A note on the upward and downward intruder segregation in granular media

The intruder segregation dependence on size and density is investigated in the framework of a hydrodynamic theoretical model for vibrated granular media. We propose a segregation mechanism based on the difference of densities between different regions of the granular system, which give origin to a buoyant force that acts on the intruder. From the analytic solution of the segregation velocity we can analyze the transition from the upward to downward intruder’s movement. The understanding of the behavior of granular segregation is relevant due to their practical importance in many industries[1,2]. When a granular mixture is subject to vertical vibrations under gravity, the grains tend to segregate with the larger particles at the top of the bed. Many studies have been devoted to this subject, usually referred as the “Brazil nut effect”[3–18]. There exists a great controversy concerning the upward to downward segregation in granular materials [19–24]. Here we address this problem using a recently proposed model for intruder size segregation in dry granular media [25]. This model characterizes the rise velocity of a large intruder particle immersed in a medium of monodisperse fluidized small particles. In this work we show explicit expressions for the segregation dynamics and the interplay of the forces which arises from the size and mass–density differences. We will derive a time–evolution equation for the segregation velocity and we will obtain a criterion for the crossover from the upward to downward intruder’s motion. We will show how the size and density dependence affect the segregation motion. Finally, we address some comments related with previous theoretical and experimental works. In Ref. [25] we have proposed a segregation mechanism based on the difference of densities between different regions of the system, which gives origin to a buoyant force that acts on the intruder. This force include an Received: 22 October 2002 Leonardo Trujillo (&), Hans J. Herrmann Physique et Mécanique des Milieux Hétérogènes École Supérieure de Physique et de Chimie Industrielles, 10, rue Vauquelin, 75231 Paris Cedex 05, France Hans J. Herrmann Institut für Computeranwendungen 1 Universität Stuttgart Pfaffenwaldring 27, 70569 Stuttgart, Germany Archimedean buoyancy force due to the differences between the intruder material density ρI and the bed density ρF , fA = (ρF − ρI)VIg. Here g is the gravity field and VI = D D r D I is the D–dimensional volume of an intruder with radius rI . The factor D = 2πD/2/ (D/2) is the surface area of a D–dimensional unit sphere. Also, we include a thermal buoyancy force caused by density variation of the granular fluid which comes from differences in the local “granular temperature” Tg. The change in the granular fluid density through the thermal expansion, produced by the difference of temperatures, is ρF = ρF (1− α Tg), where α is the thermal expansion coefficient. The thermal contribution to the buoyancy force is fT = ρFVIg, where ρF = ρF − ρF = −αρF Tg. The granular temperature Tg is defined proportional to the mean kinetic energy associated to the velocity of each particle. The granular temperature difference Tg is due to the dissipative nature of the collisions between grains. This difference is due to the fact that the number of collisions on the intruder surface increases with the size, but the local density of dissipated energy diminishes. The region with intruder is hotter than the region without intruder. This lead to the thermal buoyancy force that contributes to the intruder’s upward movement. The intruder also experiences a viscous drag force of the granular fluid. The drag force fd [26,27] is considered to be linear in the velocity of segregation u(t), and is like the Stokes’ drag force fd = −6πμrIu(t), where μ is the coefficient of viscosity. Therefore the equation of motion that governs the segregation process is