A. Goldshtein

Mechanics of collisional motion of granular materials. Part 1. General hydrodynamic equations

Collisional motion of a granular material composed of rough inelastic spheres is analysed on the basis of the kinetic Boltzmann–Enskog equation. The Chapman–Enskog method for gas kinetic theory is modified to derive the Euler-like hydrodynamic equations for a system of moving spheres, possessing constant roughness and inelasticity. The solution is obtained by employing a general isotropic expression for the singlet distribution function, dependent upon the spatial gradients of averaged hydrodynamic properties. This solution form is shown to be appropriate for description of rapid shearless motions of granular materials, in particular vibrofluidized regimes induced by external vibrations. The existence of the hydrodynamic state of evolution of a granular medium, where the Euler-like equations are valid, is delineated in terms of the particle roughness, β, and restitution, e , coefficients. For perfectly elastic spheres this state is shown to exist for all values of particle roughness, i.e. − 1≤β≤1. However, for inelastically colliding granules the hydrodynamic state exists only when the particle restitution coefficient exceeds a certain value e m (β) In contrast with the previous results obtained by approximate moment methods, the partition of the random-motion kinetic energy of inelastic rough particles between rotational and translational modes is shown to be strongly affected by the particle restitution coefficient. The effect of increasing inelasticity of particle collisions is to redistribute the kinetic energy of their random motion in favour of the rotational mode. This is shown to significantly affect the energy partition law, with respect to the one prevailing in a gas composed of perfectly elastic spheres of arbitrary roughness. In particular, the translational specific heat of a gas composed of inelastically colliding ( e = 0.6) granules differs from its value for elastic particles by as much as 55 %. It is shown that the hydrodynamic Euler-like equation, describing the transport and evolution of the kinetic energy of particle random motion, contains energy sink terms of two types (both, however, stemming from the non-conservative nature of particle collisions) : (i) the term describing energy losses in incompressibly flowing gas; (ii) the terms accounting for kinetic energy loss (or gain) associated with the work of pressure forces, leading to gas compression (or expansion). The approximate moment methods are shown to yield the Euler-like energy equation with an incorrect energy sink term of type (ii), associated with the ‘dense gas effect’. Another sink term of the same type, but associated with the energy relaxation process occurring within compressed granular gases, was overlooked in all previous studies. The speed of sound waves propagating in a granular gas is analysed in the limits of low and high granular gas densities. It is shown that the particle collisional properties strongly affect the speed of sound in dense granular media. This dependence is manifested via the kinetic energy sink terms arising from gas compression. Omission of the latter terms in the evaluation of the speed of sound results in an error, which in the dense granular gas limit is shown to amount to a several-fold factor.

Mechanics of collisional motion of granular materials. Part 2. Wave propagation through vibrofluidized granular layers

According to numerous experimental observations and theoretical models vibrated layers composed of large granules behave like a solid plastic body. In contrast, in this study experimental data are presented that reveal that, for constant vibration amplitudes A ≥ 1 cm with the frequency ω increasing from zero, all layers pass through three vibrational states, with the respective behaviours being as of (i) a solid plastic body, (ii) a liquid, (iii) a gas. In the liquid-like vibrational state transverse waves propagating along the layer width were observed. These waves were shown to be gravitational resonance waves, with the corresponding frequencies well correlated by the known formula for incompressible liquids. In the gas-like vibrational state compression (shock) and expansion waves propagating across the layer height, were observed. A theoretical model for time-periodic collisional vibrational regimes was developed on the basis of the Euler-like equations of a granular gas composed of inelastic spheres. The model shows that the vibrational granular state (bed porosity, shock wave speed, granular pressure and kinetic energy) is inter alia governed by the dimensionless parameter V = ( A ω)/( h m g ) 1/2 , with g, h m being the gravitational acceleration and the height of the resting layer, respectively. This is in contrast with the previous studies, where the behaviour of vibrated granular layers was interpreted in terms of the dimensionless acceleration Δ = ( A ω 2 )/ g . The proposed model was tested by processing the data obtained from photographs of the particle distribution within vibrated layers. Theoretical predictions of the particle average concentration compared favourably with the experimental data. Other phenomena observed in vibrated granular layers include the formation of caverns, circulatory motion of granules and synchronized periodic motion of two adjacent vibrated layers of different widths. The importance of the observed phenomena in relation to various technological processes involving bulk materials (vibromixing, vibroseparation, etc.) is discussed.

Mechanics of collisional motion of granular materials. Part 3. Self-similar shock wave propagation

(Received 2 November 1994 and in revised form 29 December 1995) Shock wave propagation arising from steady one-dimensional motion of a piston in a granular gas composed of inelastically colliding particles is treated theoretically. A selfsimilar long-time solution is obtained in the strong shock wave approximation for all values of the upstream gas volumetric concentration v,. Closed form expressions for the long-time shock wave speed and the granular pressure on the piston are obtained. These quantities are shown to be independent of the particle collisional properties, provided their impacts are accompanied by kinetic energy losses. The shock wave speed of such non-conservative gases is shown to be less than that for molecular gases by a factor of about 2. The effect of particle kinetic energy dissipation is to form a stagnant layer (solid block), on the surface of the moving piston, with density equal to the maximal packing density, vM. The thickness of this densely packed layer increases indefinitely with time. The layer is separated from the shock front by a fluidized region of agitated (chaotically moving) particles. The (long-time, constant) thickness of this layer, as well as the kinetic energy (granular temperature) distribution within it are calculated for various values of particle restitution and surface roughness coefficients. The asymptotic cases of dilute (v, 6 1) and dense (v, - vM) granular gases are treated analytically, using the corresponding expressions for the equilibrium radial distribution functions and the pertinent equations of state. The thickness of the fluidized region is shown to be independent of the piston velocity. The calculated results are discussed in relation to the problem of vibrofluidized granular layers, wherein shock and expansion waves were registered. The average granular kinetic energy in the fluidized region behind the shock front calculated here compared favourably with that measured and calculated (Goldshtein et al. 1995) for vibrofluidized layers of spherical granules.

