Labyrinth Patterns in Confined Granular-Fluid Systems

B. Sandnes,* H. A. Knudsen, K. J. Maløy, and E. G. FlekkøyDepartment of Physics, University of Oslo, P.O. Box 1048 Blindern, NO-0316 Oslo, Norway

(Received 2 March 2007; published 17 July 2007)

A random, labyrinthine pattern emerges during slow drainage of a granular-fluid system in two-dimensional confinement. Compacted grains are pushed ahead of the fluid-air interface, which becomesunstable due to a competition between capillary forces and the frictional stress mobilized by grain-graincontact networks. We reproduce the pattern formation process in numerical simulations and present ananalytical treatment that predicts the characteristic length scale of the labyrinth structure. The patternlength scale decreases with increasing volume fraction of grains in the system and increases with thesystem thickness.

DOI: 10.1103/PhysRevLett.99.038001 PACS numbers: 45.70.Qj, 47.20.Ma, 47.54.�r, 89.75.�k

Pattern and form emerge spontaneously in a diverserange of systems representing virtually every disciplineof science [1]. Common denominators are that these sys-tems are driven out of equilibrium, and that the patternformation is a consequence of instabilities governed bycompeting forces [2]. Viscous fingering, for example, de-velops as a compromise between the Saffman-Taylor in-stability that induces ever more branching, and the surfacetension of the fluid that puts an energetic penalty on thecreation of surface area [3,4]. The variation in pattern typesand length scales is striking, yet one also finds that seem-ingly unrelated processes may produce patterns with simi-lar characteristics [5].

Here we demonstrate and characterize a pattern forma-tion process where labyrinthine structures emerge duringslow drainage of a fluid-grain mixture confined in a Hele-Shaw cell. The final cluster of compacted granular materialis simply-connected, random, and has a characteristicwavelength. Experiments and simulations illustrated inFig. 1 reveal that the interface instability develops as acompromise between capillary forces and granular friction,and we present an analytic prediction of the observedcharacteristic length scales in the patterns. Similar laby-rinthine patterns have been shown to form in two-phasesystems such as magnetic and dielectric fluids under ex-ternal fields [6], and in some reaction-diffusion processes[7,8].

Polydisperse glass beads of diameter 50–100 �m aremixed into a 50% by volume glycerol-water solution andinjected into the gap between the parallel, horizontal glassplates of a Hele-Shaw cell trough a central hole in the topplate [3,9]. After injection, the fluid-grain mixture fills acircular area (�35 cm diameter) in the gap between thetwo plates (plate separations ranging from 0.3 to 1.0 mm).Each plate has the dimensions 50� 50� 1 cm. The glyc-erol is added in order to increase the viscosity of the fluidsuch that the grains remain in suspension during loading.Shortly after the cell is filled, the grains sediment out andfall to rest on the lower surface. We now start a slowdrainage of the fluid in the cell through the central outlet

using a syringe pump set to a withdrawal rate of typically0:01 ml=min ; thus, each experiment takes close to 3 daysto complete. Due to the low flow-rate, the viscous forcesthat govern viscous fingering processes, are negligible.

As the fluid is gradually drained from the cell, the fluid-air interface at the perimeter of the circular disc starts torecede. The capillary forces between the wetting fluid andthe grains gradually compile a growing layer of close-packed grains ahead of the interface as it moves. After atransient period where a layer of grains is formed all alongthe circular perimeter, the interface becomes unstable andthe fluid-grain disc is slowly invaded by fingers of air.Figure 1(a) shows consecutive images of this process,where one can observe the gradual advance and splittingof fingers resulting in the final labyrinthine pattern. Unlikepinning-dominated drying processes [10,11], the finger-splitting is due to a jamming of the interface on a lengthscale larger than the individual grains. The grains, initiallyuniformly distributed, have been moved and reorganizedby the invading fingers into a branching structure of close-packed particles. Each branch is assembled from the com-pact layer of grains pushed ahead by two adjoining fingers.While the initial circular perimeter is approximately 1 mlong, the interface has stretched and deformed such that thecircumference of the final cluster in Fig. 1(a), v measures13.3 m.

