Thesis

# Hydrodynamic Modeling of Granular Materials

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## Abstract

Granular materials are composed by macroscopic solid particles where interactions are characterized by dissipative collisions. This kind of material is present in our daily life, for example, sand, coal, nuts, rocks etc. In this thesis we study different phenomenon presents in rapid granular flows by means of hydrodynamic simulations of the Navier-Stokes granular equations. In Chapter 2, we introduce the Navier-Stokes equations and two different approaches for the transport coefficients for fluidized granular materials. One of the consequences of the inelastic interaction between the particles is that the flow developed is supersonic and sharp-profiles of the hydrodynamic fields arise. To deal with this problem, we use a high-order shock-capturing method. In Chapters 3 and 4 is shown the model used to solve the hydrodynamic equations and its successful parallelization. Due to the inelasticity, granular gases form dense clusters of particles. This phenomenon is studied for a force-free granular gas in Chapter 5. The kinetics of the cluster formation is analyzed and compared with the presently accepted mode-enslaving mechanism. We observed that the mode-enslaving theory cannot explain the process of cluster formation. Nevertheless, a direct correlation between the appearance of the shock waves and formations of clusters is observed. Another effect specific to granular systems is the collapse. Considering gravity, a granular gas will become at rest if there is no energy supply. In Chapter 6, we analyze the evolution of the granular gas throughout different stages before the collapse, where regions of supersonic and subsonic dynamics are observed. In the supersonic regions, the system develops shocks followed by sharp profiles of the temperature and the density fields moving upwards. In the last stage of the sedimentation, the energy decay has been studied and compared with previous studies. In agreement with all of them, we confirmed that the entire system collapse simultaneously. Similarly as is described for liquids, a vertical vibrated granular layer develops characteristic patterns for certain intervals of frequencies and amplitudes of oscillation. This phenomenon called Faraday instability is an interesting example of granular collective behavior. In Chapter 7, we study numerically the formation of Faraday waves using two approaches of the transport coefficients described in Chapter 2 and comparing with event-driven molecular dynamics simulations. We observed that the two approaches work quite well, although there is a discrepancy related with the expression of the heat flux.

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1. The beautiful series of forms assumed by sand, filings, or other grains, when lying upon vibrating plates, discovered and developed by Chladni, are so striking as to be recalled to the minds of those who have seen them by the slightest reference. They indicate the quiescent parts of the plates, and visibly figure out what are called the nodal lines. 2. Afterwards M. Chladni observed that shavings from the hairs of the exciting violin bow did not proceed to the nodal lines, but were gathered together on those parts of the plate the most violently agitated, i. e. at the centres of oscillation. Thus when a square plate of glass held horizontally was nipped above and below at the centre, and made to vibrate by the application of a violin bow to the middle of one edge, so as to produce the lowest possible sound, sand sprinkled on the plate assumed the form of a diagonal cross; but the light shavings were gathered together at those parts towards the middle of the four portions where the vibrations were most powerful and the excursions of the plate greatest.
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