Uphill solitary waves in granular flows

0.34 MB·English

by  E. Martinez

Uphill solitary waves in granular flows
E. Mart´inez, C. P´erez-Penichet,O. Sotolongo-Costa,1 O. Ramos, K. J. M˚aløy,2 S. Douady,3 and E. Altshuler4,∗
1“Henri Poincar´e” Group of Complex Systems, Physics Faculty, University of Havana, 10400 Havana, Cuba
2Physics Department, University of Oslo, Blindern, N-0316 Oslo, Norway
3Laboratoire Matire et Systmes Complexes (MSC),
Universit Paris 7 - Denis Diderot / CNRS, CC 7056 75251 Paris Cedex 05 - France
4“Henri Poincar´e” Group of Complex Systems and Superconductivity Laboratory,
6
Physics Faculty-IMRE, University of Havana, 10400 Havana, Cuba
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(Dated: February 6, 2008)
0
2
Wehaveexperimentallyobservedanewphenomenoninthesurfaceflowofagranularmaterial. A

n heapisconstructedbyinjectingsandbetweentwoverticalglassplatesseparatedbyadistancemuch
a largerthantheaveragegrainsize,withanopenboundary. Astheheapreachestheopenboundary,
J “soliton-like” fluctuations appear on the flowing layer, and move “up the hill” (i.e., against the

1 directionoftheflow). Weexplainthephenomenoninthecontextofstop-and-gotrafficmodels,and
3 show that soliton-like behavior is allowed within a Saint-Venantdescription for the granular flow.

] PACSnumbers: 45.70.-n,45.70Mg,45.70.Vn,81.05Rm,89.75.-k
t
f
o
s Therichdynamicsofgranularmatter–studiedforcen- funnel
.
t turies by engineers– has attracted much attention from
a
w
m the Physics community since the early 1990’s [1, 2, 3].
Granular flows, for example, have concentrated intense
-
d interest,due totheirrelevancetonaturalavalanchesand digital video y
n industrial processes, and also because they make liquid- camera
o likeandsolid-likebehaviorscoexist. Theyhavebeenthe- x
c
oreticallydescribedbasedontheexistenceoftwophases:
[
the rolling (or flowing) one, and the static one. Such

