Dilute Granular Gas Research Articles

Effects of cohesion on the flow patterns of granular materials in spouted beds

Two-dimensional spouted bed, capable to provide both dilute granular gas and dense granular solid flow patterns in one system, was selected as a prototypical system for studying granular materials. Effects of liquid cohesion on such kind of complex granular patterns were studied using particle image velocimetry. It is seen that the addition of liquid oils by a small fraction of 10(-3)-10(-2) causes a remarkable narrowing (about 15%) of the spout area. In the dense annulus, as the liquid fraction increases, the downward particle velocity gradually decreases and approaches a minimum where, at a microscopic grain scale, the liquid bridge reaches spherical regimes with a maximum capillarity. Viscous lubrication effect is observed at a much higher fraction but is really weak with respect to the capillary effect. In the dilute spout, in contrast to the dry grains, the wet grains have a lightly smaller acceleration in the initial 1/3 of the spout, but have a dramatically higher acceleration in the rest of the spout. We attribute the former to the additional work needed to overcome interparticle cohesion during particle entrainment at the spout-annulus interface. Then, using mass and momentum balances, the latter is explained by the relative higher drag force resulting from both higher gas velocities and higher voidages due to spout narrowing in the wet system. The experimental findings will provide useful data for the validation of discrete element simulation of cohesive granular-fluid flows.

Rheological effects in the linear response and spontaneous fluctuations of a sheared granular gas

The decay of a small homogeneous perturbation in the temperature of a dilute granular gas in the steady uniform shear flow state is investigated. Using kinetic theory based on the inelastic Boltzmann equation, a closed equation for the decay of the perturbation is derived. The equation involves the generalized shear viscosity of the gas in the time-dependent shear flow state, and therefore, it predicts relevant rheological effects beyond the quasielastic limit. Good agreement is found when comparing the theory with molecular dynamics simulation results. Moreover, the Onsager postulate on the regression of fluctuations is fulfilled.

Internal energy fluctuations of a granular gas under steady uniform shear flow

The stochastic properties of the total internal energy of a dilute granular gas in the steady uniform shear flow state are investigated. A recent theory formulated for fluctuations about the homogeneous cooling state is extended by analogy with molecular systems. The theoretical predictions are compared with molecular dynamics simulation results. Good agreement is found in the limit of weak inelasticity, while systematic and relevant discrepancies are observed when the inelasticity increases. The origin of this behavior is discussed.

Unsteady non-Newtonian hydrodynamics in granular gases

The temporal evolution of a dilute granular gas, both in a compressible flow (uniform longitudinal flow) and in an incompressible flow (uniform shear flow), is investigated by means of the direct simulation Monte Carlo method to solve the Boltzmann equation. Emphasis is laid on the identification of a first "kinetic" stage (where the physical properties are strongly dependent on the initial state) subsequently followed by an unsteady "hydrodynamic" stage (where the momentum fluxes are well-defined non-Newtonian functions of the rate of strain). The simulation data are seen to support this two-stage scenario. Furthermore, the rheological functions obtained from simulation are well described by an approximate analytical solution of a model kinetic equation.

Effect of Coulomb friction on orientational correlation and velocity distribution functions in a sheared dilute granular gas

From particle simulations of a sheared frictional granular gas, we show that the Coulomb friction can have dramatic effects on orientational correlation as well as on both the translational and angular velocity distribution functions even in the Boltzmann (dilute) limit. The dependence of orientational correlation on friction coefficient (μ) is found to be nonmonotonic, and the Coulomb friction plays a dual role of enhancing or diminishing the orientational correlation, depending on the value of the tangential restitution coefficient (which characterizes the roughness of particles). From the sticking limit (i.e., with no sliding contact) of rough particles, decreasing the Coulomb friction is found to reduce the density and spatial velocity correlations which, together with diminished orientational correlation for small enough μ, are responsible for the transition from non-gaussian to gaussian distribution functions in the double limit of small friction (μ→0) and nearly elastic particles (e→1). This double limit in fact corresponds to perfectly smooth particles, and hence the maxwellian (gaussian) is indeed a solution of the Boltzmann equation for a frictional granular gas in the limit of elastic collisions and zero Coulomb friction at any roughness. The high-velocity tails of both distribution functions seem to follow stretched exponentials even in the presence of Coulomb friction, and the related velocity exponents deviate strongly from a gaussian with increasing friction.

