Session R13: Granular Flows V: Fluctuations and Instabilities

Granular Impact: Predicting Dynamics with a Collisional Model

Granular impact, where a high-speed intruder strikes a granular material, occurs frequently in industrial and natural processes. Due in part to difficulty in obtaining sufficiently fast data at the grain scale, a complete description, which connects grain-scale processes to macroscopic dynamics of the intruder, is still lacking. Using photoelastic particles and a high speed camera, we perform experiments which allow us to capture the intruder dynamics and local granular force response at very fast time scales. This allows us to approach the problem both macroscopically, fitting the intruder dynamics to a fluid-like force law which is dominated by a velocity-squared drag force, and microscopically, where we observe large force fluctuations at small space and time scales. Thus, the intruder deceleration is not smooth and steady, but dominated by intermittent collisions with clusters of grains. Based on this, we present a simple collisional model which yields a shape-dependent velocity-squared drag force. Using experimental data, we show that this model captures the shape-dependence of the intruder deceleration and o-axis rotation well. This confirms the microscopic assumptions of this model, and may provide insight into other dense, driven granular flows.   

Instability in shocked granular gases

Shocks in granular media, such as vertically oscillated beds, have been shown to develop instabilities. Similar jet formation has been observed in explosively dispersed granular media. In the current study, we investigate the origin of this instability. Our previous work addresses this instability by performing discrete-particle simulations of inelastic media undergoing shock compression. By allowing finite dissipation within the shock wave, instability manifests itself as distinctive high density non-uniformities and convective rolls within the shock structure. By analyzing the time evolution of the material undergoing the shock wave compression and further relaxation, we found that the clustering instability is the dominant mechanism controlling this instability. In the present study we extend this work to investigate the instability at the continuum level. We model the Euler equations for granular gases with a modified cooling rate to include an impact velocity threshold necessary for inelastic collisions. Our results demonstrate a fair agreement between the continuum and discrete-particle models. Slight discrepancies, such as higher frequency non-uniformities in our continuum results may be attributed to the absence of viscous effects.   

Linear and nonlinear response in sheared soft spheres

Packings of soft spheres provide an idealized model of foams, emulsions, and grains, while also serving as the canonical example of a system undergoing a jamming transition. Packings' mechanical response has now been studied exhaustively in the context of ``strict linear response,'' i.e.~by linearizing about a stable static packing and solving the resulting equations of motion. Both because the system is close to a critical point and because the soft sphere pair potential is non-analytic at the point of contact, it is reasonable to ask under what circumstances strict linear response provides a good approximation to the actual response. We simulate sheared soft sphere packings close to jamming and identify two distinct strain scales: (i) the scale on which strict linear response fails, coinciding with a topological change in the packing's contact network; and (ii) the scale on which linear superposition of the averaged stress-strain curve breaks down. This latter scale provides a ``weak linear response'' criterion and is likely to be more experimentally relevant.   

Capillary-like Fluctuations of a Solid-Liquid Interface in a Non-Cohesive Granular System

One of the most noticeable collective motion of non-cohesive granular matter is clustering under certain conditions. In particular, when a quasi-two-dimensional monolayer of mono-disperse non-cohesive particles is vertically vibrated, a solid-liquid-like transition occurs when the driving amplitude exceeds a critical value. Here, the physical mechanism underlying particle clustering relies on the strong interactions mediated by grain collisions, rather than on grain-grain cohesive forces. In average, the solid cluster resembles a drop, with a striking circular shape. We experimentally investigate the coarse-grained solid-liquid interface fluctuations, which are characterized through the static and dynamic correlation functions in the Fourier space. These fluctuations turn out to be well described by the capillary wave theory, which allows us to measure the solid-liquid interface surface tension and mobility once the granular thermal kinetic energy is determined. Despite the system is strongly out of equilibrium and that the granular temperature is not uniform, there is energy equipartition at the solid-liquid interface, for a relatively large range of angular wave-numbers. Furthermore, both surface tension and mobility are consistent with a simple order of magnitude estimation.   

Nonlinear bounded convection and a phase separation instability in a dilute granular gas

A weakly nonlinear analysis using Stuart-Landau equation has been carried out to understand the onset of buoyancy-induced convection and a phase-separation instability of a dilute granular gas in a bounded domain. Previous linear stability analysis of the same base state showed that the conduction state is unstable for a range of Froude number and heat-loss parameter. A new instability mode has been found at very small values of the heat loss parameter, the origin of which is shown to be connected to the classical Rayleigh-Benard instability. The bifucation diagrams and the supercritical and subcritical nonlinear patterns with respect to inelasticity and Froude number will be discussed.   

Numerical simulation of unsteady chute flows of two-phase granular mixtures

The unsteady gravity-driven flow of a fluid-saturated granular material on an inclined plane is investigated numerically. Our studies are based on a continuum two-phase flow model for the mixtures of interest. The governing equations are integrated via a predictor-corrector algorithm that employs a generalized projection method for the computation of the phasial pressures. Further, it incorporates an interface detection and capturing method to account for the steep gradients of particle concentration across material interfaces. In our numerical setup, a dense granular layer of constant thickness is placed on the surface of an inclined plane, whereas the rest of the domain is filled with an interstitial fluid. Initially the mixture is assumed to be at rest and is accelerated by gravity. A representative sample of these simulations is presented and discussed. Since the flows of interest are susceptible to Kapitza instability, emphasis is placed on the spatio-temporal evolution of the granular layer's free surface and the interplay between inertia and gravity. Also, we discuss the flow characteristics inside the granular layer and we compare the predicted profiles for the phasial variables with those obtained from previous studies.   

Spatially modulated kinks in shallow granular layers

We report on the experimental observation of spatially modulated kinks in a shallow one-dimensional fluidized granular layer subjected to a periodic air flow. We show the appearance of these solutions as the layer undergoes a parametric instability. Due to the inherent fluctuations of the granular layer, the kink profile exhibits an effective wavelength, {\it a precursor}, which modulates spatially the homogeneous states and drastically modifies the kink dynamics. We characterize the average and fluctuating properties of this solution. Finally, we show that the temporal evolution of these kinks is dominated by a hoping dynamics, related directly to the underlying spatial structure.   

Dynamic X-ray study of the effective temperature in a three-dimensional granular gas

We carried out a high-speed x-ray imaging study of the effective temperature of a highly agitated three-dimensional (3D) granular gas in the tracer limit using tracing particles with various densities, restitution coefficients, and sizes. Both the tracing and background particles satisfy non-Gaussian velocity distributions, in addition to an absence of energy equipartition between translational and rotational degrees of freedom.