρg µr ∂∂r (cid:20) rh ∂h∂r (cid:21) , (1)where r is the radial coordinate, g the acceleration ofgravity, ρ the density, and µ the dynamic viscosity ofthe liquid. Using lubrication theory implies neglectinginertial effects. In particular this means that the closingvelocity can be derived from an (instantaneous) balanceof the gravitational force drives the closing and the vis-cous forces that counteract it, i.e. ρg ∼ µ ˙ dh ⇒ ˙ d ∼ ρgh µ , (2)in which ˙ d denotes the time derivative of the hole di-ameter and we have estimated the viscous forces in thelayer, µ ∂ u/∂z as the velocity of the rim ˙ d divided bythe squared initial layer thickness h . From this simplebalance it follows that the closing velocity should scaleas 1 /µ . If we check this for our experimental results byplotting the closing velocity ˙ d (determined from the lin-ear regime of plots as in Fig. 4) as a function of viscosity µ Fig. 7 we find a very good agreement. Remarkable isthat the plot does not only contain data without driving,but also with various driving strengths.When we assume an infinite layer of liquid, we canderive a semi-analytical self-similar solution to the closinghole problem which has the form d ( t ) = 2 η s ρgh µ ( t c − t ) , (3) d −1 −3 −2 −1 µ (Pa s) ˙ c l o s e ( m / s ) no shaking10g 50Hz20g 50Hz0.05/μ FIG. 7: Average closing velocity as a function of viscosity ina double logarithmic plot, for three different values for thedriving (no shaking; f = 50 Hz and Γ = 10; and f = 50 Hzand Γ = 20. The blue solid line is | ˙ d close | ≈ . /µ , makingthe proportionality constant in Eq. (2) equal to 0 . where η is a numerical constant and t c is the time thehole needs to close. In our case these can be thought ofas fixed by the initial hole size together with the bound-ary conditions at the sidewalls of our container. Thisself-similar solution goes to zero with a square-root de-pendence on time which is however –maybe with the ex-ception of the very end– not observable in our experi-ments (Figs. 2, 3 and 4), which is presumably connectedto the proximity of the side walls. We therefore de-cided to numerically solve Eq. (1), supplemented with R D/ h ( r, t ) rdr = constant, which expresses the conser-vation of liquid in our system.In Figs. 2 and 4 we compare our model results to theexperiments and find that the behavior is well capturedby the model.We can adapt Eq. (1) to model the modulation due tothe acceleration of the shaker as well, by simply substi-tuting g (1 + Γ sin ωt ) for g , leading to ∂h∂t = (1 + Γ sin ωt ) ρg µr ∂∂r (cid:20) rh ∂h∂r (cid:21) (4)The result is (at least in first order) the same as for thepurely gravitational case, with a continuous oscillation ontop of the gravitational result, just like we see in our ex-periments. More details of this calculation can be foundin Appendix A. IV. NON-NEWTONIAN LIQUIDS
Whereas disturbances created in a layer of a Newtonianliquid always close, independent of whether the layer isbeing vertically vibrated or not, for non-Newtonian liq-uids things are observed to be different: More specifically,for the particulate suspensions studied here [20], holesclose when the suspensions are at rest, but may eitheropen or close when vertically vibrated. d ( m ) t (s) FIG. 8: Time evolution of the diameter d of a growing hole ina suspension of σ = 40 µ m polystyrene particles in glycerine-water mixtures of three different viscosities, namely µ = 0 . · s (red line), µ = 0 .
