A texture tensor to quantify deformations: the example of two-dimensional flowing foams

8 Pages·0.14 MB·English

by  Marius Asipauskas

A texture tensor to quantify deformations: the example of two-dimensional flowing
foams
Marius Asipauskas1, Miguel Aubouy2, James A. Glazier1∗, Franc¸ois Graner3, and Yi Jiang4
1Dept. of Physics, 316 Nieuwland, University of Notre Dame, Notre-Dame, IN 46556-5670, USA
2SI3M†, D.R.F.M.C.,CEA, 38054 Grenoble Cedex 9 France
3Laboratoire de Spectrom´etrie Physique‡, BP 87, 38402 St Martin d’H`eres Cedex, France
4Theoretical Division, Los Alamos National Laboratory, New Mexico 87545, USA§
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0 Abstract: In a continuum description of materials, cialdishwashingdetergent(“Ivory”brand)and5%glyc-
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the stress tensor field σ quantifies the internal forces the erol. Its surface tension, measuredby the Nou¨y method,

n neighbouring regions exert on a region of the material. is γ =28.5 0.1 mN.m−1. Filteredair blows ata steady
a The classical theory of elastic solids assumes that σ de- flow rate of±0.08 cm3.s−1 through an 18-gauge (.084 cm
J
termines the strain, while hydrodynamics assumes that ID) stainless steel needle with a 90◦ bevel. Bubbles are
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σ determines the strain rate. To extend both successful homogeneous in size (<5% dispersity), and float to the

] theories to more general materials, which display both top of the column in randompositions. The foam is dry,
t elastic and fluid properties, we recently introduced a de- witharelativefractionoffluid<3%. Sinceabubbleedge
f
o scriptorgeneralizingtheclassicalstraintoincludeplastic (a film of soap solution) has two interfaces with air, its
s deformations: the “statistical strain”, based on averages line tension λ is λ=2γh=28.5µN.
.
t on microscopic details ( “A texture tensor to quantify Bubbles enter a horizontal channel, a Hele-Shaw cell
a
m deformations” M.Au., Y.J., J.A.G., F.G, companion pa- madeoftwoparallelPlexiglasplatesh=0.5mmapart,1
per, Granular Matter, same issue). Here, we apply such mlong,and10cmwide(Fig. 1). Theaveragebubbleve-
-
d a statistical analysis to a two-dimensional foam steadily locityis0.1cm.s−1. Thegenerationofnewbubblesforces
n flowing through a constriction, a problem beyond reach the mass of bubbles down the channel, in plug flow (free
o of both theories, and prove that the foam has the elastic slip boundary conditions) if the channel was uniform. A
c
propertiesofa(linearandisotropic)continuousmedium. 5-mmwideconstrictionneartheendofthechannelforces
[
theflowtobeheterogeneous. Beyondtheconstrictionthe

