Macroscopic liquid-state molecular hydrodynamics

Experimental evidence and theoretical modeling suggest that piles of confined, high-restitution grains, subject to low-amplitude vibration, can serve as experimentally-accessible analogs for studying a range of liquid-state molecular hydrodynamic processes. Experiments expose single-grain and multiple-grain, collective dynamic features that mimic those either observed or predicted in molecular-scale, liquid state systems, including: (i) near-collision-time-scale hydrodynamic organization of single-molecule dynamics, (ii) nonequilibrium, long-time-scale excitation of collective/hydrodynamic modes, and (iii) long-time-scale emergence of continuum, viscous flow. In order to connect directly observable macroscale granular dynamics to inaccessible and/or indirectly measured molecular hydrodynamic processes, we recast traditional microscale equilibrium and nonequilibrium statistical mechanics for dense, interacting microscale systems into self-consistent, macroscale form. The proposed macroscopic models, which appear to be new with respect to granular physics, and which differ significantly from traditional kinetic-theory-based, macroscale statistical mechanics models, are used to rigorously derive the continuum equations governing viscous, liquid-like granular flow. The models allow physically-consistent interpretation and prediction of observed equilibrium and non-equilibrium, single-grain, and collective, multiple-grain dynamics.

Equilibrium thermodynamic model for vibrated grain systems As in N-particle microscale problems, the N-grain-system equilibrium thermodynamic model requires that the effective system entropy, S, be defined in terms of the number of (athermal, mechanical) grain energy states, Ω, that are accessible to the system, specifically, the number of states within some (uncertainty-determined) interval, (E − ∆E, E + ∆E) : S = k e lnΩ, where again k e , the effective Boltzmann constant, is determined by how the effective grain temperature scale is defined. Importantly, as in microscale thermodynamics, this definition of S incorporates all of the essential properties of the traditional entropy, at least in fixed mass, fixed-composition/non-reacting systems, apropos confined, high-restitution grain systems undergoing low-amplitude vibration: i) S is maximized at equilibrium, ii) the magnitude of S depends, due to Ω s dependencies, only on N, the system (time-average) volume, V, and E, iii) due to the statistical independence of subsystems, subsystem entropies are additive, and iv) S is zero in non-vibrated systems.
Given the microcanonical definition relating S and Ω, or more generally, the canonical relation, S = −k e i p i lnp i , where p i is given above, the traditional Gibbs thermodynamic framework can be used to self-consistently calculate any effective equilibrium thermodynamic property. Specifically, assuming that the long-time average peculiar dynamics at any given point are stationary, or more to the point, that local, nominally-Maxwellian equilibrium exists, then given a realistic (Hamiltonian) model of N-grain-subsystem dynamics, an associated partition function, Q = M j exp (β e E j ) , can be determined, and in turn, used to compute local equilibrium thermodynamic properties.

Hamiltonian dynamics in high-restitution grain piles
A central assumption, on which rests the model of equilibrium statistical mechanics, is that peculiar grain dynamics are nominally Hamiltonian. This assumption, which is well-met in our experiments, at least for steel, aluminum, ceramic, and similar grain materials, requires that the characteristic grain velocity, v o = Aω, is less than the critical velocity, v c , separating nominally elastic and elastic-viscoelastic grain collisions; 4 here, A and ω are the characteristic vibration amplitude and frequency, ω = 2πf o . Under these conditions, collisions are nominally elastic and dissipationless, so that collision time-scale grain dynamics can be described in terms of a Hamiltonian.
