Original Paper The flow regime during the crystallization state and convection state on a vibrating granular bed Shih-Chang Tai, Shu-San Hsiau * Department of Mechanical Engineering, National Central University, Chung-Li 32001, Taiwan, ROC article info Article history: Received 24 September 2008 Accepted 20 January 2009 Keywords: Granular motion Occupied percentage Concentration distribution Crystallization Convection Particle segregation abstract It is possible to utilize the large particle distribution on the free surface and inside a vibrating granular bed to understand the segregation phenomenon and the granular motion states in the vibrating bed. In this study we strive to analyze different flow regimes in a binary mixture in a granular vibrating bed. The granular temperature of large particles on the surface of the vibrating bed can be used to define the motion states, the crystallization state or the convection state. When the dimensionless vibration amplitude increases from 0.25 to 1.0, the granular motion transforms from the slowly stabilizing crystal- lization state to the strong convection state. When the amplitude of the dimensionless vibration increases from 1.25 to 2.5, the granular motion transforms from the fast stabilizing crystallization state to the unstable crystallization state. The percentage of large particles occupying the free surface and the con- centration of these large particles are analyzed to understand the motion states. Ó 2009 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of Powder Technology Japan. All rights reserved. 1. Introduction The rheology of granular systems has been widely investigated over the last decade as evidenced by the many published papers [1,2]. The rheological properties of granular suspensions are nor- mally studied via molecular dynamics simulations [3] from which the ‘‘flow diagram” for the volume fraction or stress plane is derived. Recently the mechanisms for pattern formation in granular sys- tems have been experimentally and theoretically studied, especially, those concerned with the relation between particle mobility and structure, in the quasistatic regime [4,5]. Since the pioneering work of Kosterlitz and Thouless [6] the properties of the melting transition of two-dimensional solids have been extensively studied. Continuous crystallization-to-liquid melting transition has been demonstrated by the measurement of two-dimensional melting in magnetic bubble arrays [7]. Several interesting phase transitions in granular system have been reported [8–15]. Olafsen and Urbach [14] investigated the transition from a hex- agonally ordered solid phase to a disordered liquid in a monolayer vibrating spheres. Their experimental results showed strong simi- larity to simulations of the melting of hard disks in equilibrium, de- spite the fact that the granular monolayer was far from equilibrium, due to the effects of vibration force and interparticle collision and dissipation [16]. Clerc et al. [17] carried out a combined experimental, numerical and theoretical study of liquid–solid-like phase transitions that oc- curred in a vertically vibrating fluidized dense granular system. They characterized the dynamic behavior of the phase transition, while avoiding 2D effects such as curvature between the phase or crystal orientation interaction dependence. The crystallization phenomenon has also been studied by a number of researchers [18,19]. Reis et al. [18] investigated the crystallization of a uniformly heated quasi-2D granular fluid as a function of the filling fraction. They utilized the Lindemann ratio to define whether the granular bed was in the condition of crystal- lization or not. When the filling fraction U was 0.719, the Linde- mann ratio of 0.15 indicated that the granular bed entered the crystallization condition. Related references showed that when the Lindemann ratio was between 0.1 and 0.15, the granular bed would be packed as crystallization [20,21]. The Lindemann ratio L was defined as the root-mean-square displacement of particles in a crystalline solid about their equilibrium lattice positions [22], divided by their nearest neighbor distance b. L ¼ 1 b 1 M X N i¼1 ðDR i Þ 2  ! 1=2 ; ð1Þ where M is the number of particles; and the portion in brackets [...] denotes the average over the dynamic trajectories of the particles. The Lindemann criterion states that the crystal melts when L overcomes some ‘‘critical” (yet not specified a priori) value L c [20]. Obviously, one would hope this latter quantity to be approximately 0921-8831/$ - see front matter Ó 2009 The Society of Powder Technology Japan. Published by Elsevier BV and The Society of Powder Technology Japan. All rights reserved. doi:10.1016/j.apt.2009.01.003 * Corresponding author. E-mail address: sshsiau@cc.ncu.edu.tw (S.-S. Hsiau). Advanced Powder Technology 20 (2009) 335–349 Contents lists available at ScienceDirect Advanced Powder Technology journal homepage: www.elsevier.com/locate/apt