22 Pa · s (black line), and µ = 0 . · s (blue line) versus time for three opening holes in a 52%volume fraction suspensions, shaken at Γ = 28 and f = 45Hz.The three curves have been time-shifted in order of increasingviscosity. Cleary, for increasing µ the (average) growth ratedecreases. A. Experiment
As discussed in Section I, several types of non-closingholes were found in various vibrated suspensions andemulsions, including stable holes, splitting holes, andgrowing holes [1–6]. It is this last type, the growingholes, which will be the focus of this Section. Grow-ing holes are typically found in suspensions containingmonodisperse particles [21]. We therefore use monodis-perse, spherical polystyrene particles with a diameter ( σ )of 20, 40, and 80 ± µ m, and a density of 1050 kg/m (MicroBeads, TS 20-40-80). As the suspending liquidwe used various glycerine-water mixtures, with varyingviscosities and densities. Because the suspending liquidmay be either denser or less dense than the particles, wedo not attempt to density match the liquid. In all casesthe time scale at which the suspension separates is muchlarger than the time scales of the experiment. In somecases we have checked that our results did not dependon whether the liquid density would be larger or smallerthan that of the particles by adding cesium chloride tothe suspending liquid. Much care has been taken to en-sure that the packing fraction φ –the volume of the solidphase in the suspension divided by the total volume– waskept at a constant value of 0 . σ = 40 µ m particles and glycerol-watermixtures of three different viscosities. We observe that alower viscosity causes holes to open faster. This appearsto be comparable to the Newtonian liquids, where holesclose faster for lower viscosity, but one needs to be carefulin making this comparison: First of all, the viscosity ofthe suspending liquid is generally not comparable to the(non-constant) viscosity of the suspension, since there is x 10 −4 µ (mPas) ∆ d ( m ) µ m40 µ m80 µ m (a) −4 µ/σ (mPas/ µ m) ∆ d ( m ) µ m40 µ m80 µ m (b) FIG. 9: (a) The average growth ∆ d per cycle of a growing holeas a function of the suspending liquid viscosity µ . The exper-iments were done for suspensions of all three bead diameters, σ = 20 µ m (green circles), σ = 40 µ m (red diamonds), and σ = 80 µ m (blue plusses) and a packing fraction of φ = 0 . f = 45 Hz and Γ = 28. (b) Thesame data as in (a) but now plotted as a function of µ/σ , thesuspending liquid viscosity over the particle diameter. a usually non-negligible or even dominant contributionfrom the particle phase. Secondly, we are now lookingat the rate at which the hole grows against both grav-ity and the suspension viscosity, whereas for the closingholes in a Newtonian liquid gravity was the driving forceof the closure. This trend holds for all experiments weperformed. Noteworthy is that for the higher suspendingliquid viscosities and larger particles we typically observegrowth of the hole until it develops a kink (where partof the system, including part of the wall, falls dry andremains separated by a steep slope (kink) from the restof the system) whereas for small values of the suspendingliquid viscosity and large particles we also observed holesthat would go through many consecutive cycles of growthfollowed by a rapid collapse to an almost zero radius.To further quantify the dependence of the growth rateon the suspending liquid viscosity, we determined theaverage growth ∆ d of the hole diameter per cycle andplotted it against the suspending liquid viscosity µ inFig. 9(a) for all three bead sizes. We observe that allthree data sets show a clear decrease of ∆ d with in-creasing µ , confirming our observation that the growthrate decreases with increasing suspending liquid viscos-ity. The data however does not collapse onto a single −3 t (s) d ( m ) t (s) ˙ d ( m / s ) FIG. 10: (a) Overlay of many cycles of a growing hole ex-periment with 40 µ m beads suspended in a glycerine-watermixture of viscosity µ = 0 .
52 Pa · s, all shifted to start at t = 0and the initial diameter shifted to d = 0 (blue curves). Theblack plus symbols indicate the cycle-averaged hole diameter.The yellow line is a sine fit through the average (thus neglect-ing the actual growth of the hole). (b) The instantaneousvelocity of the growing hole averaged over all cycles (red plussymbols). The solid (blue) line is the derivative of the sine fitof (a). The experimental parameters are, h = 6mm, f = 45Hz, Γ = 28, and φ = 0 . curve. Therefore, in Fig. 9(b) we plot the same data asa function of µ/σ which leads to a reasonable collapse ofthe data for the two larger sizes, the significance of whichwill be discussed further down.Just like we have done for the Newtonian liquids (cf.Fig. 6), we can overlay many single cycles and computethe cycle-averaged diameter and velocity, the result ofwhich is plotted in Fig. 10. This reveals several promi-nent features: The first is that –quite unlike for the clos-ing holes in the Newtonian liquids– the signal deviatessignificantly from a sinusoidal shape. This is especiallyclear when comparing the cycle-averaged velocity to thederivative of the sine fit [Fig. 10(b)]. In this plot we finda second remarkable feature: The magnitude of the mostnegative velocity ( ˙ d ≈ − .
