The properties of dense granular systems are analyzed from a hydrodynamical
point of view, based on conservation laws for the particle number density and
linear momentum. We discuss averaging problems associated with the nature of
such systems and the peculiarities of the sources of noise. We perform a
quantitative study by combining analytical methods and numerical results
obtained by ensemble-averaging of data on creep during compaction and molecular
dynamics simulations of convective flow. We show that numerical integration of
the hydrodynamic equations gives the expected evolution for the time-dependent
fields.Comment: 10 pages, 7 figure

Esipov, Sergei E.

Poeschel, Thorsten

Saluena, Clara

English

arXiv

We analyze the properties of dense granular systems by assuming a hydrodynamical description based on conservation laws for the particle number density and linear momentum. We combine analytical methods and experimental and numerical results obtained by ensemble-averaging of data on creep during compaction and molecular dynamics simulations of convective flow

Saluneña, Clara

Pöschel, Thorsten

OPEN FAU Online-Publikationssystem der Friedrich-Alexander-Universität Erlangen-Nürnberg

In: R.P. Behringer, J. T. Jenkins (Eds.), Powders & Grains’97, Balkema (Rotterdam, 1997), p.341-344Hydrodynamic fluctuations and averaging problems in dense granular flowsClara SalueñaJames Franck Institute and the Department of Physics, University of Chicago, US, andDepartament de F́ısica Fonamental, Universitat de Barcelona, SpainSergei E. EsipovJames Franck Institute and the Department of Physics, University of Chicago, USThorsten PöschelInstitut für Physik, Humboldt-Universität zu Berlin, GermanyABSTRACT: We analize the properties of dense granular systems by assuming a hydrodynamical de-scription, based on conservation laws for the particle number density and linear momentum. We combineanalytical methods and experimental and numerical results obtained by ensemble-averaging of data oncreep during compaction and molecular dynamics simulations of convective flow.1 INTRODUCTIONIn spite of the early and in many aspects suc-cessful hydrodynamical approach applied by Haff(P. K. Haff 1983 1986) to granular systems, therestill remain many unsolved questions, since the be-havior of a granular mass is fundamentally differ-ent from that of typical fluids. For instance, inaddition to the ordinary source of hydrodynamicfluctuations –which in this case can’t be removedby taking the thermodynamic limit, the fact thatsolid grains in a dense arrangement can’t be re-garded as points in any length scale leads to asecond source of fluctuations; it operates at dis-tances much larger than the typical diameter ofthe particles and is related to the appearance ofextensive arrangements inside the system. Givena mean density not far away from the close-packinglimit, different actual realizations in the configura-tion space of the particles may lead to departingdynamic properties and, ultimately, in generatingcompletely different time-sequences. This contri-bution is what we call non-local noise. Both, localand non-local noise, are always present but theirrelative importance strongly depends on the forc-ing applied to the system, measured by the param-eter Fg = f/ρg, f being the volume density of theforcing and g the acceleration of gravity.We propose the existence of two regimes; in theweak-forcing limit, Fg<∼ 1, the non-local com-ponent of the noise dominates, and consequentlyone expects very long relaxation rates and non-self-averaging quantities. In this limit, only ensembleaveraging is meaningful, as in every possible rea-lization the system explores a small region of theconfigurational space. The glass-like behavior ismost apparent. In the opposite limit, Fg À 1,the system is not easily trapped in an immobilearrangement, and one can safely assume that ina sufficiently long time it explores the represen-tative part of the configuration space. Time ave-raging can substitute ensemble averaging only inthis limit. Consistently with this picture, the cri-tical density, ρc, is not unique in general, but is adistributed quantity depending on the configura-tional state, Γ. Actually, we shall see that expe-rimental data on compaction at weak forcing dis-play quenched behavior, and the final density mayvary over more than 10%. Instead, our numericalstudy of granular convection under strong forcingindicates a much narrower histogram of maximalachieved densities, ρc, despite the fact the numberof particles and the used number of samples foraveraging were much smaller.We show that it is possible to understand bothlimits within the frame of hydrodynamic equa-tions, which are in general stochastic equationsincluding both local and non-local noise. It is be-yond of the scope of this paper a detailed analysisof the properties of such equations, which is ex-tensively done elsewhere. Our aim is to presentsome results of our study of the evolution of themean hydrodinamic fields, comparing them withtheir observed behavior in a sample of cases.2 HYDRODYNAMIC EQUATIONSBy hydrodynamic equations we mean balanceequations for mean mesoscopic hydrodynamicfields. The lack of an intermediate length scale–contrary to what happens in simple fluids for ins-tance, containing a sufficiently large number