The final labyrinth pattern is characterized by a wave-length that is uniform throughout the area occupied by thestructure. The pattern forming process is a result of localmass transport in a direction normal to the advancinginterface. The shape of the initially circular interface iscontinuously changing, but because the interface is stabi-lized and kept separate from other fingers due to frictionaljamming, no pinch-off of the interface occurs during theexperiment, and the interface remains a deformed circle.Consequently both the fingers and grain cluster are topo-logically simply connected.

The visual randomness seen in the final pattern arisesfrom the symmetry-breaking associated with fingers goingleft or right. Disorder is always present in the experiments

PRL 99, 038001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending20 JULY 2007

0031-9007=07=99(3)=038001(4) 038001-1 © 2007 The American Physical Society

in form of, e.g., small inhomogeneities in mass distribu-tion, and the dynamics of the pattern formation is governedby this, in addition to the history of the moving interface.However, simulations with no initial disorder (other than

the floating point round-off errors intrinsic to the calcula-tions) show that randomness in the pattern inevitablydevelops.

Changing the initial volume fraction of grains ’ in theexperiment influences the process, where the most notableeffect is that the characteristic wavelength in the patternbecomes smaller as the volume fraction of grains increases[Fig. 2(a)]. The volume fraction ’ is normalized such that’ � 1 for the close-packed grains. Using image analysiswe identify the simply connected grain cluster in eachpicture and measure the characteristic wavelength as � �2Adisc=O, where O is the circumference of the cluster andAdisc is the area of the initial circular disc of fluid. In theexperiments � ranges from 9.6 to 40.1 mm. The measure-ments also show a slight thickening of the grain brancheswith increasing volume fraction.

As in most other pattern forming systems, the instabilitydevelops as a compromise between competing forces. Inthe case of the granular labyrinth, the two main players aresurface tension and friction. Narrow invading air fingersgather less grains and therefore encounter less friction, butthe higher curvature associated with the finger tip interfacegives a higher capillary force. Broader fingers, on the otherhand, will encounter a smaller capillary resistance but willgather more grains along the interface, leading to increasedfriction. The width of the accumulated layer of grains, L, isin general dependent on the initial volume fraction ofgrains ’, the finger width and the history of the advancingfinger. In steady state the finger gathers as much particlesas it leaves behind, and the width of the layer of grains isgiven by L � ’�=2.

Surface tension produces a pressure difference across acurved fluid interface. This difference, the Laplace pres-sure, is given by �Pcap � ��1=r1 � 1=r2�, where � issurface tension and r1 and r2 are the principal radii ofcurvature of the fluid meniscus. In the Hele-Shaw cell, theout-of-plane curvature is fixed by the parallel glass platessuch that 1=r � 2 cos�=�z, where � is the fluid/glasswetting angle and �z is the plate separation, while thein-plane curvature may vary freely. Figure 3(a) illustrates atop view of a finger of air with a tip of curvature 1=R

(a)

i ii

iii iv v

(b)

FIG. 1 (color online). (a) Experiments. Pictures i–iv show atime sequence of fingers of air invading the initially circulargranular-fluid system. Picture v shows the resulting branchingstructure of close-packed grains (white cluster). The Hele-Shawcell plate spacing was 0.4 mm and the volume fraction of grains20% [14]. (b) Simulations. The pattern forming process has beenreproduced by modeling the competing capillary and frictionalforces that govern the instability. The white areas represent air,the black compact mass, and gray the fluid with sedimentedgrains [15].

FIG. 2 (color online). Characteristiclength scale as a function of normalizedvolume fraction. (a) Experiments. Thevolume fraction of grains ’ in the mix-ture increases from left to right aslabeled in the pictures. All experimentswere conducted with a plate spacing of0.4 mm. The dimensions of each pictureframe are 40� 40 cm. (b) Simulations.The observed decrease in the lengthscale of the pattern with increased vol-ume fraction is reproduced in the simu-lations.

PRL 99, 038001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending20 JULY 2007

038001-2

invading the fluid-grain disc. In the situation where grainsinteract with the fluid meniscus, the pressure difference notonly maintains curvature, but also imposes a stress� on thecompact grain layer. At the point when the friction isovercome and a segment of the interface is set in motion,the yield pressure can to a first approximation be written

�P � ��2 cos�

�z�

1

R

�� �; (1)

where the stress required to move the grains is equal to theopposing frictional stress between the grains and the glasssurfaces.