90 m/s) is larger than that ofthe most positive velocity ( ≈ .
65 m/s), which is sur-prising since the hole on average is growing, i.e., for thetime average we have h ˙ d i >
0. When determining theduration of the opening and closing parts of the cycle,we find that they tend to lie very close to one another,implying that large closing velocities occur in a narrowtime interval, whereas large opening velocities are foundin a broader period of time. Indeed, the second, closinghalf of the cycle is sharply peaked, compared to a wider,somewhat closer to sinusoidal, shape during the openinghalf. This is clearly observed in Fig. 10(b).The above asymmetry is visible in all of our exper-iments, as can be seen in Fig. 11, where we plot thedifference between the magnitudes of the largest open-ing and closing velocities ∆ V ≡ | max( ˙ d ) | − | min( ˙ d ) | =max( ˙ d ) + min( ˙ d ). The fact that ∆ V is always negativeexpresses that the magnitude of the most negative veloc-ity is larger than that of the most positive. Just like theaverage growth ∆ d per cycle decreased with increasingviscosity, so does the magnitude of the velocity difference −0.8−0.6−0.4−0.20 ∆ V ( m / s ) µ (mPas) µ m40 µ m80 µ m (a) −0.8−0.6−0.4−0.20 µ/σ (mPas/ µ m) ∆ V ( m / s ) µ m40 µ m80 µ m (b) FIG. 11: (a) The difference ∆ V ≡ max( ˙ d ) + min( ˙ d ) in themaximum opening and closing velocities in the growing holestate, averaged over all cycles as a function of the suspend-ing liquid viscosity µ , again for suspensions of all three beaddiameters, σ = 20 µ m (green circles), σ = 40 µ m (red dia-monds), and σ = 80 µ m (blue plusses) and a packing fractionof φ = 0 .
52. As before, the driving parameters are f = 45 Hzand Γ = 28. (b) The same data as in (a) but now plotted asa function of µ/σ . ∆ V , which becomes less negative as µ becomes larger. Inaddition we find that the data for the different hole sizesare rather scattered in the ∆ V versus µ plot, but appearto collapse when plotted against µ/σ .Finally, we can determine the phase shift ∆ ψ betweenthe driving and the hole although this is slightly moredifficult than in the Newtonian liquid case (as well assuffering from some arbitrariness) because of the devia-tions from the sinusoidal shape. The results are plottedas a function of µ/σ in Fig. 12: Again the horizontaloscillation of the hole lags behind the vertical containerposition but now by a phase shift that is slightly largerthan π/ µ/σ (mPas/ µ m) ∆ ψ µ m40 µ m80 µ m FIG. 12: The phase difference ∆ ψ between the vertical po-sition of the container (vibrated at f = 45 Hz and Γ = 28)and the diameter of a growing hole, averaged over all cycles.This quantity is plotted as a function of the suspending liquidviscosity µ , again for suspensions of all three bead diameters, σ = 20 µ m (green circles), σ = 40 µ m (red diamonds), and σ = 80 µ m (blue plusses) and a packing fraction of φ = 0 . B. Interpretation
It is now time to make an inventory of what we believehappens when we create a hole-shaped disturbance in aliquid layer in a container which is oscillated vertically: • For highly viscous fluids (even if not Newtonian),the velocity of the hole walls is in phase with theacceleration the liquid layer experiences. • A viscous Newtonian liquid follows the accelera-tion perfectly, i.e., for a sinusoidal acceleration alsothe velocity is sinusoidal. This stands to reasonsince for a viscous fluid forcing (acceleration) andthe response of the liquid (the velocity profile inthe layer) are proportional, with viscosity µ as theproportionality constant (Section III). • Consequently, if the liquid is non-Newtonian theproportionality factor itself depends on the forcingand therefore the response of the liquid to a sinu-soidal acceleration is a deformed signal. However,if stress depends monotonously on strain rate (like,e.g., in a power-law fluid) the deformation will besymmetric, i.e., sinusoidal with a superposition ofonly odd higher harmonics. • For our vertically vibrated suspension layers wefind a non-symmetric velocity cycle. The sec-ond, negative velocity part is strongly deformed,whereas the first, positive velocity half