Inelastic collisional effect on a dilute granular shock layer with a heated wall

The inelastic collisional effect on a shock layer of a dilute granular gas with a heated wall is numerically studied. To investigate the inelastic collisional effect via the gain term in the inelastic Boltzmann equation on the shock layer, an inelastic Bhatnagar-Gross-Krook (BGK) type equation, whose loss term is equivalent to that in the inelastic Boltzmann equation, is formulated on the basis of the kinetic theory of the granular gas. The inelastic BGK-type equation formulated for a hard-sphere particle is generalized to that for an inverse power law (IPL) molecule. Numerical results in a weakly inelastic regime confirm the nonequilirium contribution to the cooling rate, when the collision frequency depends on the particle velocity. The profile of the negative high-velocity tail of the distribution function in the generation regime of the shock wave obtained by the Direct Simulation Monte Carlo method is higher than that obtained by the proposed BGK-type equation when the collision frequency depends on the particle velocity because of the inelastic collisional effect via the gain term in the inelastic Boltzmann equation, which is not included in the proposed BGK-type equation.

Sonine approximation for collisional moments of granular gases of inelastic rough spheres

We consider a dilute granular gas of hard spheres colliding inelastically with coefficients of normal and tangential restitution α and β, respectively. The basic quantities characterizing the distribution function f(v,ω) of linear (v) and angular (ω) velocities are the second-degree moments defining the translational (Ttr) and rotational (Trot) temperatures. The deviation of f from the Maxwellian distribution parameterized by Ttr and Trot can be measured by the cumulants associated with the fourth-degree velocity moments. The main objective of this paper is the evaluation of the collisional rates of change of these second- and fourth-degree moments by means of a Sonine approximation. The results are subsequently applied to the computation of the temperature ratio Trot/Ttr and the cumulants of two paradigmatic states: the homogeneous cooling state and the homogeneous steady state driven by a white-noise stochastic thermostat. It is found in both cases that the Maxwellian approximation for the temperature ratio does not deviate much from the Sonine prediction. On the other hand, non-Maxwellian properties measured by the cumulants cannot be ignored, especially in the homogeneous cooling state for medium and small roughness. In that state, moreover, the cumulant directly related to the translational velocity differs in the quasi-smooth limit β→−1 from that of pure smooth spheres (β=−1). This singular behavior is directly related to the unsteady character of the homogeneous cooling state and thus it is absent in the stochastic thermostat case.

Class of dilute granular Couette flows with uniform heat flux

In a recent paper [F. Vega Reyes et al., Phys. Rev. Lett. 104, 028001 (2010)] we presented a preliminary description of a special class of steady Couette flows in dilute granular gases. In all flows of this class the viscous heating is exactly balanced by inelastic cooling. This yields a uniform heat flux and a linear relationship between the local temperature and flow velocity. The class (referred to as the LTu class) includes the Fourier flow of ordinary gases and the simple shear flow of granular gases as special cases. In the present paper we provide further support for this class of Couette flows by following four different routes, two of them being theoretical (Grad's moment method of the Boltzmann equation and exact solution of a kinetic model) and the other two being computational (molecular dynamics and Monte Carlo simulations of the Boltzmann equation). Comparison between theory and simulations shows a very good agreement for the non-Newtonian rheological properties, even for quite strong inelasticity, and a good agreement for the heat flux coefficients in the case of Grad's method, the agreement being only qualitative in the case of the kinetic model.