ofparticles such that local fluctuations fade away,adds on the above exposed problem of non-localfluctuations due to the intrinsic granular nature ofthe system, operating at length scales where hydro-dynamic fields can already be defined. The formercan be modelled as the usual additive stochasticcontribution to the dissipative flows and comes re-lated to the existence of some kind of ”tempera-ture”, the latter enters via distributed kinetic co-efficients (depending on the configurational stateΓ). In the continuum approach, conservation ofmass and linear momentum read∂tρ+∇(ρv) = 0, (1a)ρ∂tvi =∂σij∂xj+ fi, (1b)in the Stokes approximation, and where fi arethe components of the volume density of forcing.As for the terms constituting the stress tensor σija few comments are in order. Provided that thegranular particles may be modelled as sufficientlyhard spheres, we neglect any elastic contributionother than that introduced by pressure effects, andassume that the non-equilibrium part of σij is alocal functional of the derivatives of the local ve-locity,σij = −pδij + η(ρ,Γ)( ∂vi∂xj+∂vj∂xi−23δij∇ · v)(2a)+ ζ(ρ,Γ)δij∇ · v + ξij.where p is the pressure and η and ζ the shear andvolume viscosities, respectively.2.1 The role of temperature and the pressure termOur molecular dynamic simulations indicate thatthe evolution of the velocity field in dense granularflow is nearly independent from the granular, tem-perature, defined as the mean fluctuational part ofthe velocity. Effectively, well inside the bulk, thequantity δv/v, measuring such fluctuational devi-ations from the mean velocity v, is typically aboutsix orders of magnitude smaller than close to theboundary. This and other evidences allow us tosuppose that granular flows in dense systems can’tbe sustained by a temperature-based mechanismalone –unlike Rayleigh-Benard convection in sim-ple fluids, for instance. Note that the equation forthe energy balance has been omitted; consistentlywith the observation that the granular tempera-ture plays no significant role, its evolution appearsdecoupled from the previous system of equations.Similarly, one cannot account for elastic contri-butions in the high density limit only by usinga thermal pressure. Therefore, one has to modelsuch terms by means of an artificial equation ofstate which must help to resolve the delicate limitρ→ ρc. We assumed the most simple dependence,p = p0/(1 − ρ/ρc), where p0 represents a certainconstant, in our numerical integration of the hy-drodynamic equations in the strong forcing regime.2.2 The viscosityAccordingly, the model that we adopt for the vis-cosity is not thermal, but glass-like. In dense clus-ters, in order to move, a complex rearrangement ofparticles has to occur making use of voids. Similarproperties are exhibited by glasses. Available ex-perimental data and our numerical results indicatethe presence of a factor exp[c/(1 − ρ/ρc))] in themean flow rates, where c is a dimensionless num-ber. This formula is related to the Vögel-Fultcherlaw for glasses (see N. F. Mott & E. A. Davis 1979);it measures the number of attempts needed for onestep in the direction of average flow in a dense gra-nular system. Sufficiently close to ρc, we then ex-pect a shear viscosity of the formη(ρ,Γ) = η0(ρ,Γ) exp( c1− ρ/ρc(Γ)). (3)and a similar dependence for the bulk viscosity,ζ(ρ,Γ).3 RESULTSWe focus on two different examples which are rep-resentative of each regime. For the weak forcinglimit, we use data on compaction of sand duringtapping experiments (J. B. Knight et al. 1995).For the strong forcing limit we perform exten-sive ensemble averaging of samples generated bymolecular dynamic simulations of vertical shaking,–other geometries can be studied, see for exam-ple (S. E. Esipov et al., 1996) for an study underhorizontal shaking, comparing them with resultsfrom the integrated hydrodynamic equations forthe mean fields.3.1 Weak forcing limit. Application to compactionexperimentsThese results provide additional support for theexistence of non-local noise, and evidence that themean flows are of hydrodynamic nature even invery dense limits. Beginning with a loosely packedsand at volume fraction ρ0 = 0.57 the authors re-port a logarithmic density growth,ρc − ρ(t) =A1 +B ln(1 + t/τ), (4)where A,B, τ, ρc are four fitting parameters. Itcan be shown that it is possible to retrieve such adependence by integration of hydrodynamic equa-tions. Omitting further details about calculationsand average over non-local noise, one finds afterintegration of the 1-dimensional version of (1) atlate timesρ̄c − ρ(t) =cln[(∫ t0dtF )/cη̄0], (5)ρ̄c and c̄, already averaged over non-local noise, areconstants which depend on, say, the amplitude offorcing, but do not change over time. The quan-tity F is related to the integral of the density offorcing and is left unspecified. Equation (5) canbe compared with experimential fit, (see Figure1) assuming∫ t0dtF = t〈F 〉, where the average istaken over a period of repeated tapping. We findA/B = c̄, τ = cη̄0/〈F 〉.This is a three parameter fit. It demonstratesthat hydrodynamics may be used for analyzingexperimental data. Different fitting values of ρ̄c,c̄ support the assumption that granular configura-tions with different ρc, c (different states Γ) do notcommunicate at weak forcing.3.1 Strong forcing limitThis case is object of a more complete study. Itis well known that sand under periodic verticalshaking and gravity in a container develops typi-cal convective rolls, with sand going upwards in-side, and downwards along the vertical walls. Themotion is evidenced, for example, by the bulgingcolored stripes resulting from NMR experiments(E. E. Ehrichs et al. 1995), which represent cycle-averaged displacements. Again, it is possible toshow that the system of equations (1) can be inte-grated to give a good fit of the experimental data(S. E. Esipov et al., 1996). A major understandingof the motion requires, however, a time-resolvedanalysis, and we show how this can be done viamolecular dynamics simulations.The details of the simulations will be omittedhere, it suffices to say that we used a polydispersesample of 2000 soft spheres in a tall rectangularcontainer, and the chosen values for the frictionparameters reproduce correctly the experimentalNMR images mentioned above. Fg was about 2.The following steps summarize the procedure weused:1. Time discretization. Each period of shakingis divided in an equal number of frames, where po-sitions of particles are recorded.2. Spatial discretization and averaging. Usinga high resolution grid, the container is divided incells of the size of the order of one particle. Timeaveraging –which can replace ensemble averagingin this case, is performed with data of the corres-ponding frames of more of 100 periods of shaking.3. Mean density, velocity and temperature(mean fluctuational velocity) are obtained and dis-played. As an example, we reproduce in Figure2a the horizontal component of the velocity, vx.Observe that the motion is complex and unex-pected, in the sense that one cannot infer fromthe sequence of pictures of vx (neither from vy, notshown) the direction of the global motion.4. Test of the hydrodynamic equations. By usingthe mean hydrodynamic fields obtained in 3., thesystem of equations (1) is checked. Selected framesprovide fitting values for c=0.15 and ρc = 1.01max(ρ). η0 was found to be about 300 cpoise bycomparing histograms of the tangential force andthe velocity gradient close to the walls, whereas theobservation that the flow is mostly divergence-freeallows us to neglect the effects of ζ.5. Study of boundary conditions. We obtainedeffective boundary conditions for the flow that re-produce to some extent the assumptions of micro-scopic friction during collisions, but we also foundthat the motion of sand along the vertical wallscomes accompanied by dramatic periodic changesin the density and the stress.6. The previous results are used to integrate nu-merically the system of hydrodynamic equations.In Figure 2b we show comparatively the sequenceobtained for vx.4. CONCLUSIONS1. Hydrodynamic equations provide an adequatetheoretical frame for the study of dense granularsystems.2. Temperature-based mechanisms can be prac-tically dismissed in the description of dense granu-lar flows.3. Averaging of data from molecular dynamicsimulations is an useful tool to reveal the details ofthe evolution of hydrodynamic fields.4. Results of numerical integration of the sys-tem of hydrodynamic equations show a qualita-tive agreement with the evolution of hydrodynamicfields.REFERENCESEhrichs E. E., H. M. Jaeger, G. S. Karczmar, V. Yu. Ku-perman, & S. R. Nagel 1995. Granular convection obser-ved by Magnetic Resonance Imaging. Science 267: 1632-1634.Esipov, S. E., C. Salueña, & T. Pöschel 1996. Granularglasses and fluctuational hydrodynamics (preprint).Haff, P. K. 1983. Grain flow as a fluid mechanical problem.J. Fluid Mech. 134: 401-430.Haff, P. K. 1986. A physical picture of kinetic granularflows. J. Rheology 30: 931-948.Knight, J. B., C. G. Fandrich, C. N. Lau, H. M. Jaeger, &S. R. Nagel, 1995. Density relaxation in a vibrated gra-nular material. Phys. Rev. E 51: 3957-3963.Mott N. F. & E. A. Davis 1979. Electronic Properties ofNon-Crystalline Materials. NY: Oxford U. Press.

Hydrodynamic fluctuations and averaging problems in dense granular flows

: We analize the properties of dense granular systems by assuming a hydrodynamical description, based on conservation laws for the particle number density and linear momentum. We combine analytical methods and experimental and numerical results obtained by ensemble-averaging of data on creep during compaction and molecular dynamics simulations of convective flow. 1 INTRODUCTION In spite of the early and in many aspects successful hydrodynamical approach applied by Haff (P. K. Haff 1983 1986) to granular systems, there still remain many unsolved questions, since the behavior of a granular mass is fundamentally different from that of typical fluids. For instance, in addition to the ordinary source of hydrodynamic fluctuations --which in this case can&apos;t be removed by taking the thermodynamic limit, the fact that solid grains in a dense arrangement can&apos;t be regarded as points in any length scale leads to a second source of fluctuations; it operates at distances much larger than the typical..

Clara Salueña

Sergei E. Esipov

Thorsten Poschel

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Hydrodynamic Fluctuations and Averaging Problems in Dense Granular Flows

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<title>Hydrodynamic fluctuations and averaging problems in dense granular flows</title>