In order to consider this crucial question in detail, we note that a Hamiltonian statistical mechanical model of high-restitution grains undergoing low amplitude vibration neglects two sources of dissipation: sliding frictional contact between colliding grains, and thermal dissipation of elastoacoustic energy within grains. A straightforward scale analysis of the equation of motion of indi-vidual grains within the vibrating pile, stated in the container-fixed, non-inertial frame: shows that, for the conditions in our experiments: i) the inertial force, F inertia = m g Aω 2 , is roughly an order of magnitude larger than grain weight: ii) the ratio of characteristic elastic to inertial forces, F elast,z m g Aω 2 −1 = O 10 −4 , where, for , is the inertial pressure exerted on the grain; and most importantly, iii) the ratio of characteristic friction to inertial force, F f ric,z m g Aω 2 −1 = O 10 −2 , where F f ric,z ≈ µ k m g (Aω) 2 d −1 g , and where µ k ≈ 10 −1 is the coefficient of kinetic friction.
Thus, corresponding ratios involving the characteristic grain kinetic energy, elastic potential energy, and frictional energy dissipation, m g (Aω) 2 , F elast,z ∆z e , and F f ric,z A, respectively, exhibit either identical magnitudes, or in the case of F elast,z m g (Aω The above estimates apply to grain dynamics that include, and are dominated by, the solidlike vibration of the grain pile-container system. As shown in Fig. 5

Grain pile solidification
Vibrating grain beds solidify, i.e., long-time-scale granular flow ceases, at depths exceeding a critical depth, z c . This depth can be estimated for a vertically vibrating grain column of constant crosssectional area by balancing the total characteristic kinetic energy of the unsolidified grain layers above z c against the total elastic potential energy of the solidified and elastically strained grain layers below z c : where L o = N tot d g is the total static height of the grain column, N tot is the total number of grain layers, each of thickness d g , within the column, N the number of solidified layers below z c , m o is the total mass of the grains within any layer, andg = g + Aω 2 is the sum of the gravitational and

Necessary experimental conditions
In order to use a vibrated grain system as a molecular liquid or dense gas analog, two experimental conditions should be met. First, in order to take advantage of the existing theoretical machinery developed for molecular hydrodynamic systems, couched as it is in terms of the pair correlation function, g(r) 5, 6 one should ensure that peculiar grain dynamics, over regions that are several times larger than the PIV interrogation area (2D problems) or volume (3D), satisfy nominal translational and rotational invariance. Second, in order to reliably reproduce molecular-scale interactions, it is necessary to design intergranular potentials, u(r), that capture those believed to exist in the molecular liquid of interest. For example, the hard sphere potential has served as a realistic, zeroth order model of intermolecular interactions in simple atomic liquids. 5,6 For these liquids, the elastic collisions that characterize the dynamics of high restitution grain piles under low amplitude vibration are expected to produce realistic, predictive single atom dynamics and collective N-atom hydrodynamics. 2) Contrary to theoretically predicted sub-collision time scale, single atom dynamics in simple liquids and dense gasses, 9 sub-collision time scale, individual grain dynamics are not ballistic, but overwhelmingly determined by collective, long time scale hydrodynamics. We propose a simple hydrodynamic model to explain the dynamics manifested in Fig. 4.
3) In order to further enforce physical consistency between atomic microscale systems and macroscale analogs, any proposed macroscale statistical mechanical model must allow calculation, via the Kubo relations, 8,9,10 of non-equilibrium transport coefficients. In vibration-driven grain flows, the effective kinematic viscosity, ν e , comprising the key transport property, must be calculable. We present a rough scaling estimate of ν e and show that the estimate is comparable to experimentally measured effective viscosities. 11 4) On hydrodynamic time and length scales, dynamical consistency between atomic liquid/dense gas systems and analog grain systems requires that vibration-induced grain flows correspond to those produced in identically vibrated liquids/dense gasses. As described in the Article and as detailed below, hydrodynamic-scale consistency is tested in preliminary fashion by solving the NS equations for vibration-driven flow of a constant viscosity Newtonian fluid in a vibrated hemisphere.
A non-trivial problem that we also treat concerns development of continuum boundary conditions appropriate to smooth, vibrated grain container walls.