Figure 3(b) illustrates a cross-sectional view of the air/fluid interface and a layer of close-packed grains of widthL that is gathered ahead of the front. The gap between theplates is several particle diameters wide, and we adopt aJanssen’s description as a good approximation for stressdistribution in the packing [12]. The horizontal compres-sive force exerted by the meniscus on the frontline particlesmobilizes a contact network between the close-packedgrains that redistributes stress vertically to the plates suchthat �zz � ��xx, where � is the Janssen’s proportionalityconstant, and x and z denote the in-plane and transversedirections, respectively. This yields a maximum frictionalstress at the interface that increases exponentially with thewidth L of the compacted layer

� ��g�z

2�

��1� ��� exp

�2��L

�z

�� 1

�: (2)

Here � is the density contrast between grains and fluid, g isthe gravitational acceleration, and � is the friction coeffi-cient [13].

The capillary and friction forces have been implementedin an algorithm that appears to capture the essential fea-tures of the pattern formation process. The model is basedon a one-dimensional representation of the interface,where thousands of points discretize the perimeter of thefluid-grain area. At each point a local curvature is calcu-lated from the position of the neighboring points, and avector pointing normal to the interface (into the fluid)represents the width of the compacted layer of grains.The yield pressure given by the curvature and friction,[Eqs. (1) and (2)], is calculated for each point, and forevery time step the point with the lowest yield pressure ismoved ahead in a direction normal to the interface. The

moved length is small compared to the distance betweenneighboring points. The simulation is carried out with asmall amount of disorder in the background mass density(distributed on a square grid). As the interface advances,the width of the compact layer is recalculated to accountfor the accumulated mass based on the invaded area and thebackground volume density of grains. As the simulationprogresses, the interface stretches. When the distance be-tween neighboring points exceeds a set limit, a new point isinserted halfway between the two original points such thatthe curvature of the line segment is preserved. Where twofingers meet, the respective points along the interface areimmobilized as this represents the formation of a close-packed branch, and the simulation runs until the entire areadefined by the initially circular disc has been compacted inthis manner.

The visual similarity between simulated and experimen-tal labyrinth patterns is evident (Fig. 1). One notabledifference is the number of invading fingers across theinitial circular interface (one in the simulations, severalin the experiments). This can be attributed to a differencein disorder along the initial circular perimeter, which ismuch larger in the experiments compared to simulations.Figure 2(b) shows final labyrinth patterns generated in aseries of simulations which reproduce the decreasing pat-tern length scale with volume fraction that we observe inthe experiments. Figure 4 shows the measured character-istic wavelengths in the patterns generated in the experi-ments and simulations as a function of the volume fractionof grains in the systems.

The effective friction coefficient in the confined granularsystem is inherently difficult to estimate accurately. Wehave used the value � � 0:47, which is within the realisticrange, and fits the experimental results at high ’. At low ’

FIG. 4. Characteristic length scale as a function of volumefraction of granular material. The measured wavelengths areplotted for experiments (filled squares) and simulations (opencircles). The line represents characteristic wavelengths predictedby the analytic scaling law. The inset shows pattern length scaleas a function of plate spacing with ’ kept constant at 20%.

FIG. 3 (color online). Sketch of the interface. (a) Top view ofan invading finger of air with built-up mass along the front.(b) Layer of compacted grains of width L ahead of the interface.

PRL 99, 038001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending20 JULY 2007

038001-3

the simulations produce somewhat higher �, possibly dueto the friction law providing a less accurate descriptionwhen L becomes comparable to the plate spacing. Themost convincing result is that the overall ’ dependenciesof the simulations and experiments are very close, indicat-ing that the rules implemented in the simulations capturethe essence of the physical mechanisms. The friction law[Eq. (2)] is governed by the redistribution of stress alongcontact networks within the granular packing. This givesthe exponentially increasing friction with L, which isimportant for reproducing the experimental results. Wheninstead a linear friction law is implemented in the simula-tions, the results do not come close to agreement.