Navier-Stokes hydrodynamics of thermal collapse in a freely cooling granular gas

We show that, in dimension higher than one, heat diffusion and viscosity cannot arrest thermal collapse in a freely evolving dilute granular gas, even in the absence of gravity. Thermal collapse involves a finite-time blowup of the gas density. It was predicted earlier in ideal, Euler hydrodynamics of dilute granular gases in the absence of gravity, and in nonideal, Navier-Stokes granular hydrodynamics in the presence of gravity. We determine, analytically and numerically, the dynamic scaling laws that characterize the gas flow close to collapse. We also investigate bifurcations of a freely evolving dilute granular gas in circular and wedge-shaped containers. Our results imply that, in general, thermal collapse can only be arrested when the gas density becomes comparable with the close-packing density of grains. This provides a natural explanation to the formation of densely packed clusters of particles in a variety of initially dilute granular flows.

Fluctuating hydrodynamics for dilute granular gases: A Monte Carlo study

We investigate hydrodynamic noise in a dilute granular gas during the homogeneous cooling state, by means of a proper application of the direct simulation Monte Carlo (DSMC) algorithm. The DSMC includes a source of randomization which is not present in molecular dynamics (MD) for inelastic hard disks. Notwithstanding this difference, a fair quantitative agreement is found, including a violation of the fluctuation-dissipation relation for the noise amplitude of the same order observed in MD. This study suggests that deterministic collision dynamics is not an essential ingredient to reproduce, up to a good degree of approximation, hydrodynamic fluctuations in dilute granular gases.

Theory of Dilute Binary Granular Gas Mixtures

A computer-aided method for accurately carrying out the Chapman-Enskog expansion of the Boltzmann equation, including its inelastic variant, is presented and employed to derive a hydrodynamic description of a dilute binary mixture of smooth inelastic spheres. Constitutive relations, formally valid for all physical values of the coefficients of restitution, are calculated by carrying out the pertinent Chapman-Enskog expansion to sufficient high orders in the Sonine polynomials to ensure numerical convergence. The resulting hydrodynamic description is applied to the analysis of a vertically vibrated binary mixture of particles (under gravity) differing only in their respective coefficients of restitution. It is shown that even with this “minor”difference the mixture partly segregates, its steady state exhibiting a sandwich-like configuration.

Velocity distribution function and correlations in a granular Poiseuille flow

Probability distribution functions of fluctuation velocities (P(ux) andP(uy), whereuxanduyare the fluctuation velocities in thex- andy-directions, respectively; the gravity is acting along the periodicx-direction and the flow is bounded by two walls parallel to they-direction) and the density and the spatial velocity correlations are studied using event-driven simulations for an inelastic smooth hard disk system undergoing gravity-driven granular Poiseuille flow (GPF). It is shown that for GPF with smooth and/or perfectly rough walls the Maxwellian/Gaussian is the leading-order distribution over a wide range of densities in the quasi-elastic limit, which is a surprising result, especially for a dilute granular gas for which the Knudsen number belongs to the transitional flow regime. The signature of wall-roughness-induced dissipation mainly shows up in theP(ux) distribution in the form of a sharp peak for negative velocities in the near-wall region. BothP(ux) andP(uy) distributions become asymmetric with increasing dissipation at any density, and the emergence of density waves, which appear in the form of sinuous wave/slug at low-to-moderate values of mean density, makes these asymmetries stronger, especially in the presence of a slug. At high densities, the flow degenerates into a dense plug (where the density approaches its maximum limit and the shear rate is negligibly small) around the channel centreline and two shear layers (where the shear rate is high and the density is low) near the walls. The distribution functions within the shear layer follow the characteristics of those at moderate mean densities. Within the dense plug, the high-velocity tails of bothP(ux) andP(uy) appear to undergo a transition from Gaussian in the quasi-elastic limit to power-law distributions at large inelasticity of particle collisions. For dense flows, it is shown that although the density correlations play a significant role in enhancing the velocity correlations when the collisions are sufficiently inelastic, they do not induce velocity correlations when the collisions are quasi-elastic for which the distribution functions are close to Gaussian. The combined effect of enhanced density and velocity correlations around the channel centreline with increasing inelastic dissipation seems to be responsible for the emergence of non-Gaussian high-velocity tails of distribution functions.