Hydrodynamic conservation equations
We adapt the molecular-scale arguments in 7,8,12 to derive the instantaneous, grain-scale equations of mass, momentum, and energy conservation. Thus, consider a large volume of elastic grains undergoing low-amplitude, single frequency vibration within a smooth-walled, rigid container. In a reference frame attached to the container, and over time scales that are long relative to the characteristic inter-granular collision rate, τ = f −1 o , where f o is the vibration frequency, assume that an equilibrium state exists in which no bulk displacement of the grains occurs. At some instant, a small perturbing force acts on one, or a small number of grains simultaneously, producing mass, momentum, and energy currents in the surrounding grains. Under low-amplitude vibration, the perturbing force is assumed large enough to produce a collective response, on length scales that are large relative to characteristic grain size, d g , but small enough that long-time-scale advective transport remains negligible. Both assumptions are crucial since they allow physically reasonable definitions of local, spatially varying densities and currents.
Introduce definitions of the local, instantaneous grain number density, n (r, t) , momentum density, g (r, t) , and energy density, e (r, t) : where r κ (t) and P κ (t) , are the instantaneous position and momentum of grain κ, and where the delta function acts as a counter for grains that are within some spherical radius, r o (t), of r : Due to the large size of the grains, N cannot be arbitrarily large. Rather, constraints must be met when defining N , as well as the time interval, ∆t, over which we recast system dynamics into continuum form: i) The characteristic size of the system, L ≈ N 1/3 d g , must be small relative to characteristic size, L c , of the container holding the grains. ii) ∆t must be long enough to ensure local equilibrium, i.e., ∆t >> τ c = f −1 o , but short enough that thermal dispersion (produced by grain peculiar kinetic energy) of the N'-grain system remains negligible. The latter constraint ensures validity of momentum conservation, as applied to the system at any instant, t, and can be understood by a physical/intuitive derivation of the Navier-Stokes equations. [A balance of instantaneous surface and body forces acting on an arbitrary fluid particle, combined with introduction of an assumed linear/Newtonian constitutive relationship between surface stresses and local velocity gradients, leads directly to these equations. Importantly, the derivation fails if the time step used to determine the particle's momentum change is so large that significant thermal particle dispersion occurs.] The time scale for thermal dispersion of the N grain fluid particle is on the order of connected through the Einstein and Stokes relations (see, e.g., 13 ), D = k e T e (6πd g,e µ e ) −1 , to the effective grain diameter, d g,e , (see below), and the effective grain viscosity, µ e (see below). Note that where, for example, in strongly two-dimensional, locally isotropic grain flows (like those near the grain pile surface in our experimental system), i = 1, 2.
[Aside: For modeling purposes, it is often convenient to replace non-spherical grains with equivalent spherical grains. A geometric conversion, expected to provide only semi-quantitative predictions of grain dynamics, follows by, e.g., setting the actual grain volume equal to an equivalent sphere.
This then leads to an effective d g,e .] Conservation equations, governing the instantaneous continuum dynamics of an arbitrary system of N grains lying withinr of r, are derived in wavenumber space, following. 12 This approach assumes a small disturbance to the equilibrium state of an N'-grain system, lying within a large equilibrium reservoir, and explicitly captures the long wavelength, slow time-scale (hydrodynamic) system response. An alternative approach, illustrated here via the mass conservation equation: i) bypasses Fourier transforms, ii) explicitly demonstrates mass, momentum, and energy conservation, and iii) shows that each term in each continuum conservation law represents a volumetric average over the instantaneous grain system volume, V o (t) . It is important to note that in both approaches, the smeared delta function, δ (r − r κ (t)) , is the mathematical device that spatially homogenizes discrete number, momentum, and energy densities, allowing recasting of the conservation laws in continuum form.
Thus, consider a system of N grains lying within a deforming, translating volume V o (t), having translates with the local mean velocity, u (r, t) = κ v κ , and where an equilibrium ensemble is used for averaging. The instantaneous mass of grains lying within V o (t) is given by Taking the time derivative inside the integral gives Next, calculate the net mass flux through the surface and use the divergence theorem to obtain: Since δ ,rκ = −δ ,r at all instantaneous grain positions, r κ (t) , where subscripts denote partial derivatives, then Three results follow from Eq. (9). First, by the Liebnitz theorem, the two terms on the left are identically equal to proving that the mass of the N −grain system is conserved. Second, by application of the mean value theorem, we obtain the differential mass conservation law: which holds at some location, r * , within V o (t) . Third, it is clear that each term in Eq. (10) can be interpreted as the instantaneous volume-average of that term, over V o (t) .
The hydrodynamic momentum and energy conservation laws are most easily derived by transforming to wavenumber space. 9,12 The resulting equations are given by: 8 and ∂e ∂t + ∇ · j e = 0 (12) where τ is the momentum current density, or equivalently, the stress tensor: and where j e is the energy current density: Here, φ |r κλ |, t = φ |r κ (t) − r λ (t)| is the elastic potential energy between pairs of directly contacting grains, as well as between pairs of separated grains, instantaneously connected through randomly connecting and disconnecting chains of elastic contact points. In order to take advantage of the machinery developed for pairwise potentials, we express the elastic potential in this fashion.
Likewise, φ i r κλ is the i th component of the corresponding elastic force between directly contacting and indirectly contacting grain pairs.
Given exact continuum conservation laws, the generalized Navier-Stokes equations follow by expanding long-time/ensemble-averaged (local) mass, momentum, and energy current densities in terms of both non-derivative reactive terms, which follow from Galilean transformation between the local bulk fluid motion and the local rest state, and derivative dissipative terms which capture, e.g., Fourier thermal conduction and Newtonian frictional/viscous stresses. See. where : where d, is the dimension of the hydrodynamic (response) flow.
Turning to calculation of the single grain velocity autocorrelation function, wherer = r −r o , andr o is the grain's initial position. In other words, the single-grain ensemble average velocity at timet is approximated as the grain fluid velocity at the same instant, while the initial grain velocity is approximated as the fluid velocity at the instant of vibration-induced Finally, due to the short time scale, single grain displacement is small, i.e.,r o ≈ 0, so that thus providing a purely hydrodynamic mechanism -pure diffusion of vibration-induced momentum injection -for explaining the short-time, single grain dynamics exposed in Fig. 4.

Consistency check 1: Scaling estimate of the effective grain viscosity via a macroscale Kubo relation
Here, a Kubo relation 10 is used in a scaling estimate of the grain fluid's effective shear viscosity, µ e , which, in turn, is compared against experimentally measured viscosities. 11 This represents a significant consistency check since the Kubo relations: i) assume existence of hydrodynamic-scale conservation laws associated with conserved microscale variables, 7, 9 ii) assume existence of a Hamiltonian N-particle system, initially in thermodynamic equilibrium, but subject to small, nonequilibrium fluctuations or forcing, iii) rigorously determine hydrodynamic transport coefficients in terms of correlation functions of current densities, iv) require equilibrium ensemble averaging, and v) connect long-time-and long-length-scale particle dynamics, typically beyond direct observation in microscale systems, to directly observable continuum dynamics.
At least three approaches have been used to derive the Kubo relation for µ e . 8,9,10,12,14,15 Using any of these approaches, the shear viscosity can be expressed as the cumulative, long-time evolution of the grain-scale transverse momentum current, i.e., shear stress, arising in response to a perturbation to the transverse momentum density: 9 where the transverse grain-scale stress tensor component is given by Here, the random or imposed perturbation acts in the x−direction, y is a transverse direction, r κλ = |r λ − r κ | is the spatial separation between grains κ and λ, v κ (t) and P λ (t) are the instantaneous velocity and momentum of grains κ and λ, relative to the local mean velocity, and u κλ represents the separation-dependent elastic potential between grain pairs. See above for a detailed version of τ.