The inset in Fig. 4 shows wavelengths measured fordifferent plate separations at constant ’. The wavelengthincreases with increasing system thickness, and again theresults from experiments, simulations and theory are inagreement. The experiments at 0.8 and 1.0 mm plate spac-ing are less well defined as the hydrostatic pressure be-comes significant, and the meniscus is unable to pull alongall the particles, leaving behind beads on the lower plate.

To obtain a theoretical prediction of the characteristicwavelength, we assume that the moving finger tip has aconstant curvature, R, in which case mass conservationimplies Ltip � R’=�1� ’�. Using this relationship andcombining Eqs. (1) and (2) give a pressure as a functionof R with a global minimum (surface tension dominates athigh curvature, friction at low). We further assume that thefingers will form at a width that minimizes the pressuredifference across the interface, i.e., @�P=@R � 0.Inverting this equation gives the critical fingertip curvature

Rc ��z�1� ’���’

W��

���2’

�g�1� ’��1� �����z�2

�1=2�;

(3)

where W is Lambert’s W function. Now, we assume thatthe finger boundaries are straight and parallel some dis-tance behind the curved tip. The pressure difference is thesame all along the interface, such that �P�Rc� � �P�L�,where L is the width of the compact layer of grains alongthe straight interface segments. From this we find thecharacteristic wavelength as � � 2L=’. The result is plot-ted as a function of the volume fraction in Fig. 4, and matchclosely the corresponding wavelength seen in the experi-ments and the simulations.

In conclusion, we have introduced a system that formslabyrinth patterns, and that is well understood in terms ofonly two competing forces: friction and the capillary pres-sure force. The simulations and the analytic theory capture

the characteristic length scale in the experiments as afunction of initial volume fraction of granular materialand system thickness. The simulations also reproduce theevolution and qualitative shape of the labyrinths. Thedynamics of the labyrinth formation is governed by aself-organized sequential selection of lowest interfaceyield pressure along the interface. The similarity of thepresent patterns to those of patterns formed in highlydifferent systems, such as ferrofluids and reaction-diffusion systems, raises the question of a deeperconnection.

We thank Dragos-Victor Anghel, Stephane Santucci,Grunde Løvoll, and Joakim Bergli for discussions. Thiswork was supported by the Norwegian Research Councilthrough Petromaks, Nanomat and SUP grants.

*bjornar.sandnes@fys.uio.no[1] D. Thompson, On Growth and Form (Cambridge

University Press, Cambridge, England, 1961).[2] M. C. Cross and P. C. Hohenberg, Rev. Mod. Phys. 65, 851

(1993).[3] P. G. Saffman and G. Taylor, Proc. R. Soc. A 245, 312

(1958).[4] L. Paterson, J. Fluid Mech. 113, 513 (1981).[5] P. Ball, The Self-Made Tapestry. Pattern Formation in

Nature (Oxford University, New York, 1999).[6] R. E. Rosensweig, M. Zahn, and R. Shumovich, J. Magn.

Magn. Mater. 39, 127 (1983).[7] K. J. Lee, W. D. McCormick, Q. Ouyang, and H. L.

Swinney, Science 261, 192 (1993).[8] D. M. Petrich and R. E. Goldstein, Phys. Rev. Lett. 72,

1120 (1994).[9] H. S. Hele-Shaw, Nature (London) 58, 34 (1898).

[10] Y. Yamazaki and T. Mizuguchi, J. Phys. Soc. Jpn. 69, 2387(2000).

[11] Y. Yamazaki, S. Komura, and K. Suganuma, J. Phys. Soc.Jpn. 75 043001 (2006).

[12] H. A. Janssen, Zeitschrift des Vereines DeutscherIngenieure 39, 1045 (1895).

[13] See EPAPS Document No. E-PRLTAO-99-054729 for aderivation of the friction law. For more information onEPAPS, see http://www.aip.org/pubservs/epaps.html.

[14] See EPAPS Document No. E-PRLTAO-99-054729 fora movie of the pattern formation during the experiment.For more information on EPAPS, see http://www.aip.org/pubservs/epaps.html.

[15] See EPAPS Document No. E-PRLTAO-99-054729 for amovie of the simulation of the pattern formation. For moreinformation on EPAPS, see http://www.aip.org/pubservs/epaps.html.

PRL 99, 038001 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending20 JULY 2007

038001-4