Nonlinear dynamics of density waves in granular flows through narrow vertical channels

We study granular flows through narrow channels driven by gravity in the framework of the kinetic theory for dissipative dense gases. We derive equations of motion for quasi-one-dimensional systems. In a certain range of flow density, the steady homogeneous regime is found to be unstable against the formation of density waves. We show moreover that near the onset of the instability, the governing equation for the flow density is a mixture of the Korteweg-de-Vries equation, which leads to soliton, and the Bürger equation which exhibits spatio-temporal chaos. The competition between chaos and solitons may lead either to regular spatially ordered density waves or to chaotic dynamics. We argue that these two types of dynamics can be encountered experimentally according to the channel width and the dissipative properties of the granular media. Received: 11 March 1998 / Revised and Accepted: 3 July 1998     

Density waves and the effect of wall roughness in granular Poiseuille flow: Simulationand linear stability

The formation of density waves and the effect of wall roughness on them are studied using molecular dynamics simulations of gravity-driven granular Poiseuille flow. Three basic types of structures are found in moderately dense flows: a plug, a sinuous wave and a slug; a new varicose wave mode has been identified in dense flows with channels of large widths at moderate dissipations; only clump-like structures appear in dilute flows. The simulation results are contrasted with the predictions of a linear stability analysis of the kinetic-theory continuum equations for granular Poiseuille flow. The theoretical predictions on the form of density waves are in qualitative agreement with simulations in denser flows, however, there are discrepancies between simulation and theory in dilute flows.     

稠密颗粒射流撞击壁面颗粒膜表面波纹特征

采用高速摄像仪对稠密颗粒射流撞击有限尺寸壁面的流动过程进行了实验研究,重点研究了颗粒膜及其表面波纹特征,考察了颗粒粒径、射流速度和固含率等因素对颗粒膜形态和表面波纹的影响.研究结果表明,随着颗粒粒径增大,稠密颗粒撞壁流由颗粒膜向散射模式转变.与液体射流撞壁液膜相比,颗粒膜扩展角较大,射流速度对其影响不显著.稠密颗粒射流撞壁颗粒膜表面波纹存在明显的叠加现象,颗粒膜表面波纹频率比液膜大约低一个数量级.颗粒膜表面波纹主要由射流脉动引起,表面波纹频率与射流脉动频率具有相同的数量级.     

Nonlinear instability of elementary stratified flows at large Richardson number

Elementary stably stratified flows with linear instability at all large Richardson numbers have been introduced recently by the authors [J. Fluid Mech. 376, 319-350 (1998)]. These elementary stratified flows have spatially constant but time varying gradients for velocity and density. Here the nonlinear stability of such flows in two space dimensions is studied through a combination of numerical simulations and theory. The elementary flows that are linearly unstable at large Richardson numbers are purely vortical flows; here it is established that from random initial data, linearized instability spontaneously generates local shears on buoyancy time scales near a specific angle of inclination that nonlinearly saturates into localized regions of strong mixing with density overturning resembling Kelvin-Helmholtz instability. It is also established here that the phase of these unstable waves does not satisfy the dispersion relation of linear gravity waves. The vortical flows are one family of stably stratified flows with uniform shear layers at the other extreme and elementary stably stratified flows with a mixture of vorticity and strain exhibiting behavior between these two extremes. The concept of effective shear is introduced for these general elementary flows; for each large Richardson number there is a critical effective shear with strong nonlinear instability, density overturning, and mixing for elementary flows with effective shear below this critical value. The analysis is facilitated by rewriting the equations for nonlinear perturbations in vorticity-stream form in a mean Lagrangian reference frame. (c) 2000 American Institute of Physics.     

稠密颗粒射流倾斜撞击颗粒膜特征

采用高速摄像仪对稠密颗粒射流倾斜撞击形成的类液体颗粒膜特征进行实验研究,考察了颗粒粒径、射流速度以及射流含固率等因素对颗粒膜形态及动态特征的影响.结果表明:随着颗粒粒径增大,稠密颗粒倾斜撞击流由颗粒膜向散射模式转变;随着射流速度增加,气固不稳定增强,射流流量出现脉动,正面与侧面分别表现为颗粒膜的非轴对称振荡和表面波纹结构;颗粒膜非轴对称振荡的振荡频率和振荡幅度随射流速度的增大而增大;表面波纹速度和波长沿传播方向增大,波纹间存在叠加现象.颗粒膜出现非轴对称振荡主要是因为喷嘴出口由气固不稳定性引起的射流流量脉动,射流流量脉动频率与撞击面振荡频率基本相当.     