Constitutive relations in dense granular flows

We use simulations to investigate constitutive relations in dry granular flow. Our system is comprised of polydisperse sets of spherical grains falling down a vertical chute under the influence of gravity. Three phases or states of granular matter are observed: a free-fall dilute granular gas region at the top of the chute, a granular fluid in the middle and then a glassy region at the bottom. We examine a complete closed set of constitutive relations capable of describing the local stresses, heat flow, and dissipation in the different regions. While the pressure can be reasonably described by hard sphere gas models, the transport coefficients cannot. Transport coefficients such as viscosity and heat conductivity increase with decreasing temperature in the fluid and glassy phases. The glass exhibits signs of a finite yield stress and we show that the static sand pile is a limit of our glassy state.

Non-Newtonian Granular Hydrodynamics. What Do the Inelastic Simple Shear Flow and the Elastic Fourier Flow Have in Common?

We describe a special class of steady Couette flows in dilute granular gases admitting a non-Newtonian hydrodynamic description for strong dissipation. The class occurs when viscous heating exactly balances inelastic cooling, resulting in a uniform heat flux. It includes the Fourier flow of ordinary gases and the simple or uniform shear flow (USF) of granular gases as special cases. The rheological functions have the same values as in the USF and generalized thermal conductivity coefficients can be identified. These points are confirmed by molecular dynamics simulations, Monte Carlo simulations of the Boltzmann equation, and analytical results from Grad's 13-moment method.

Binary granular gas mixtures: Theory, layering effects and some open questions

A theory of dilute binary granular gas mixtures comprising smooth spheres, which is formally valid for all physical values of the pertinent coefficients of restitution, is presented. Constitutive relations are obtained using the Chapman Enskog procedure, in the framework of which relatively high orders in the Sonine polynomial expansion are employed. The latter is made possible by a computer-aided method. The transport coefficients are shown to converge as a function of the number of Sonine polynomials employed, typically requiring 3 polynomials, but in some cases up to 6 polynomials are required for convergence to within 1%. A comparison with results obtained using the Grad method helps reveal the limitations of the Chapman-Enskog expansion. In particular, in some cases both the Chapman Enskog expansion and the Grad method give unphysical results, though they agree with each other. These issues are related to the lack of scale separation in granular gases. Using the constitutive relations we obtain a novel segregation pattern in vertically vibrated granular mixtures, which comprises up to five layers; this is an extension of a previously obtained three-layer configuration. Finally, the question of effective hydrodynamic boundary conditions at the transition to the Knudsen regime is discussed: in particular, it seems that in some cases the boundary condition for the heat flux in the hydrodynamic regime is “unphysical”.

The Fourier state of a dilute granular gas described by the inelastic Boltzmannequation

The existence of two stationary solutions of the nonlinear Boltzmann equation for inelastichard spheres or disks is investigated. They are restricted neither to weak dissipation nor tosmall gradients. The one-particle distribution functions are assumed to have a scalingproperty, namely that all the position dependence occurs through the density and thetemperature. At the macroscopic level, the state corresponding to both is characterized byuniform pressure, no mass flow, and a linear temperature profile. Moreover, the stateexhibits two peculiar features. First, there is a relationship between the inelasticity ofcollisions, the pressure, and the temperature gradient. Second, the heat flux can beexpressed as being linear in the temperature gradient, i.e. a Fourier-like law is obeyed.One of the solutions is singular in the elastic limit. The theoretical predictionsfollowing from the other one are compared with molecular dynamics simulationresults and a good agreement is obtained in the parameter region in which theFourier state can be actually observed in the simulations, namely not too stronginelasticity.