1 ideahasbeencastedintoad hoc phenomenologicalequa- d
v
tions [4, 5, 6], into a Saint-Venant hydrodynamic ap-
5
8 proach conveniently modified to take into account the
6 particularities of granular matter [7, 8], and eventually FIG. 1: Simplified diagram of theexperimental setup
1 by defining an order parameter characterizing the local
0
state of the system [9, 10]. Finally, granular flows have
6 wiched between two square glass plates with inner sur-
been described by “microscopic” equations based on the
0
faces separated by a distance w in the range from 0.3cm
/ newtonianmotionofindividualgrainssubmittedtograv-
t to 3cm (Fig. 1). The lengths of the base and the ver-
a ity, shocks, and trapping events [11, 12].
m tical wall were approximately d 23cm. The sand was
In particular, researchers have studied both exper- ≈
poured vertically into the cell near the vertical wall us-
- imentally and theoretically granular flows on inclined
d ing tiny funnels with several hole diameters, in order to
planes and tubes, and granular heaps, finding a whole
n obtaindifferentflux values. Asthe sandwaspouredinto
o jungle ofpatterns, spatio-temporalstructures, andother the cell, a heap grew until it reached the open bound-
c nontrivial phenomena, such as fingering [13], avalanches
ary, where the grains were allowed to fall freely (Fig.
v: extending both downhill and uphill [14], “logitudinal 1). Digital videos were taken using a High Speed Video
i vortices” [15, 16], “bubbling flows” [17, 18],“revolving
X Camera Photron FASTCAM Ultima-APX model 120K
rivers” [19] and even “singing” dunes [20].
in the rangefrom 50fpsto 4000fps,with a resolutionof
r
a So it looks very unlikely that one can find further un- 1024 1024 pixels.
exploredphenomenainexperimentsassimpleaspouring Fig×. 2 contain images from experiments with w =
sandon a heap with an open boundary. However,we re- 0.7cm and F = 0.6cm3/s. Fig. 2 (a) is a picture taken
portheretheexistenceofsoliton-likewavesmovinguphill from a video recordedat 50fps, and show a closeup of a
in such experiments, i.e., the spontaneous appearance of section of the surface at the central region of the heap,
bump-likeinstabilitiesintheflowinglayerthatpropagate where our basic findings are easily identified: “bump-
uphill, contrary to the flow of sand. shaped” instabilities appearing at random places of the
We used sand with a high content of silicon oxide and surface move “up the hill”, as indicated by the white
anaveragegrainsizeof100µmfromSantaTeresa(Pinar arrows. These bumps couldtraveleither alone or in tan-
del R´ıo, Cuba) [19]. The sand was poured into a cell dem. The bumps maintain their ”identity” within vari-
consisting in a horizontal base and a vertical wall, sand- abletimes, whichcanreachmore than10seconds. After
2
FIG. 2: Basic experimental findings. (a) Uphill bumps moving in tandem, identified by arrows. (b) Difference between
two pictures of a single bump, separated by 50ms. In both pictures, the smallest ticks in the scales correspond to 1mm.
(c) Spatial-temporal diagram of bump dynamics, where x grows to the right and t grows upward, spanning 20cm and 2s,
respectively.
that, they flatten out. We have also observed that, if s) 55
m/ 50
two of them move at speeds different enoughto interfere m
with each other, they can be still identified after the in- v)(up4405
teraction. However, these are rare events, and we have s ( 35
p
no videorecordofthem. All inall,our observationshint m 30
u
at a soliton-like behavior in the observed instabilities. b 25
Fig. 2(b) shows the difference between two pictures d of 20
e 15
from a single bump taken from a video recorded at spe 10
500fps,which are separated50ms fromeach other. The hill 5
darker horizontal band allows to visualize the downhill Up 0
motion. Careful examination of picture 2(b) allows to 0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4
separate an approximately 1mm-depth band of “flying” Input flux (F)(cm3/s)
grains,and an approximately1mm depth layer of “flow-
ing” grains underneath. The height of the perturbation FIG. 3: Uphill speeds of bumps measured as a function of
input flux,for w=0.7mm
abovetheunperturbedstreamisdifficulttomeasure,but
it can be roughly estimated as 1/5 of the flowing layer.
Fig. 2(c) shows a spatial-temporaldiagramof the free
surface. A horizontal line of the video record was taken this sequence of images reveals the mechanism of move-
just at the (average) free surface, so lower parts appear ment of the instability at the single grain level. The
black, and higher parts appear white. From the picture grains involved in the flowing layer in picture (a) show
itcanclearlybe seenfourshock-waves,passingsuddenly twotypesofbehavior: whilemostofthegrainskeepflow-
from black to white, moving upward with a well defined ingfromlefttorightatanaveragespeedv 10cm/s
flow
∼
velocity (even if it fluctuates a little on the right). On asthebumppassesunderneathfromrighttoleft,afrac-
thelefttheappearanceofanewsolitoncanbeseen,first tion of the grains at the lower part of the flowing layer
increasing in height before starting suddenly to propa- show a different behavior. The white circle in Fig. 4(a),
gate. represents one of those grains, which is moving from left
The average speed of the uphill motion of the bumps to right. From 4(b) to 4(d), the grain has incorporated
