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Granular Matter 5, 71–74c Springer-Verlag 2003DOI 10.1007/s10035-003-0127-9A texture tensor to quantify deformations: the exampleof two-dimensional ﬂowing foamsMarius Asipauskas, Miguel Aubouy, James A. Glazier, Fran¸cois Graner, Yi JiangAbstract In a continuum description of materials, thestress tensor ﬁeldσ quantiﬁes the internal forces theneighbouring regions exert on a region of the material.The classical theory of elastic solids assumes thatσ de-termines the strain, while hydrodynamics assumes thatσ determines the strain rate. To extend both successfultheories to more general materials, which display bothelastic and ﬂuid properties, we recently introduced a de-scriptor generalizing the classical strain to include plasticdeformations: the “statistical strain,” based on averagesof microscopic details (“A texture tensor to quantify de-formations” M.Au., Y.J., J.A.G., F.G, companion paper,Granular Matter, this issue). Here, we apply such a statis-tical analysis to a two-dimensional foam steadily ﬂowingthrough a constriction, a problem beyond reach of bothtraditional theories, and prove that the foam has theelastic properties of a (linear and isotropic) continuousmedium.Received: 1 November 2002M. AsipauskasDept. of Physics, 316 Nieuwland, University of Notre Dame,Notre-Dame, IN 46556-5670, USAJ. A. GlazierIndiana Univerisity, Department of Physics,Swain West 159, 727 East Third Street,Bloomington, IN 47405-7105, USAM. AubouySI3M∗, D.R.F.M.C., CEA, 38054 Grenoble Cedex 9 France∗U.M.R. n◦5819: CEA, CNRS and Univ. Joseph-FourierF. GranerLaboratoire de Spectrom´etrie Physique∗∗,BP87,38402 St Martin d’H`eres Cedex, France∗∗U.M.R. n◦5588 : CNRS and Univ. Joseph-FourierAddress for correspondence: [email protected]: (+33) 4 76 63 54 95, BP 87, 38402Y. JiangTheoretical Division, Los Alamos National Laboratory,New Mexico 87545, USAThanks are due to S. Courty, B. Dollet, F. Elias, E. Jani-aud for discussions. YJ is supported by the US DOE undercontract W-7405-ENG-36. JAG acknowledges support fromNSF, DOE and NASA contracts NSF-DMR-0089162, DOE-DE-FG0299ER45785 and NASA-NAG3-2366. We thank S. Coxfor access to unpublished simulations.Keywords Dispersed material, Foam, Stress-strainrelation, Shear modulusA “plastic” deformation means that microscopic rear-rangements take place in the material, so that the mi-croscopic pattern does not return to its initial conditioneven after the applied force has ended. An example of ahighly heterogeneous one in a viscoelastic material is atwo-dimensional foam steadily ﬂowing through a constric-tion (Fig. 1). This apparently simple example is utterlyintractable from the perspective of both elasticity theo-ry [1] and Navier-Stokes [2] treatments. In this paper, wepresent a new approach to analyse complex ﬂows of dis-ordered materials.We prepare the foam by blowing air into the bottom ofa column of soap solution. The solution is 10% commer-cial dishwashing detergent (“Ivory” brand) and 5% glyc-erol. Its surface tension, measured by the No¨uy method,is γ =28.5 ± 0.1 mN.m−1. Filtered air blows at a steadyﬂow rate of 0.08 cm3.s−1through an 18-gauge (0.084 cmID) stainless steel needle with a 90◦bevel. Bubbles arehomogeneous in size (<5% dispersity), and ﬂoat to thetop of the column in random positions. The foam is dry,with a relative fraction of ﬂuid <3%. Since a bubble edge(a ﬁlm of soap solution) has two interfaces with air, itsline tension λ is λ =2γh =28.5 µN.Bubbles enter a horizontal channel, a Hele-Shaw cellmade of two parallel Plexiglas plates h =0.5 mm apart,Fig. 1. Two-dimensional foam ﬂowing through a constriction.The 10 cm (551 pixel) wide ﬁeld of view shows only the end ofthe 1 m long horizontal channel721 m long, and 10 cm wide (Fig. 1). The average bubble ve-locity is 0.1 cm.s−1. The generation of new bubbles forcesthe mass of bubbles down the channel, in plug ﬂow (freeslip boundary conditions) if the channel were uniform. A5-mm wide constriction near the end of the channel forcesthe ﬂow to be heterogeneous. Beyond the constriction thebubbles reexpand into the full width channel for a shortdistance and then fall from the open channel end (no stressboundary conditions) into a collection vessel. The volumevariations during the transit time (50 s) across the ﬁeld ofview, due to pressure differences or diffusion of gas fromone bubble to its neighbours, are below our pixel induceddetection limit of 2%.All measurements we present use 60 × 60 pixel slidingboxes (“representative volumes”), meaning that we do notevaluate them within 30 pixels of the channel walls; andaverage over 2800 successive images of a 30 Hz movie. Tomeasure the “Eulerian” velocity ﬁeld, we track the cen-ter of mass of each bubble between two successive imagesand add the velocities of all bubbles in the same volume.The velocity ﬁeld is smooth and regular (Fig. 2), quali-tatively indicating that the foam behaves as a continuousmedium.To obtain a more quantitative characterization, wemeasure the stress in the foam. Stress has dissipative andelastic components; the pressure inside the bubbles andthe network of bubble edges contribute to the latter. Sincethe pressure stress is isotropic, it does not contributeto the elastic normal stress differenceσxx− σyy(as mea-sured for granular materials [3]) or the shear stressσxy,entirely due to the network. Dimensionally,σ is of theorder as the line tension λ (which, in a 2D foam, is thesame for all edges: here 28.5 µN) divided by a typicalbubble size.We measure locally the network stress in each “repre-sentative volume element,” that is, a square box, centeredFig. 2. Velocity ﬁeld in the foam (in arbitrary units, the samescale for each arrow)around the point of measurement, of a mesoscopic size:larger than a bubble, but much smaller than the channelwidth. We proceed as follows. We identify the bubble edg-es which cross the boundaries of the volume [4]. We de-termine the tension τ = λˆe of each edge, where ˆe is theunit vector tangent to the edge. We determine the averageforcef on a boundary element dS by vectorially addingthese tensions and obtainσ deﬁned by: fi= σijdSj[1,5]; equivalently, we may use an average over all links toimprove the statistics [6,7].Clearly, the stress ﬁeld is strongly heterogeneous (Fig.3). The upstream inﬂuence of the constriction becomesvisible as the lobe whereσxx−