Numerical modelling of free interaction of shock waves in corner flows

The inviscid supersonic flows in corners between intersecting compression wedges were studied numerically. Under usual conditions, the flows in such corner configurations are conically self-similar. Besides, shock waves formed by wedges are plane and they interact with one another in these flows and the downstream development of the shock interaction structure at that occurs in a region which is bounded in cross-sectional directions by the walls and contracts towards the corner rib, and in this sense the above interaction of shocks may be subjected to influence of the rib. The corner flows of another type with an interaction of shocks not subjected to the “rib effect”, in other words free, were considered. This was ensured by that the corner configurations were designed with a gradually expanding gap along the rib in a way that the reflected shocks arising as a result of interaction passed through the gap without falling onto the walls. This eliminated the influence of a local flow near the corner rib on the interaction of plane shock waves formed by the wedges. The inviscid flows symmetric with respect to the bisector plane of the corner dihedral angle were considered. The computations showed that, in the gapped corner configurations with the free interaction of shocks, if these interactions were irregular, the cross-flow patterns were practically the same as in the original corner configurations without a gap. The conditions for the flows over corner configurations were also considered under which the interaction of shocks corresponded to the Neumann criterion of mechanical equilibrium. These conditions are equivalent to those, which take place in two-dimensional steady flows, when a hysteresis phenomenon is possible, which manifests itself in changing the moment of transition from a regime of regular reflection of shocks to an irregular one and conversely at a direct and inverse courses of varying the flow parameters—the Mach number and the inclination angle of the wedges. The conducted computations showed that a hysteresis is possible for considered corner flows in gapped configurations, but it must be followed by a detachment of shocks from the corner apex and a breakdown of the conical flow structure.     

Acoustic probing of the jamming transition in an unconsolidated granular medium

Experimentally determined dispersion relations for acoustic waves guided along the mechanically free surface of an unconsolidated granular packed structure provide information on the elasticity of granular media at very low pressures that are naturally controlled by the gravitational acceleration and the depth beneath the surface. The experiments confirm recent theoretical predictions that relaxation of the disordered granular packing through nonaffine motion leads to a peculiar scaling of shear rigidity with pressure near the jamming transition corresponding to zero pressure.     

Stress distribution and surface instability of an inclined granular layer

Static granular materials may avalanche suddenly under continuous quasi-static drives. This phenomenon, which is important in many engineering applications, can be explained by analyzing the stability of the elastic solutions. We show this for a granular layer driven by its inclination angle in gravity, where the elastic problem can be solved generally and analytically. It is found that a loss of stability may occur only at the free surface of the layer. This result is considered to be relevant for understanding surface avalanches and the flows observed experimentally.     

Temporally heterogeneous dynamics in granular flows

Granular simulations are used to probe the particle scale dynamics at short, intermediate, and long time scales for gravity-driven, dense granular flows down an inclined plane. On approach to the angle of repose, where motion ceases, the dynamics become intermittent over intermediate times, with strong temporal correlations between particle motions-temporally heterogeneous dynamics. This intermittency is characterized through large-scale structural events whereby the contact network periodically spans the system. A characteristic time scale associated with these processes increases as the stopped state is approached. These features are discussed in the context of the dynamics of supercooled liquids near the glass transition.     

Some remarks on the rheology of dense granular flows

The paper reviews some peculiar properties exhibited by granular flows. We emphasize the inability of kinetic theory and of Bagnolds heuristic approach to describe the rapid regime of densely packed flows, characterized by the breakdown of the binary collision picture and by multibody long-lasting contacts. We suggest that deformation waves through the continuous paths of contacts can be effective to transport momentum and energy through the bulk, in a time very short compared to the inverse of the shear rate. This mechanism could explain some key rheological features encountered in dense granular materials.Received: 25 June 2004, Published online: 31 August 2004PACS: 45.70.-n Granular systemsJ. Rajchenbach: On leave from LMDH (CNRS-UMR 7603), Université P. et M. Curie, 75005 Paris, France.     

Bragg interaction of surface magnetostatic waves with a periodic granular HTSC structure

The Bragg interaction of surface magnetostatic waves with periodic granular HTSC structure has been investigated. The dispersion equation for the coupled waves has been obtained. Resonant absorption of waves near the critical temperature involving the granular structure of the superconductor has been found. The possibility of using the observed effect for making frequency-selective structures and high-speed bolometric photodetectors is shown. Tomsk University. Radio Electronic and Control Systems. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 94–98, October, 1997.     

Bragg Interaction of Surface Magnetostatic Waves with a Periodic Granular High-Temperature Superconducting Structure

Bragg interaction of surface magnetostatic waves with a periodic granular high-temperature superconducting structure is studied. A dispersion equation for coupled waves describing the characteristics of surface magnetostatic waves is derived. Resonance wave absorption near the critical temperature is revealed. The magnetostatic wave reflection coefficients of semi-infinite and finite periodic high-temperature superconducting structures are calculated. The results obtained can be used in designing tunable frequency selectors and transient bolometric photodetectors.     