as a function of the input flux, F was estimated from intothestaticlayer(deposition),whilethebumpispass-
the analysis of the motion of several instabilities using ingfromrighttoleftaboveit. InFig. 4(e),thegrainhas
picturesanalogousto thoseusedtoobtainFig2(b). Fig. reincorporated into the flowing layer (erosion), moving
3 presents the dependence of the average uphill speed again from left to right.
versus input flux. The solid line corresponds to vup Exchange of particles between the static and fluid
∼
F1/4, which will be discussed below. phases has been considered in several models of surface
Fig. 4 shows a sequence of pictures of a bump sepa- flows since the 1990’s [4]. We believe that our bumps
ratedbyintervalsof0.125s,extractedfromavideotaken form when a fluctuation implies an extra deposition of
at4000fps. Carefulinspectionofthe videoassociatedto grainsfromtheflowinglayeronthestaticone,whichthen
3
y
v
up
z zzz (x,t)
h hhh (x,t)
h v
0 H(x,t) flow
x
flowing phase static phase
FIG. 5: Diagram showing the neighborhood of a “bump”
which includes some parameters of the Saint-Venant model
described in thetext.
surface of the heap (see Fig. 5) . The first equation
corresponds to mass conservation:
∂ ζ+∂ (ΓH2/2)=0 (1)
t x
FIG. 4: Sequence of pictures of an bump moving uphill,
with thesame parameters described forFig. 2(e). Thewhite whereζ istheheightofthefreesurface,H isthethick-
circle represents a grain of sand (a) moving downhill as part ness of the flowing layer, and Γ is the velocity gradient
of theflowing layer,(b)-(d)trappedin thestaticlayeras the transversal to the flowing layer, which is assumed con-
bump passes by from left to right, and (e) moving again as stant. Then, the momentum q =ΓH2/2 evolves with H
partoftheflowinglayer. Thehorizontallengthofthepicture
due to the erosion/deposition process, like the one illus-
corresponds to 22mm in reality
trated for one grain in Fig. 4. The second equation cor-
responds to the conservation of momentum, which can
be seen as an equation governing the evolution of the
propagates backwardsthrough an “stop-and-go”mecha-
thickness of the flowing layer, H:
nism observed in traffic dynamics [22]. However, in our
case the usual traffic models should be modified to take
into account the free surface. A way to do it could be a g
∂ H +∂ (ΓH2/2)= (tanθ µ(H)) (2)
two-lane system consisting in a fast lane where cars flow t x Γ −
steadily,andaslowlanewherecarsstopandgo,produc-
where µ(H) is the friction acting on the layer and
ing a backward wave (or train of waves) within the slow
tanθ = ∂ ζ is the slope of the free surface.
lane, well described in the literature [22]. x
−
Letusassumethatthatthefrictioncoefficientdepends
All in all, the situation can be pictured as a solitary
on H in the general form:
wavecrawlinguphillunderneathashallowstreamofsand
flowing downhill. In fact, the stop-and-go traffic model
has been mapped into the “classical” Kortweg-de Vries
γ
(KdV) equation after a few approximations, resulting in µ=α+(β+ H)(h20−H2)qδ−ε(h20−H2) (3)
soliton-like solutions [23]. In the context of the KdV
equation, the speed of the soliton can be estimated as where the values of constants α, β, γ, δ and ǫ can be
v √gh whereg andh aretheaccelerationofgravity expressedintermsofexperimentalparametersv ,v ,
s 0 0 up flow
∼
and the depth of the unperturbed stream, respectively h0, g, and a characteristic length L0. Then, equations
[24]. If one assumes the well accepted result that the (1)and(2)canbetransformedintothe“textbook”KdV
depth of the flowing layer is proportional to F1/2 [21], equation [24]
we then get v F1/4, which follows the experimental
s
∼
result shown in Fig. 3.
v
Knowing that the KdV equation can be derived from L20vflowη′′+6(vup−vflow)η+30 fhlowη2 =0 (4)
0
mass and momentum conservation equations applied to
an incompressible fluid flowing down a shallow channel In (4), η = ζ h and η′′ = d2η, where ϕ= x+v t.
[24], we could expect that soliton solutions can be found − 0 dϕ2 up
One particular solution of that equation, is
if we describe our system through hydrodynamics Saint-
Venant equations modified to take into account the par-
ticularitiesofgranularmatter[7,8]whichwearewriting h x+v t
ζ =h + 0sech2( up ) (5)
relative to a reference frame parallel to the average free 0 5 L
0
4
where we have taken L = 10h , v = 1Γh = ments and image processing, and E. Cl`ement, A. Daerr,
0 0 flow 2 0
√gh , and v = 1v in order to match our experi- S.Franz,H.Herrmann,J.Mar´ın,D.Mart´ınez,R.Mulet,
0 up 3 flow
mental observations(i.e., mainly that the uphill speed is D. Stariolo and J.E. Wesfreid for useful discussions and
roughly 1/3 of the flow speed, and the height of the per- comments. G.QuinteroandJ.Ferna´ndezcollaboratedin
turbationisapproximately1/5ofthethicknessoftheun- numerical calculations. We appreciate financial support
perturbed flowing layer). Formula (5) describes a “text- from the “Abdus Salam” ICTP during the last stage of
book”,bell-shapedsolitonmovingupward. Thissolution this project.
holdsforafrictioncoefficientgivenbyequation(3)which
is a positive, decreasing function of H when reasonable
experimental parameters are introduced, in agreement
with previous models [8]. Although we do not discard
∗
[email protected]
that “non-textbook” solitonic equations might describe
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We thank Ø. Johnsen and C. Noda for help in experi-