In order to apply (21) to a grain pile undergoing low-amplitude vibration, we first use the equipartition theorem to estimate β e , the Lagrange multiplier determined by satisfying the canonical ensemble energy constraint. Thus, model pair-wise elastic contact between grain pairs via equivalent elastic springs, note that the corresponding N-grain Hamiltonian is quadratic in its 3N momenta and 3N spatial coordinates, and use the equipartition theorem -see, e.g., 16 for a generic developmentto obtain: where A and ω are respectively the vibration amplitude and frequency. Thus, as in microscale problems, the macroscale single-grain energy scale, β −1 e , is on the order of the average kinetic energy of the short, random, collision-time-scale motion of individual grains. Clearly, an equivalent grain temperature could be defined, based on (23). Note too that we obtain the same scaling, β −1 e ∼ A 2 ω 2 , for Hamiltonians that incorporate grain rotational energy.
Next, assume that the grain-scale transverse stress autocorrelation function, τ xy (t) τ xy (0) , decays exponentially, τ xy (t) τ xy (0) = τ xy (0) τ xy (0) exp −t/τ c , and assume that the time constant, τ c , is on the order of the grain collision time scale, which in turn corresponds to the inverse vibration ρ e is the effective density of N grains, each of mass m g , occupying volume V, and carrying out the integral, finally leads to where ν e = µ e ρ −1 e is the effective momentum diffusivity, i.e., the kinematic viscosity. In order to test the rough estimate in (24), we compare it against experimentally measured kinematic viscosities, obtained for grain piles comprised of one of nine different grain shapes, each shape composed of either high density plastic, ceramic, or high density ceramic. 11 These experiments, to be reported in a another paper, 11 directly measure the total, time averaged drag force produced on an instrumented cylinder, submerged in vibrating grain piles; see 11 for details.
As shown in Fig. 1 iii) azimuthal back-and-forth vibration. In a non-inertial reference frame fixed to the bowl, the non-dimensional, constant viscosity NS equations, stated in terms of vorticity, ω, take the form: where Re = v s R/ν is the Reynolds number, v is the local grain fluid velocity, v s ≈ 2 10 −2 ms −1 , is the observed long-time scale average velocity (as measured by PIV or direct observation), ν ≈ 4 10 −3 m 2 s −1 , is the characteristic kinematic viscosity, as measured in 11 and as estimated above, R ≈ 10 −1 m, is the hemisphere/bowl radius, and D/Dt is the material derivative operator.
With regard to an assumed constant dynamic viscosity, µ,

Vorticity boundary condition
The walls of the vibrational bowl in our experiments are smooth, and in flow field measurements about fixed smooth plates, we observe almost pure slip flow, with essentially no sticking. Focusing for illustration on the boundary condition associated with precessive bowl rotation, and referring to where, for small amplitude vibration, the diameter of the RG is significantly larger than A; ii) over one vibration cycle, the RG traces out a bowl-grain interaction envelope, while a fixed point on the wall traces out a (much) smaller circle of radius r (where the latter can be readily determined as a function of vibration amplitude, A, and position, r,θ , on the bowl wall); iii) fill the latter circle with a virtual solid disk of grain material, i.e., recognize that over time scales long relative to f −1 o , the circle is, on average, occupied by grains (colliding randomly with the wall); and iv) on the same long time scale, assume that the virtual grain disk rotates at the precessive frequency, f o .
Next, focus on the long-time-scale dynamics of the virtual disk and assume that the total, time- ∆m v 2 (r) r dt = r r,θ τ r 2πr r,θ d g whereT >> f −1 o is the averaging period, ∆m = ρ or ∆r∆θd g , v 2 (r)r −1 = Ω 2 or , r = r r,θ , and in turn calculation, via the Biot-Savart relation, the steady time-averaged velocity field. Further details will be reported in a forthcoming paper.
As shown in Fig. 6, this relatively simple model captures the non-trivial helical grain flow that we observe. Importantly, and consistent with observed short-time-scale, single grain dynamics -