1 bubbles reexpand into the full width channel for a short
v distance and then fall from the open channel end (no
7 A “plastic” deformation means that microscopic rear-
stress boundary conditions) into a collection vessel. The
1 rangements take place in the material, so that the mi-
volume variation during the transit time (50 s) across
0 croscopic pattern does not return to its initial condition
1 even after the applied force has ended. An example of the field of view, due to pressure differences or diffusion
0 of gas from one bubble to its neighbours, are below our
a highly heterogeneous one in a viscoelastic material is
3 pixel induced detection limit of 2%.
a two-dimensional foam steadily flowing through a con-
0
striction (Fig. 1). This apparently simple example is
/ FIG.1. Two-dimensional foam flowing througha constric-
at utterly intractable from the perspective of both elastic- tion. The 10 cm (551 pixels) wide field of view shows only
m ity theory [1] and Navier-Stokes [2] treatments. In this
theend of the1 m long horizontal channel.
paper, we present a new approach to analyse complex
-
d flows of disordered materials.
All measurements we present use 60 60 pixels sliding
n Wepreparethefoambyblowingairintothebottomof ×
boxes (“representative volumes”), meaning that we do
o
a column of soapsolution. The solutionis 10%commer-
not evaluate them within 30 pixels of the channel walls
c
: ; and average over 2800 successive images of a 30 Hz
v
movie. Tomeasurethe“Eulerian”velocityfield,wetrack
i
X thecenterofmassofeachbubble betweentwosuccessive
r ∗Present address: Department of Physics, Swain West 159, images and add the velocities of all bubble in the same
a 727 East Third Street, Indiana University, Bloomington, IN volume. The velocity field is smooth and regular (Fig.
47405-7105, USA 2), qualitatively indicating that the foam behaves as a
†U.M.R. n◦ 5819: CEA, CNRS and Univ. Joseph-Fourier. continuous medium.
‡U.M.R. n◦ 5588 : CNRS and Univ. Joseph-Fourier. Ad-
dressforcorrespondence: [email protected]. Fax: (+33) FIG. 2. Velocity field in the foam (in arbitrary units, the
4 76 63 54 95. same scale for each arrow).
§Acknowledgments: ThanksareduetoS.Courty,B.Dollet,
F. Elias, E. Janiaud for discussions. YJ is supported by the To obtain a more quantitative characterization, we
USDOEundercontractW-7405-ENG-36. JAGacknowledges measure the stress in the foam. Stress has dissipative
support from NSF, DOE and NASA. M. As. acknowledges and elastic components; the pressure inside the bubbles
hospitality at theLSP. andthenetworkofbubbleedgescontributetothelatter.
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Since the pressure stress is isotropic, it does not con- FIG. 4. Experimental measurement of M(R~), superim-
tribute to the elastic normal stress difference σxx σyy posed on a snapshot of the foam (Fig. 1). Since M is a
−
(as measured for granular materials [3]) or shear stress real symmetric tensor with strictly positive eigenvalues, we
σxy, entirely due to the network. Dimensionally, σ is of display it as an ellipse, the length of each axis (in arbitrary
the order of the line tension λ (which, in a 2D foam, is units, the same scale for each ellipse) being proportional to
thesameforalledges: here28.5µN)dividedbyatypical an eigenvalue. We omit data on boxes touching the channel
bubble size. wall.
We measure locally the network stress in each “repre-
sentativevolume element” thatis a squarebox,centered We have defined the “statistical strain tensor” [7]:
around the point of measurement, of a mesoscopic size:
1
largerthana bubble, but muchsmallerthan the channel U(R~)= logM(R~) logM .
2(cid:16) − 0(cid:17)
width. We proceed as follows. We identify the bub-
ble edges which cross the boundaries of the volume [4]. HerethereferencevalueM ischosenintheundeformed,
0
We determine the tension ~τ = λeˆ of each edge, where
isotropic foam far upstream of the constriction (a choice
eˆ is the unit vector tangent to the edge. We determine
that plays no role in elastic properties): M = ℓ2 I/2,
the average force f~ on a boundary element dS~ by vec- 0 h 0i
where I is the identity tensor, here in two dimension.
torially adding these tensions and obtain σ defined by:
f =σ dS [1,5];equivalently,weuseanaverageoverall By definition, we calculate the tensor logM by rotating
i ij j
links to improve the statistics [6,7]. M to a basis where it is diagonal, take the logarithm of
Clearly,the stress field is strongly heterogeneous(Fig. its eigenvalues, then rotate it back into the initial basis
3). The upstream influence of the constriction becomes (this definition is more convenient than the equivalent
visible as the lobe where σxx σyy changes sign. one using a development as an infinite series). Hence
−
logM hasthesameaxesasM,andisrealandsymmetric.
FIG.3. Snapshotofthefoam(Fig. 1)withregionscolored
The “statistical strain” tensor U reduces to the usual
accordingtothesignoftheexperimentalvalueofthenormal
definition of strain u in the validity limits of classical
elastic stress difference σxx−σyy: blue ×, negative; black ◦,
zero within error; red +, positive; green ⋄, values we omit in elasticity[7]. Itis purely geometric,anddoes notexplic-
itly depend on stresses and forces. It applies to a whole
Fig. 5.
generalclassofmaterialswithbothelasticityandplastic
rearrangements,whether 2Dor3D,wheneverwecanex-
Tocharacterizethedeformationofthebubbles,wecon-
sider a volume (R~) around the position R~. In this vol- perimentally measure the relevant information: whether
V detailed(listofmicroscopicpositionsandlinks)ormeso-
ume, we list all pairs (~r,r~′) of positions of neighboring
scopic averages (quantities related to M). It is a state
vertices connected by one bubble edge. From the vector