is closerto sinusoidal [Fig. 10(b)]. It appears that duringthe second half of the cycle the suspension be-haves strongly non-Newtonian[22] whereas duringthe first half the response is closer to that of a New-tonian fluid.The behavior in this last point may be summarized bysaying that the behavior of the liquid is highly hysteretic. This is in (qualitative) agreement with the phenomeno-logical model proposed by Deegan [6], who argued that ahysteretic rheology would be necessary to explain the ex-istence of stable or growing holes in a vertically vibratedliquid layer.Now, let us speculate about what could cause the sus-pension to respond in this manner. In the second (i.e.,closing) half of the driving the suspension layer expe-riences a downward acceleration and, consequently, thesuspension layer will be pushed against the bottom ofthe container and when set in motion by the presenceof the hole it will do so with the typical non-Newtonian(shear-thinning) behavior that characterizes suspensions.In the first (opening) half of the driving, inertia actuallycreates a low pressure between the layer and the con-tainer bottom. Now suppose that this pressure gradientwould be able to displace the liquid slightly with respectto the particle phase such that a thin layer of liquid –with a thickness comparable to the particle diameter σ –forms between the bottom and the suspension. Such alayer could act as a lubrication layer,i.e., during this firsthalf period the layer would move on top of this layer andthe entire velocity gradient would be in this thin layer ofNewtonian liquid, i.e., it would be a shear band.This in turn would explain why the suspension layer inthe first half behaves closer to a Newtonian fluid, namelybecause this thin liquid layer is a Newtonian fluid. Morespecifically, if we balance the gravitational energy of thesuspension layer and the dissipation in the lubricationlayer we obtain ρ s gh ∼ µ ˙ dσ ⇒ ˙ d ∼ ρgh σµ , (5)i.e., the velocity ˙ d in the second half of the driving wouldscale as ( µ/σ ) − . This is consistent with the fact thatmany of the observables (∆ V and δd ) that character-ize the growth of the hole, show a better collapse whenplotted against µ/σ rather than µ itself. Conversely, onecould state that dependence on µ/σ indicates the exis-tence of a shear layer of suspending liquid (with viscosity µ ) and thickness ∼ σ .Incidentally, the presence of such a thin shear layermay also account for the convection rolls that have beenobserved in the rim of these structures [2, 3]: In the sec-ond, closing half of the driving the suspension respondswith a flow profile in the layer in with the largest veloc-ity on top and zero velocity at the bottom. In the first,opening half the layer slides back as a whole, on top ofthe thin shear layer. Consequently, the displacement percycle of a fluid element near the bottom is different fromthat near the top, giving rise to a convection roll. V. CONCLUSIONS
We have comparatively studied the dynamics of holesin a vertically vibrated layer of viscous Newtonian liquidson the one hand and of dense particle suspensions on theother. We find that all the holes oscillate with a phaseshift of ∆ ψ = π/ Appendix A: Modeling of hole closure in a viscouslayer
A lubrication model of an axisymmetric viscous layer h ( r, t ) starts with the axisymmetric Stokes’ equation inthe thin layer limit, with pressure given by the hydro-static pressure in the layer p = ρg [ h ( r, t ) − z ]. Neglectinggradients in the radial direction in comparison to thosein the vertical direction we than integrate µ ∂ u r ∂z = ∂p∂r ⇒ u r = ρg µ ∂h∂r z ( z − h ) , (A1)where we have used the no-slip boundary condition atthe bottom ( u r (0) = 0) and the free-slip condition at thefree surface ( ∂u r /∂z ( h ) = 0). Continuity, integrated overthe layer height gives ∂h∂t = − r ∂∂r " r Z h u r ( z, t ) dz , (A2) which with Eq. (A1) immediately leads to Eq. (1) ∂h∂t = ρg µr ∂∂r (cid:20) rh ∂h∂r (cid:21) . (A3)If we look to compute the closure of a hole of initialdiameter d (at t = 0 s) in an infinite layer of liquid ofthickness h , we can find a similarity solution to Eq. (A3).To