Vertical colunm of beads under external vibrations

We present results from computational experiments of onedimensional granular fluid. With characteristic propertie s lying between those of liquids and solids, this class of material consists of ass emblies of solid particles which interact via mechanical forces (contact and fr iction) and are maintained together by a gravitational field. The system is compo sed of a vertical column of spherical beads driven by a sinusoidally vibratin g plate. Results for N-beads experiments show the fluidization and condensation phe omena, both dependent on the amplitude and frequency of the driving plat e. Some scaling properties arising from the dynamics of the beads are also sh own. Resumo. Usando a t́ ecnica de Evento-Dirigido, apresentamos resultados de simulaç̃oes computacionais da din âmica de um fluido granular unidimensional. Com caracteŕ ısticas semelhantes aos s ólidos e aos ĺ ıquidos, esses tipos de materiais s̃ao formados por um conjunto de part ı́culas śolidas que interagem via forças de contato mec ânicas e de fricç̃ ao, sendo mantidas juntas por um um campo gravitacional. O sistema comp õe-se de uma coluna vertical de esferas submetidas̀a aç̃ao de uma base vibrat ória. Os resultados das simulaç ões para o caso geral comN esferas apresentam fen ômenos de fluidizaç ̃ o e condensaç ão, ambos dependentes da amplitude e frequ ência da base. Leis de escalas, incluindo a posiç̃ao ḿedia do centro de massa e a dilataç ão da coluna, resultantes da din̂amica das esferas, tamb ém s̃ao verificadas. 1. Introdução Granular materials are present in everyday life playing a cr itical role in several fields of applications such as pharmaceutical, cosmetic and housing industries. The studies about granular proprieties can improve procedures involving ext raction, transport, storage and mixing process . Each process can present desired and undesi red phenomena ocasioned by intrisic features of the granular dynamic. Moreover, und erstanding these proprieties is essential to optimize costs of granular handling such as w aste of energy for example. Granular flow systems can present clogging and mixture proce dures that may lead to segregation given wrong conditions [Duran 2000]. Initiall y, due to the large and practical applicability, this field was first restricted to engineerin g studies. However, as granular systems can also exhibit period doubling followed by chaoti c behavior, they have been investigated by physicists and computational mathematici ans. The great challenge to understand the behavior of granular s ystem is to determine the state of matter of the system. Granular matter can be have either like fluid or like a solid. Therefore new theoretical ideas beyond clas sic l mechanics are needed [Kadanoff 1999]. Solid matter has initial shape called unstressed shape whic h can be modified by a shear stress. The modified quantity from the rest shape is cal led defomation and it is proportional to Young modulus. Granular matter can not be consi dered as solid because they can deform continually under an applied shear stress regard less of how small the applied stress is [Anderson 1990]. In addition, granular matter can flow. Granular matter display similar phenomena as in the ordinary fluids such as freezing, plasticity and hysteresis [Kadanoff 1999]. By contrast, granular matter is composed o f discrete and macroscopical units and present substantial change of density histeresis [Du et al. 1995]. The density of the fuid is either null (void), or material density (ρ) breaking the continuum mechanics approximations [Landau and Lifshitz 2004]. Therefore t h y can not be considered as fluid and can not be modeled by the Navier-Stokes equation. Moreover, granular matter and molecular gases are composed of discrete units and behave according to Newton’s laws of mechanics. However, be yond the diferences in the size of particles, collisions between units dissipate ener gy leading to emergence of some phenomena exclusively found on granular systems. Granular gases exhibit a tendency to coagulate into cluster without the aid of attractive intera ctions [Goldhirsch 1993]. This phenomenon are consequence of the dissipative interaction s and they do not have correspondence in the elastic world. Collapses occur in groups between near-interacting particles. It happens when there is a high number of collisio n between particles in short interval of time and the overall state of the system does not p resent relevant changing. The assembly of particles has a small relative velocity lead ing to the emergence of particles strings with vanishing relative velocities. Colapse a re consequence of local interation [Goldhirsch 1999]. Other important phenomena is clusterin g. Consider a uniform distribution of particles over a domain. As a result of statisti cal fluctuation, the density in certain region is increasead locally without change in gran ul r temperature. High density leads to a high number of collisions and dissipation which in tur makes the temperature to rise in the region of high density as a result of energy diss ipation. As collisions are inelastic, the granular temperature and consequently the p ressure in the dense regime decays faster than neighboring region. Other particles are at t acted to the denser region in attempting to equalize global pressure. The number of parti cles and collisions increases and self-amplify the process [Goldhirsch 2003]. The present work focuses on the condensation and fluidizatio n phenomena verified in one-dimmensional granular system. Explaining the mo del and the numerical procedure, section 2 presents the simulation of a colunm of bead s driven by a vibrating plate. Section 3 gives some results and discussions about fluidizat ion nd condensation state as well as some scaling laws related to center of mass. Conclusi ons are presented in section 5. 2. The Numerical Procedure The simulation system consists of a vertical container with sp eres with radius r equally spaced and driven by a sinosuidally plate of infinitely large mass. Spheres are allowed to move only in the vertical coordinate and therefore they ca n not exchange postions. Dissipation only occurs during collisions between particl es with a coeficient of restitution 0 < e < 1. Dissipation may also occur between the first sphere and the p lat , according to coeficient of restitutionew. Collisions between spheres are instantaneous and head to he ad since there is no angular momentum transfer between particles. The post coll isions velocities obey the following relation [Luding et al. 1994]: