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In: R.P. Behringer, J. T. Jenkins (Eds.), Powders & Grains'97, Balkema (Rotterdam, 1997), p.341-344

Hydrodynamic uctuations and averaging problems in dense granular ows

Clara Salue~a n James Franck Institute and the Department of Physics, University of Chicago, US, and Departament de F sica Fonamental, Universitat de Barcelona, Spain Sergei E. Esipov James Franck Institute and the Department of Physics, University of Chicago, US Thorsten Poschel Institut fur Physik, Humboldt-Universitat zu Berlin, Germany

ABSTRACT: 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 ow. 1 INTRODUCTION In spite of the early and in many aspects successful hydrodynamical approach applied by Ha (P. K. Ha 1983 1986) to granular systems, there still remain many unsolved questions, since the behavior of a granular mass is fundamentally di erent from that of typical uids. For instance, in addition to the ordinary source of hydrodynamic uctuations {which in this case can't be removed by taking the thermodynamic limit, the fact that solid grains in a dense arrangement can't be regarded as points in any length scale leads to a second source of uctuations; it operates at distances much larger than the typical diameter of the particles and is related to the appearance of extensive arrangements inside the system. Given a mean density not far away from the close-packing limit, di erent actual realizations in the con guration space of the particles may lead to departing dynamic properties and, ultimately, in generating completely di erent time-sequences. This contribution is what we call non-local noise. Both, local and non-local noise, are always present but their relative importance strongly depends on the forcing applied to the system, measured by the parameter F g = f= g, f being the volume density of the forcing and g the acceleration of gravity. We propose the existence of two regimes; in the weak-forcing limit, F g < 1, the non-local component of the noise dominates, and consequently one expects very long relaxation rates and non-selfaveraging quantities. In this limit, only ensemble averaging is meaningful, as in every possible realization the system explores a small region of the con gurational space. The glass-like behavior is most apparent. In the opposite limit, F g 1, the system is not easily trapped in an immobile arrangement, and one can safely assume that in a su ciently long time it explores the representative part of the con guration space. Time averaging can substitute ensemble averaging only in this limit. Consistently with this picture, the critical density, c , is not unique in general, but is a distributed quantity depending on the con gurational state, ?. Actually, we shall see that experimental data on compaction at weak forcing display quenched behavior, and the nal density may vary over more than 10%. Instead, our numerical study of granular convection under strong forcing indicates a much narrower histogram of maximal achieved densities, c, despite the fact the number of particles and the used number of samples for averaging were much smaller. We show that it is possible to understand both limits within the frame of hydrodynamic equations, which are in general stochastic equations including both local and non-local noise. It is be-

yond of the scope of this paper a detailed analysis of the properties of such equations, which is extensively done elsewhere. Our aim is to present some results of our study of the evolution of the mean hydrodinamic elds, comparing them with their observed behavior in a sample of cases. 2 HYDRODYNAMIC EQUATIONS By hydrodynamic equations we mean balance equations for mean mesoscopic hydrodynamic elds. The lack of an intermediate length scale {contrary to what happens in simple uids for instance, containing a su ciently large number of particles such that local uctuations fade away, adds on the above exposed problem of non-local uctuations due to the intrinsic granular nature of the system, operating at length scales where hydrodynamic elds can already be de ned. The former can be modelled as the usual additive stochastic contribution to the dissipative ows and comes related to the existence of some kind of "temperature", the latter enters via distributed kinetic coe cients (depending on the con gurational state ?). In the continuum approach, conservation of mass and linear momentum read

ow is nearly independent from the granular, temperature, de ned as the mean uctuational part of the velocity. E ectively, well inside the bulk, the quantity v=v, measuring such uctuational deviations from the mean velocity v, is typically about six orders of magnitude smaller than close to the boundary. This and other evidences allow us to suppose that granular ows in dense systems can't be sustained by a temperature-based mechanism alone {unlike Rayleigh-Benard convection in simple uids, for instance. Note that the equation for the energy balance has been omitted; consistently with the observation that the granular temperature plays no signi cant role, its evolution appears decoupled from the previous system of equations. Similarly, one cannot account for elastic contributions in the high density limit only by using a thermal pressure. Therefore, one has to model such terms by means of an arti cial equation of state which must help to resolve the delicate limit ! c . We assumed the most simple dependence, p = p0=(1 ? = c ), where p0 represents a certain constant, in our numerical integration of the hydrodynamic equations in the strong forcing regime.

2.2 The viscosity

@t + r( v) = 0; @tvi = @ ij + fi; @x

j

(1a) (1b)

in the Stokes approximation, and where fi are the components of the volume density of forcing. As for the terms constituting the stress tensor ij a few comments are in order. Provided that the granular particles may be modelled as su ciently hard spheres, we neglect any elastic contribution other than that introduced by pressure e ects, and assume that the non-equilibrium part of ij is a local functional of the derivatives of the local velocity, c @v @v = ?p ij + ( ; ?) @xi + @xj ? 2 ij r v (2a) and a similar dependence for the bulk viscosity, ij j i 3 ( ; ?). + ( ; ?) ij r v + ij : where p is the pressure and and the shear and volume viscosities, respectively. 2.1 The role of temperature and the pressure term Our molecular dynamic simulations indicate that the evolution of the velocity eld in dense granular 3 RESULTS We focus on two di erent examples which are representative of each regime. For the weak forcing limit, we use data on compaction of sand during tapping experiments (J. B. Knight et al. 1995). For the strong forcing limit we perform exten-

Accordingly, the model that we adopt for the viscosity is not thermal, but glass-like. In dense clusters, in order to move, a complex rearrangement of particles has to occur making use of voids. Similar properties are exhibited by glasses. Available experimental data and our numerical results indicate the presence of a factor exp c=(1 ? = c ))] in the mean ow rates, where c is a dimensionless number. This formula is related to the Vogel-Fultcher law for glasses (see N. F. Mott & E. A. Davis 1979); it measures the number of attempts needed for one step in the direction of average ow in a dense granular system. Su ciently close to c, we then expect a shear viscosity of the form c (3) ( ; ?) = 0( ; ?) exp 1 ? = (?) :

sive ensemble averaging of samples generated by molecular dynamic simulations of vertical shaking, {other geometries can be studied, see for example (S. E. Esipov et al., 1996) for an study under horizontal shaking, comparing them with results from the integrated hydrodynamic equations for the mean elds. 3.1 Weak forcing limit. Application to compaction experiments These results provide additional support for the existence of non-local noise, and evidence that the mean ows are of hydrodynamic nature even in very dense limits. Beginning with a loosely packed sand at volume fraction 0 = 0:57 the authors report a logarithmic density growth,

This is a three parameter t. It demonstrates that hydrodynamics may be used for analyzing experimental data. Di erent tting values of c , c support the assumption that granular con gurations with di erent c , c (different states ?) do not communicate at weak forcing. 3.1 Strong forcing limit This case is object of a more complete study. It is well known that sand under periodic vertical shaking and gravity in a container develops typical convective rolls, with sand going upwards inside, and downwards along the vertical walls. The motion is evidenced, for example, by the bulging colored stripes resulting from NMR experiments (E. E. Ehrichs et al. 1995), which represent cycleaveraged displacements. Again, it is possible to show that the system of equations (1) can be integrated to give a good t of the experimental data (S. E. Esipov et al., 1996). A major understanding of the motion requires, however, a time-resolved analysis, and we show how this can be done via molecular dynamics simulations. The details of the simulations will be omitted here, it su ces to say that we used a polydisperse sample of 2000 soft spheres in a tall rectangular container, and the chosen values for the friction parameters reproduce correctly the experimental NMR images mentioned above. F g was about 2. The following steps summarize the procedure we used: 1. Time discretization. Each period of shaking is divided in an equal number of frames, where positions of particles are recorded. 2. Spatial discretization and averaging. Using a high resolution grid, the container is divided in cells of the size of the order of one particle. Time averaging {which can replace ensemble averaging in this case, is performed with data of the corresponding frames of more of 100 periods of shaking. 3. Mean density, velocity and temperature (mean uctuational velocity) are obtained and displayed. As an example, we reproduce in Figure 2a the horizontal component of the velocity, vx. Observe that the motion is complex and unexpected, in the sense that one cannot infer from

c?

A (t) = 1 + B ln(1 + t= ) ;

(4)

where A; B; ; c are four tting parameters. It can be shown that it is possible to retrieve such a dependence by integration of hydrodynamic equations. Omitting further details about calculations and average over non-local noise, one nds after integration of the 1-dimensional version of (1) at late times

c?

(t) =

c ; ln ( 0 dtF )=c 0]

Rt

(5)

constants which depend on, say, the amplitude of forcing, but do not change over time. The quantity F is related to the integral of the density of forcing and is left unspeci ed. Equation (5) can be comparedR with experimential t, (see Figure 1) assuming 0t dtF = thF i, where the average is taken over a period of repeated tapping. We nd A=B = c, = c 0=hF i.

c and c, already averaged over non-local noise, are

the sequence of pictures of vx (neither from vy , not shown) the direction of the global motion. 4. Test of the hydrodynamic equations. By using the mean hydrodynamic elds obtained in 3., the system of equations (1) is checked. Selected frames provide tting values for c=0.15 and c = 1.01 max( ). 0 was found to be about 300 cpoise by comparing histograms of the tangential force and the velocity gradient close to the walls, whereas the observation that the ow is mostly divergence-free allows us to neglect the e ects of . 5. Study of boundary conditions. We obtained

e ective boundary conditions for the ow that reproduce to some extent the assumptions of microscopic friction during collisions, but we also found that the motion of sand along the vertical walls comes accompanied by dramatic periodic changes in the density and the stress. 6. The previous results are used to integrate numerically the system of hydrodynamic equations. In Figure 2b we show comparatively the sequence obtained for vx.

4. CONCLUSIONS 1. Hydrodynamic equations provide an adequate theoretical frame for the study of dense granular systems. 2. Temperature-based mechanisms can be practically dismissed in the description of dense granular ows. 3. Averaging of data from molecular dynamic simulations is an useful tool to reveal the details of the evolution of hydrodynamic elds. 4. Results of numerical integration of the sys-

tem of hydrodynamic equations show a qualitative agreement with the evolution of hydrodynamic elds. REFERENCES

Ehrichs E. E., H. M. Jaeger, G. S. Karczmar, V. Yu. Kuperman, & S. R. Nagel 1995. Granular convection observed by Magnetic Resonance Imaging. Science 267: 16321634. Esipov, S. E., C. Salue~a, & T. Poschel 1996. Granular n glasses and uctuational hydrodynamics (preprint). Ha , P. K. 1983. Grain ow as a uid mechanical problem. J. Fluid Mech. 134: 401-430.

Ha , P. K. 1986. A physical picture of kinetic granular ows. 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 of Non-Crystalline Materials. NY: Oxford U. Press.

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