Segregation and Fingering Instability in Granular Media

Abstract

Granular systems are ubiquitous in both industry and nature, yet we still know comparatively little of the physics governing their dynamics. Owing to this lack of insight, there is still no continuum description of granular systems that can describe accurately phenomena that are commonly encountered such as segregation. Hence, this doctoral thesis is concerned with obtaining a better understanding of phenomena that occur in bi-disperse granular systems, i.e. the dynamics and segregation of intruders and the formation of Rayleigh-Taylor-like fingering structures. First, the doctoral thesis investigates the motion of a single intruder in dynamic granular systems, i.e., a dense shear flow and a vibro-fluidized bed. To this end, we develop a granular buoyancy model using a generalization of the Archimedean formulation of the buoyancy force inspired by previous work on buoyancy in chute flows. The first model system that was studied is a convection-free vibrated system, allowing us to calculate the buoyancy force through three different approaches, i.e., a generalization of the Archimedean formulation, the spring force of a virtual spring and through the granular pressure field. The buoyancy forces obtained through these three approaches agree very well, providing strong evidence for the validity of the generalization of the Archimedean formulation of the buoyancy force which only requires information about the solid fraction of the intruder (defined as ratio of its volume to its Voronoi volume), hence allowing for a calculation of the buoyancy force that is computationally efficient as coarse-graining is avoided. In a second step to increase the complexity of the granular system, convection is introduced. In such a system, the lift force acting on the intruder is composed of granular buoyancy and a drag force. Using a drag model for the slow velocity regime, the lift force, directly measured through a virtual spring, is predicted accurately via the proposed generalization of the Archimedean formulation of granular buoyancy. The lift force model developed here allowed us in turn to rationalize the dependence of the lift force on the density of the bed particles and the intruder diameter, and the independence of the lift force on the intruder density and the vibration strength (once a critical value is exceeded). Next, we develop a new lift force model to describe the motion of intruders in dense, granular shear flows. Our derivation is based on the thermal buoyancy model of Trujillo and Hermann (2003) but takes into account both granular temperature and pressure differences in the derivation of the net buoyancy force acting on the intruder. We further extend the model to take into account also density differences between the intruder and the bed particles. The model predicts very well the rising and sinking of intruders, the lift force acting on intruders as determined by discrete element model (DEM) simulations and the neutral-buoyancy limit of intruders in shear flows. Phenomenologically, we observe that the presence of an intruder leads to a cooling effect and a local flattening of the shear velocity profile (lower shear rate). The cooling effect increases with intruder size and explains the sinking of large intruders. On the other hand, the introduction of small to mid-sized intruders, i.e., up to 4 times the bed particle size, leads to a reduction in the granular pressure compared to the hydrostatic pressure, which in turn explains the rising of small to mid-sized intruders. Lastly, we turn to a newly discovered Rayleigh-Taylor (RT)-like fingering instability in binary granular media. Fingering instabilities akin to the Rayleigh-Taylor (RT) instability in fluids have been observed in a binary granular system in which dense and small particles are layered on top of lighter and larger particles, when the system is subjected to vertical vibration and fluidizing gas flow. Using observations from experiments and numerical modelling we explore whether the theory developed to describe the Rayleigh-Taylor (RT) instability in fluids is also applicable to binary granular systems. Our results confirm multiple key properties of the RT instability theory for binary granular systems: (i) The characteristic wavenumber is constant with time, (ii) the amplitude of the characteristic wavenumber initially grows exponentially and (iii) the dispersion relation between the wavenumbers k of the interface instability and the growth rates n(k) of their amplitudes holds also for binary granular systems. Our results show that inter-particle friction is essential for the RT instability to occur in granular media. For zero particle friction the interface instability bears a greater resemblance to the Richtmyer-Meshkov instability. We further define a yield criterion Y by treating the granular medium as a viscoplastic material; only for Y > 15 fingering occurs. Interestingly, previous works has shown that instabilities in the Earth’s lower mantle, another viscoplastic material, also occur for similar values of Y.