Fingering instabilities in particle systems

Abstract

In many multiphase systems, material interfaces can be destabilized by shocks. Small disturbances at these interfaces can grow in size to form large-scale fingers. We consider a shock propagating through a granular system that consists of two types of particles, of different mass, that are initially separated by an interface. The finger instabilities are studied by numerical simulations. We have developed a code to simulate granular system efficiently in both two and three dimensions. Our numerical algorithm takes the advantages of two popular numerical methods. We use the Discrete Element Method that computes the forces to propagate particles in high volume fraction regions and the collision-detection method to propagate particles in low volume fraction regions. An underlying mesh is used to achieve a fast detection of particle collisions and calculation of inter-particle forces. We have designed and developed a suitable data structure and implementation that ensures that forces and collisions only occur between particles in adjacent mesh blocks. This gives rise to a highly efficient and accurate numerical method for general particle systems. This approach leads to a numerical method that scales linearly with number of particles. The code takes a very general form and can be used to study a diverse range of granular systems. In the classical case of immiscible fluids, the finger of heavy fluid propagating into the light fluid grows faster and becomes much thinner than the finger of light fluid propagating into the heavy fluid. This typically leads to a rapidly moving, long thin finger of heavy fluid and a slowly moving, wide finger of light fluid. This is extremely important in understanding the role that the shock plays in promoting mixing. Heavy fluid can penetrate deep into the light fluid whereas light fluid typically penetrates much shorter distances. We consider a system that consists of two types of discrete particles. The particles have different mass and are initially separated by an interface. When two particles collide, we assume that the energy loss can be characterized by a constant coefficient of restitution. We show that when a shock propagates through such a material collisions at the interface between particles of different type lead to shock refraction and focusing. If collisions between particles conserve energy, then the relative sizes and growth rates of the fingers are similar to that in the analogous shock-induced fingering instability in fluids. This causes a flux of particles that leads to initial finger growth that is qualitatively similar to the fluid case. We also show that microscopic mixing can have an important impact on finger growth. On the other hand, if particle collisions lose energy, the non-linear growth phase of inelastic systems is very different from that of elastic systems. The simulation results show that energy loss during particle collisions, even when very small, causes the qualitative features of the finger growth to be completely opposite to the fluid case. The fingers formed by light particles grow faster and become longer and narrower than the fingers formed by heavy particles. In addition, the finger composed of light particles collapses into an extremely compact, tortuous filament and diffusive mixing between particle types at the particle scale is heavily suppressed. We show that this qualitative change is due to a shear driven compression of the light particles. Finger growth in elastic systems is relatively insensitive to the perturbation amplitude, in contrast the simulation results show that the finger development does depend strongly on the interface amplitude for inelastic systems. These surprising phenomena show that elasticity is the critical feature for these systems. The initial finger growth also contains some interesting characteristics. When a shock passes from the light-particle phase to the heavy-particle phase the shock focusing is such that perturbations on the interface grow directly. On the other hand, when a shock passes from heavy-particle phase to the low-particle phase the shock focusing occurs at different locations and perturbations on the interface first become flattened and then grow in the opposite direction. We refer to this as phase inversion. We found that the phase inversion only occurs when the density ratios are larger than a critical value. This has important implications for mixing and this critical value represents the weakest possible mixing for both elastic and inelastic systems.

Date of Award	15 Feb 2007
Original language	English
Awarding Institution	City University of Hong Kong
Supervisor	Jonathan James WYLIE (Supervisor) & Qiang ZHANG (Co-supervisor)