Discrete and continuum descriptions of shaken granular matter

The subject of this thesis is the dynamics of granular materials. Granular matter is defined as collections of macroscopic, dissipative particles. The size of the individual particles (grains) must be large enough so that thermal fluctuations may be ignored. The loss of kinetic energy at every grain-grain collision implies the need of an external energy source to keep grains in movement. This thesis centres on a specific energy injection method: vibrated systems, where the grains container is shaken such that particles gain energy through collisions with the walls. As farfrom-equilibrium dissipative systems, vibrated granular matter presents many distinct out-of-equilibrium stable states and complex transitions between them. In this thesis both particle simulations and different continuum models are used to investigate further the relation between discrete and continuum descriptions of particle systems, a subject of fundamental scientific interest. A collective, semi-periodic movement of the grains inside vertically vibrated containers is for the first time identified and characterized. A simulational study of these oscillations is presented in Chapter 2, and Chapter 3 mainly describes an experimental observation of it. The oscillations take place in density-inverted states, such as the granular Leidenfrost effect, where grains separate in a high temperature region near the moving bottom wall and a dense region on top. The quasiperiodic movement is usually orders of magnitude slower than the energy injection shaking frequency, thus they are named low-frequency oscillations (LFOs). Furthermore, from the equations of mass and momentum conservation in continuum media an expression for the typical natural oscillation frequency is derived, in good agreement with both simulations and experiments in the high energy injection limit. Increasing the energy input and system size takes the system from the granular Leidenfrost state to a buoyancy-driven convective state. Chapter 4 presents an indepth study of this transition, revealing the existence of fluctuating convective flows far before the transition, as also suggesting a reinterpretation of the dynamics that includes the influence of LFOs. The characteristic length and time-scales of precursory fluctuations are measured, and the amplitude of the critical mode is observed to be consistent with a quintic supercritical amplitude equation. The last two chapters of this thesis study the granular Leidenfrost to convection transition using granular hydrodynamics. Chapter 5 deals directly with the relation of discrete and continuum descriptions of granular systems. A methodology is proposed to quantify the finite-number effects on fluctuations, involving scalings that leave the granular hydrodynamic equations invariant while varying the total number of particles. This method allows us to conclude that LFOs are a finite-number, discretization phenomena, although the mode of oscillation is seen to be a macroscopically determined quantity. In Chapter 6 the granular hydrodynamic equations are numerically solved, and the solutions compared with particle simulations. Deviations of the continuum model due to finite-size and higher order effects are discussed. Finally, it is observed that to first order the transition can be understood as a Rayleigh-Bernard instability described by Navier-Stokes-like equations in the Boussinesq approximation.