Higher-order quadrature-based moment methods for kinetic equations

Kinetic equations containing terms for spatial transport, body forces, and particle–particle collisions occur in many applications (e.g., rarefied gases, dilute granular gases, fluid-particle flows). The direct numerical solution of the kinetic equation is usually intractable due to the large number of independent variables. A useful alternative is to reformulate the problem in terms of the moments of the velocity distribution function. Closure of the moment equations is challenging for flows sufficiently far away from the Maxwellian limit. In previous work, a quadrature-based third-order moment closure was derived for approximating solutions to the kinetic equation for arbitrary Knudsen number. A key component of quadrature-based closures is the moment-inversion algorithm used to find the non-negative weights and velocity abscissas. Here, a robust inversion procedure is proposed for three-component velocity moments up to ninth order. By reconstructing the velocity distribution function, the spatial fluxes in the moment equations are treated using a kinetic-based finite-volume solver. Because the quadrature-based moment method employs the moment transport equations directly instead of a discretized form of the kinetic equation, the mass, momentum and energy are conserved for arbitrary Knudsen and Mach numbers. The computational algorithm is tested for the Riemann shock problem and, for increasing Knudsen numbers (i.e. larger deviations from the Maxwellian limit), the accuracy of the moment closure is shown to be determined by the discrete representation of the spatial fluxes.

Fluctuating hydrodynamics for dilute granular gases

Starting from the kinetic equations for the fluctuations and correlations of a dilute gas of inelastic hard spheres or disks, a Boltzmann-Langevin equation for the one-particle distribution function of the homogeneous cooling state is constructed. This equation is the linear Boltzmann equation with a fluctuating white-noise term. Balance equations for the fluctuating hydrodynamic fields are derived. New fluctuating forces appear as compared with the elastic limit. The particular case of the transverse velocity field is investigated in detail. Its fluctuations can be described by means of a Langevin equation, but exhibiting two main differences with the Landau-Lifshitz theory: the noise is not white, and its second moment is not determined by the shear viscosity. This shows that the fluctuation-dissipation relations for molecular fluids do not straightforwardly carry over to inelastic gases. The theoretical predictions are shown to be in good agreement with molecular-dynamics simulation results.

Fluctuations of energy flux in a simple dissipative out-of-equilibrium system

We report the statistical properties of the fluctuations of the energy flux in an electronic RC circuit driven with a stochastic voltage. The fluctuations of the power injected in the circuit are measured as a function of the damping rate and the forcing parameters. We show that its distribution exhibits a cusp close to zero and two asymmetric exponential tails, with the asymmetry being driven by the mean dissipation. This simple experiment allows one to capture the qualitative features of the energy flux distribution observed in more complex dissipative systems. We also show that the large fluctuations of injected power averaged on a time lag do not verify the fluctuation theorem even for long averaging time. This is in contrast to the findings of previous experiments due to their small range of explored fluctuation amplitude. The injected power in a system of N components either correlated or not is also studied to mimic systems with large number of particles, such as in a dilute granular gas.

Knudsen temperature jump and the Navier-Stokes hydrodynamics of granular gases driven by thermal walls

Thermal wall is a convenient idealization of a rapidly vibrating plate used for vibrofluidization of granular materials. The objective of this work is to incorporate the Knudsen temperature jump at thermal wall in the Navier-Stokes hydrodynamic modeling of dilute granular gases of monodisperse particles that collide nearly elastically. The Knudsen temperature jump manifests itself as an additional term, proportional to the temperature gradient, in the boundary condition for the temperature. Up to a numerical prefactor O(1) , this term is known from kinetic theory of elastic gases. We determine the previously unknown numerical prefactor by measuring, in a series of molecular dynamics (MD) simulations, steady-state temperature profiles of a gas of elastically colliding hard disks, confined between two thermal walls kept at different temperatures, and comparing the results with the predictions of a hydrodynamic calculation employing the modified boundary condition. The modified boundary condition is then applied, without any adjustable parameters, to a hydrodynamic calculation of the temperature profile of a gas of inelastic hard disks driven by a thermal wall. We find the hydrodynamic prediction to be in very good agreement with MD simulations of the same system. The results of this work pave the way to a more accurate hydrodynamic modeling of driven granular gases.