Anomalous wave reflection at the interface of two strongly nonlinear granular media

Granular materials exhibit a strongly nonlinear behavior affecting the propagation of energy and information. Dynamically self-organized strongly nonlinear solitary waves are the main information carriers in granular chains. We report the first experimental observation of the dramatic change of solitary wave reflectivity from the interface of two granular media triggered by a magnetically induced precompression. It may be appropriate to name this phenomenon the "acoustic diode" effect. We explain this effect by the high gradient of particle velocity near the interface.     

Helical normal waves near a cylindrical cavity in an elastic medium

The properties of helical waves arising near a cylindrical cavity in an elastic medium are described. These waves are a manifestation of aperiodic (in angle) solutions to the dynamic elasticity equations. The dispersion characteristics of the waves are determined, and the spatial structure of the waves is described.     

On nonlinear periodic capillary-fluctuation wave flow in a thin liquid film on a solid substrate

Analytical calculation of a nonlinear periodic wave flow on the free surface of a charged layer of an ideal incompressible conducting liquid resting on a solid substrate is carried out for the case when fluctuation-induced forces (the dispersion component of the wedging pressure) have a decisive effect on the system. It is shown that wave flows emerge in the liquid in calculations of the second order of smallness in the wave amplitude, which is assumed to be small compared with the thickness of the liquid layer. These flows result from nonlinear interaction as nonlinear corrections to the waves set at the zero time. The field of fluctuation-induced forces displaces these flows toward the periphery of the area of influence of these forces. This effect takes place both in the presence of an external electric field near the free surface and in its absence. The sign and value of the nonlinear corrections depend on whether an electric field is present near the free surface of the liquid. In the presence of an electric field, the curvature of the crest of the nonlinear waves increases; in its absence, the curvature decreases.     

Kinetic theory for sheared granular flows

Rapid granular flows are far-from-equilibrium-driven dissipative systems where the interaction between the particles dissipates energy, and so a continuous supply of energy is required to agitate the particles and facilitate the rearrangement required for the flow. This is in contrast to flows of molecular fluids, which are usually close to equilibrium, where the molecules are agitated by thermal fluctuations. Sheared granular flows form a class of flows where the energy required for agitating the particles in the flowing state is provided by the mean shear. These flows have been studied using the methods of kinetic theory of gases, where the particles are treated in a manner similar to molecules in a molecular gas, and the interactions between particles are treated as instantaneous energy-dissipating binary collisions. The validity of the assumptions underlying kinetic theory, and their applicability to the idealistic case of dilute sheared granular flows are first discussed. The successes and challenges for applying kinetic theory for realistic dense sheared granular flows are then summarised.     

Intermittency of energy in rapid granular shear flows

Hard-disk simulations are used for two-dimensional rapid granular shear flows of circular disks between two rotating cylinders. The intermittency effects associated with the rate of the energy dissipation of collisions are studied. The statistics of intermittent signals of energy dissipation reveals that a power law governs the dynamics of rapid shear granular flows. A dynamical system approach based on the Gledzer-Ohkitani-Yamada shell model of turbulence is employed to reproduce signals for energy dissipation that are statistically consistent with those from simulations. The results suggest that rapid granular flows can be analyzed by appropriate turbulent models.     

Analytical approach to continuous and intermittent bottleneck flows

We propose a many-particle-inspired theory for granular outflows from a hopper and for the escape dynamics through a bottleneck based on a continuity equation in polar coordinates. If the inflow is below the maximum outflow, we find an asymptotic stationary solution. If the inflow is above this value, we observe queue formation, which can be described by a shock wave equation. We also address the experimental observation of intermittent outflows, taking into account the lack of space in the merging zone by a minimum function and coordination problems by a stochastic variable. This results in avalanches of different sizes even if friction, force networks, inelastic collapse, or delay-induced stop-and-go waves are not assumed. Our intermittent flows result from a random alternation between particle propagation and gap propagation. Erratic flows in congested merging zones of vehicle traffic may be explained in a similar way.     

Excitation of surface electrostatic waves in semibounded layered superconductors by a nonrelativistic electron beam

Excitation of potential surface waves by a nonrelativistic electron beam traveling in a vacuum space near the boundary of a layered superconductor is studied theoretically. Dispersion relations for surface waves at an arbitrary angle between superconductor layers and interface are obtained. Allowance is made for an arbitrary direction of wave propagation in the interfacial plane. Increments of kinetic and hydrodynamic instabilities are found. It is shown that absolute instability may occur.