Agglomeration of submicrometer particles in weak periodic shock waves

Agglomeration of submicrometer particles in the presence of acoustic fields was treated experimentally. Fast agglomeration of aerosol particles in weak periodic shock waves was observed. The rate of particle agglomeration in shock waves was found to be much faster than in continuous sound waves.

Resonance gas oscillations in closed tubes

The problem of gas motion in a tube closed at one end and driven at the other by an oscillating poston is studied theoretically. When the piston vibrates with a finite amplitude at the first acoustic resonance frequency, periodic shock waves are generated, travelling back and forth in the tube. A perturbation method, based on a small Mach number. M and a global mass conservation condition, is employed to formulate a solution of the problem in the form of two standing waves separated by a jump (shock front). By expanding the equations of motion in a series of a small parameter e = M ½ , all hydrodynamic properties are predicted with an accuracy to second-order terms, i.e. to e 2 . It is found that the first-order solution coincides with the previous theories of Betchov (1958) and Chester (1964), while additional terms predict a non-homogeneous time-averaged pressure along the tube. This prediction compares favourably with experimental results from the literature. The importance of the phenomenon is discussed in relation to different transport processes in resonance tubes.

Heat interaction in a resonance tube

Gas oscillations in a resonance tube are investigated experimentally and numerically. We found that for large amplitudes, heat interaction with the tube wall affects strongly gas flow for frequencies within the resonance band. A proposed model, accounting for heat interaction, shows good agreement with the experimental data.

Mechanics of collisional motion of granular materials. Part 4. Expansion wave

The problem of expansion into a vacuum of a semi-infinite layer composed of chaotically moving inelastic rough spherical particles is solved analytically. A variant of the matched asymptotic expansion scheme is used to obtain a matched composite solution, which is valid for both small and large times in the wave head, wave tail and intermediate domains of the disturbed part of the layer. The effects of granular initial energy and particle collisonal properties on their hydrodynamic velocity, temperature, density and pressure are studied in the limit of low initial density of the granular gas. The total granular mass, M , within the disturbed region was found to change with time as log t . This is in contrast with the comparable classical result M ∼ t obtained for conservative (molecular) gases. This logarithmic dependence stems from the influence of kinetic energy losses, which reduce the granular temperature and speed of sound in the wave head region. The ultimate escape energy and momentum (i.e. those achieved for long times by the expanding part of the layer) are shown to be finite quantities, dependent on the particle restitution coefficient, roughness and the initial granular temperature. The estimated mass of the escaping part of the layer, calculated here for dilute gases, serves as an upper bound on this quantity for all (also dense) comparable granular gases. This mass is determined by the collisional losses, as embodied within the particle restitution and roughness coefficients.

Resonance Oscillations in Granular Gases

The paper is devoted to investigation of a new phenomenon: resonance oscillations in granular gases produced by a piston vibrating at one end of a closed tube. The main feature of this phenomenon, predicted by a hydrodynamic model and confirmed by computer simulations, are: (i) a column of a granular gas oscillates periodically with the frequency f/n, where f is the vibrationalfrequency, n is a positive integer number, (ii) the oscillation patterns are governed by the shock waves propagating across the column; (iii) the averaged kinetic energy per particle is proportionalto f 2, and it strongly depends on the vibrational amplitude; (iv) the maximalv alue of this kinetic energy pumped by these externalvibrations is of order f 2 L 2, where L is the tube length

A model for the collapse of a fluidized bed

A one-dimensional hydrodynamic model for the collapse of a fluidized bed following the instantaneous shut-off of fluidizing air was developed in terms of the sedimentation of a dense system of monodisperse particles. This model is based on the continuity and momentum equations written for the fluid and solid phases. The solution of the problem was formulated in terms of two distinct regions separated by a discontinuity. All hydrodynamic properties in these regions together with the speed of propagation of the discontinuity were calculated analytically. The predicted pressure distribution compares favorably with experimental data from the literature.

Resonance oscillations with thermal effects of an inviscid gas in a closed tube

The problem of gas motion inside a resonance tube, closed at one end by a plug and fitted at the other with an oscillating piston is treated analytically and numerically. An analytical model is derived for arbitrary piston motion and gas oscillations about the first resonance frequency, where the gas flow is characterized by a shock wave travelling periodically back and forth in the tube. The model is obtained by a perturbation analysis in terms of a small-amplitude parameter e. All the hydrodynamic properties of the gas are predicted with accuracy up to the second-order terms of e. Isentropic and adiabatic problem formulations are addressed. Expressions for spatial distributions of the time-averaged hydrodynamic gas properties are derived for any frequency within the resonance band