find it, we first nondimensionalize h , r , and t with thelength and time scale in the problem, namely h and t ≡ µ/ ( ρgh ) respectively. If we now use a selfsimilaransatz e h = e t α H ( e r/ e t β ) in Eq. (A3), we find a solutionprovided that α = 0, β = 1 / h ( r, t ) = h H s µr ρgh ( t c − t ) ! , (A4)where t c is the time at which the hole closes and H ( η ) isa solution of d H dη + 1 η dH dη = − η dH dηH ( ∞ ) = 1 ; H ( η ) = 0 . (A5)The fact that η needs to be a constant implies that therim diameter d ( t ) should scale as d ( t ) = d s ( t c − t ) t c = 2 η s ρgh ( t c − t )3 µ . (A6)Note that the problem is not uniquely determined by pro-viding d , and that in addition the closure time t c needsto be supplied to obtain a full solution to the problem.Then, η can be determined as η = [3 µd / (4 ρgh t c )] / and Eq. (A5) has a unique solution. Note, that in thiscase the initial profile is also fixed by the self-similar so-lution. Alternatively, one could therefore also start fromthe initial profile, match it to the solution of Eq. (A5)for a certain η which then fixes t c . (This can be doneprovided that the initial profile is compatible with theequations.)To obtain solutions of Eq. (A3) that are more realisticgiven the experimental setup that we use we turn to nu-merical simulations. Here, we replace the actual bound-ary conditions at the side wall (zero radial velocity andno-slip) –which are impossible to incorporate into the lu-brication model– with the following integral statement ofmass conservation in the system Z D/ r =0 h ( r, t ) rdr = constant , (A7)where D is the diameter of the container, or equivalently,taking the time derivative of Eq. (A7) and using Eq. (A3) D (cid:20) h ∂h∂r (cid:21) r = D/ = 0 ⇒ ∂h∂r (cid:12)(cid:12)(cid:12)(cid:12) r = D/ = 0 , (A8)where it was used that h ( D/ , t ) >
0. Eq. (A3) is ofa type that is known as a non-linear diffusion equation,which is of a very stable type that renders them easyto solve numerically. The equations are therefore solvedwith a simple forward integration scheme which lead toresults that compare well to the experiments (see Sec-tion III B).Actually it is conceptually straightforward to incorpo-rate the driving into the equations, as the only thing oneneeds to do is to substitute the gravitational acceleration g with g + a ( t ) where a ( t ) is the instantaneous accel-eration of the container, a ( t ) = aω sin ωt = Γ g sin ωt (with ω = 2 πf ). This however has enormous implica-tions for the numerical solvability of the equations, sincefor Γ > g + a ( t ) <
0. In this interval Eq. (A3) becomesa non-linear diffusion equation with a negative diffusioncoefficient, which is terribly unstable and consequentlyextremely difficult to solve numerically. For the currentproblem there exists a workaround however, for whichwe need some additional understanding of the equationsfirst.To this end let us first examine a modified Eq. (A3)without gravity ∂h∂t = Γ sin( ωt ) ρg µr ∂∂r (cid:20) rh ∂h∂r (cid:21) . (A9)Clearly, a solution to Eq. (A9) must have the same peri-odicity as the driving, i.e., h ( r, t + T ) = h ( r, t ). Now,the simplest form that such a solution could have is h ( r, t ) = h s ( r ) + A ( r ) exp[ iωt + ϕ ( r )], which correspondsto neglecting non-linear effects in Eq. (A9). Now h s ( r )can be any profile that satisfies the non-driven Eq. (A9),i.e., ∂h s /∂t = 0, which can be any well-behaved functionof r . Inserting this form into Eq. (A9) and linearizingleads to ωA ( r ) e i ( ωt + ϕ ( r )+ π/ = Γ ρg µr ∂∂r (cid:20) rh s ∂h s ∂r (cid:21) e iωt , which needs to hold for any t , leading to A ( r ) = Γ ω ρg µr ∂∂r (cid:20) rh s ∂h s ∂r (cid:21) ,ϕ ( r ) = − π . (A10)with which h ( r, t ) = h s ( r ) + Γ ω ρg µr ∂∂r (cid:20) rh s ∂h s ∂r (cid:21) exp[ i ( ωt − π/ , (A11)The full equation, including both the driving and grav-ity, is equal to ∂h∂t = (Γ sin( ωt ) + 1) ρg µr ∂∂r (cid:20) rh ∂h∂r (cid:21) . (A12) t (s) d ( m ) −7 r (m) h / h Δ − t c T − t c T − t c T + t c T + t c T + t c − T + t c − t c T − t c T + t c (a)(b) FIG. 13: (a) Time evolution of the hole diameter in a layerof a liquid of viscosity µ = 1 .