~ℓ=r~′ ~r,weconstructthetensor~ℓ ~ℓ=(ℓ ℓ ). Thisten- variable, constant in a steady flow, and is not necessar-
− ⊗ i j ily homogeneous through the whole sample, allowing a
soraveragedover (R~)definesthelocal“texturetensor”
V thermodynamic description of non-equilibrium complex
M(R~): fluids [8].
Do the elastic stress and the statistical strain relate
Mij =hℓiℓjiV(R~), deterministically? In Fig. 5 we plot the normal stress
difference σ σ versus U U ; in this represen-
xx yy xx yy
see details in the companion paper [7]. This symmet- − −
tation, the isotropic contributions (of M in U, and of
ric tensor has two strictly positive eigenvalues, both of 0
order ℓ2 , the largest being in the direction in which thepressurestressinσ)playnorole,hencewereferonly
(cid:10) (cid:11) to the foam’s current state. Each data point is a mea-
bubbles elongate. It reflects at large scales the relevant
surement which derives from averages at one position of
featuresoftheactualmicrostructureofthematerial. For
instance, Fig. 4 shows an example of M(R~) measuredin the foam: data sources are the same as in Figs. 3 and
4. That is, in principle Fig. 5 measures a constitutive
the flowingfoamexperiment ofFig. 3. This texture ten-
relationfromasingleimage;inpractice,weaverage2800
sor M(R~) thus quantifies the qualitative impression of
similar images to improve the precision.
compression or elongation we obtain by looking at the
Since M, and hence U, is completely independent of
bubbles in each region of Fig. 1. The smooth and con-
tinuous variations of the statistical strain tensor field in σ [7], the high correlation between U and σ apparent in
spacevalidatetreatingthefoamasacontinuousmedium. Fig. 5 reflects the physical constitutive relation required
to treat the foam as a continuous medium, in which de-
tails of the microstructure appear only through meso-
scopic averages. Since different applied strains u can
appl
correspondto the same σ, such relation does not appear
in classical σ vs.u plots [6].
appl
2
FIG. 5. Plot of U versus σ. The color code is the same [2] L. D. Landau, E. M. Lifschitz, Fluid Mechanics (Reed
as in Fig. 3, showing the sign of the normal stress difference Educational and Professional Publishing, Oxford, ed.2,
σxx−σyy: blue, negative; black, zero within error bars; red, 2000).
positive. We omit data from the boxes touching the channel [3] A. Meheboob, and S. Luding, Rheology of bidisperse
wall. Theblacklineisalinearfit. Onthesamegraph,weplot granular mixtures via event driven simulations (2002),
σxy (violet) versus Uxy, shifted by −1 mN.m−1 for clarity. to appear in J. Fluid Mech.
[4] Inpractice,weapproximateeˆbyusingtheunitvectorof
thechordconnectingthetwovertices.Thisiscorrectfora
Moreover, the relation between U and σ is linear over
2Dfoamatequilibrium[F.Graner,Y.Jiang,E.Janiaud,
the whole range covered by this experiment. The slope
C. Flament, Phys. Rev. E, (2001) 63 pp. 011402/1-13].
of Fig. 5 measures 2µ, where µ is the shear modulus of
In a 2D foam at low flow rate, the error we introduce is
the foam, much beyond the validity regime of classical
negligible.
elasticity. We find, for the xx yy component:
[5] L. E. Malvern, Introduction to the Mechanics of a Con-
−
tinuous Medium (Prentice-Hall, Inc., NewJersey, 1969).
µxx−yy =3.00±0.005mN.m−1. [6] D. A. Reinelt, A.M. Kraynik, J. Rheol. 44, 453 (2000).
[7] M. Aubouy, Y. Jiang, J. A. Glazier, F. Graner, “A tex-
The foam is nearly isotropic, since we find almost the
turetensortoquantifydeformations”, companionpaper,
same value for the xy component:
Granular Matter, same issue.
µ =3.06 0.01mN.m−1. [8] G. Porte, J.-F. Berret, J. Harden, J. Phys. II France 7,
xy ± 459 (1997).
[9] D. Weaire, S. Hutzler, Physics of Foams (Oxford Univ.
Note that we find similar values, 3.00 0.01 and 3.01
± ± Press, Oxford, 1999).
0.01 respectively, for a foam flowing relatively to a fixed
[10] H. M. Princen, J. Coll. Interf. Sci. 91, 160 (1983).
round obstacle (data not shown). A regular “honey-
[11] S. Khan, R. Armstrong, J. Non-Newtonian Fluid Mech
comb” hexagonal lattice has an isotropic shear modu-
22, 1(1986).
lus µh = λ/(ℓ√3), hence with the same tension and av- [12] S. Alexander, Phys. Reports 296, 65 (1998).
erage bubble area A it would have a shear modulus
µ = 0.465 λ/ Ah =i 2.52 mN.m−1 [9–11]. Our value
h
ph i
µ = 3.0 is higher: µ/µ = 1.2. This difference is likely
h
due to the side length disorder ℓ2 / ℓ 2 = 1.22 of the
(cid:10) (cid:11) h i
foam. How µ/µ depends on the microstructure and its
h
disorder is an open problem [12].
Wehaveperformedthefollowingtestsofthevalidityof
ourmethod. Wehavecheckedtheagreementbetweenour
measurementofstatisticalstrainandtheclassicalstrain,
first analytically on a honeycomb lattice, then numeri-
cally on a disordered simulated foam kept in the elastic
regime. We have checked that if we increase the size of
the volume element, we decrease the fluctuations, and
thus the upper and lower values of σ and U in Fig. 5,
but the slope and hence µ are unchanged. Finally, the
value of the shear stress tensor field or shear modulus,
resulting from our image analysis, predicts the force ex-
ertedbytheflowingfoamonanobstacle. Wehavetested
it against an independent, macroscopic measurement of
the force, and found that they agree(S. Courtyet al., in
preparation).
In summary, we have measured the stress, the texture
tensor and the statistical strain for a 2D flowing foam.
We have shown that the 2D foam behaves like a linear
and isotropic continuous material.
[1] L.D.Landau,E.M.Lifschitz,TheoryofElasticity(Reed
Educational and Professional Publishing, Oxford, ed. 3,
1986).
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σ −σ (mN/m) vs U − U →
2 xx yy xx yy
1
0
−1
−2
← σ (mN/m) vs U
−3 xy xy
−4
−0.6 −0.4 −0.2 0 0.2 0.4 0.6