23 Pa · s, density ρ = 1 . · kg/m and thickness h = 6 mm, driven at f = 50 Hz andΓ = 20, obtained by numerically integrating Eq. (A12) usingthe procedure described in the text (blue solid line). Theblack lines are solutions of the corresponding gravitationalclosure problem Eq. (A3), one starting from the same initialcondition as the blue line (solid) and the other is the onethat is used in the integration procedure of the blue curve.The inset illustrates the integration procedure. (b) Differencebetween the profile at t = T / t c obtained starting from t = − t c by direct integration of Eq. (A12) and by integratingEq. (A3) to t = T − t c and integrating Eq. (A12) backwardsin time to t = T / t c . The inset illustrates this procedure. Note that, since we are dealing with Γ ≫
1, gravityis a small perturbation to Eq. (A9). This implies thatthe gravitational timescale at which the profile decays( t g ∼ µ/ ( ρgh )) is typically much larger than that ofthe driving. I.e., if h g ( r, t ) is a solution to Eq. (A3), onthe timescale of a single period it does not significantlychange and may hardly interfere with the oscillation. Us-ing Eq. (A11), this suggests a solution to the full problemof the form h ( r, t ) = h g ( r, t ) + Γ ω ∂h g ∂t exp[ i ( ωt − π/ , (A13)where we have used that h g ( r, t ) is a solution to Eq. (A3)to simplify the expression for the amplitude of the oscil-lation.This line of reasoning comes to the rescue when numer-ically solving Eqs. (A9) and (A12). A first useful trickis to realize that we can numerically integrate Eq. (A9)from t = 0 to T / τ = − t we obtain a minus sign on the0left hand side which exactly compensates for the minussign of the coefficient in the interval [ − T / , t = 0 to − T /
2, such that we obtain the solutionon [ − T / , T / t = − ( T / π ) arcsin(1 / Γ) ≡ − t c (where the coefficientchanges sign), obtain a numerical solution on [ − T / t c , T / t c ], i.e., also on a full period of the driving. And,of course, it is impossible to extend this interval becauseit is bounded by an interval where the coefficient is pos-itive on the negative side –such that backwards integra-tion is not possible– and similarly by an interval wherethe coefficient is negative on the positive side. We canhowever take the solution (cq. initial condition) h ( r, − t c )and integrate it using Eq. (A3), i.e., the equation thatonly contains gravity, from − t c to T − t c . The solu-tion h g ( r, T − t c ) is now subsequently used as an initialcondition for the full problem Eq. (A12) which we thenintegrate on the interval [ T / t c , T / t c ]. If there is any truth in the analytical approximation Eq. (A13) thetwo solutions should (approximately) match at the pointwhere the two intervals meet, i.e., in t = T / t c . Thisprocedure can be iterated until the solution is obtainedon the full time interval [see the inset of Fig. 13(a)].In Fig. 13(a) we plot the result of this procedure for alayer of liquid with viscosity µ = 1 .
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