(70bp) Convective and Diffusive Motions of Granular Materials in a Vibrated Bed

The use of vibrated beds in the treatment of granular materials, including granular mixing, has intrigued researchers in recent years. There are many complicated phenomena, such as heaping, fluidization, surface wave and arching, can be generated in a granular bed under external vibrations. The convection of granular materials is an important mechanism to drive these interesting phenomena. For a vibrated granular bed with frictional sidewalls, the flow of granular materials between top and bottom occurs in the form of symmetric convection rolls under some vibrating conditions. One of the purposes in this study is to investigate the influence of vibrating condition on the formation of symmetric convection flow.

In a vibrated granular bed with different size particles, the most typical phenomenon is size segregation, which is also driven by convective motion of granular materials. However, by using identical particles, the granular mixing may be benefited from the convective flow. In addition, the experimental studies had showed that the granular mixing was a diffusive process in a vertical vibrated bed. Therefore, convective displacements and diffusive movements of particles are the two important factors which influence the mixing behaviors in a vibrated granular bed.

The current study uses the three-dimensional soft particle discrete element method (DEM) to simulate the behavior of particles in a vibrated granular bed. The DEM is able to give intimate detailed events and conditions within the bed, including contact forces and velocities. In the simulations of the study, all the granular materials are frictional and inelastic spherical glass beads with mass density of 2500 kg/m³ and diameter of d =1 mm. The granular bed is energized by vertical sinusoidal oscillations under different vibrating conditions: four different vibration frequencies f (12, 20, 30, 40 Hz) and various dimensionless vibration accelerations G. The dimensionless vibration acceleration is defined as G = aw²/g, where a is the vibration amplitude, w is the vibration angular frequency and g is the gravity acceleration. The three-dimensional granular bed is a rectangular tank composed of one frictional bottom plane and four frictional sidewall planes. The frictional coefficients of particle-particle and particle-wall are assumed to be a constant, m=0.8. The total number of glass beads in the three-dimensional bed is 1728 (12*12*12).

In this study, the long-term average velocity method is used to calculate the velocity field in the vibrated granular bed, which is divided into 2880 (12*20*12) cubic bins with (0.001m)³. Under the vibration frequency f of 20 Hz and the dimensionless vibration acceleration G of 3, the simulation velocity field demonstrates that the flow exhibits a symmetric circulation pattern with an upward velocity at the center and a downward motion along the sidewalls, like a fountain jetting from the center and spraying around the sidewalls.

For a nearly perfect symmetric convection flow, the horizontal (x and z directions) velocities are almost zero in the horizontal plane crossing the convection center, and only the vertical velocity component V_y exists. Hence, if we divide the granular bed into several horizontal layers with the height of

= d, the averaged V_y of the layer near the convection center is the largest value compared to that of other layers. Furthermore, the vertical mass flow rate of each layer
 can be defined by

where
 is the solid fraction. Note that the second term of the right hand side in Eq. (1),
, is used to subtract the averaged bulk motion of the layer. For a perfectly symmetric convection flow, the averaged bulk motion of each layer is zero, since the mass flow rate in every direction is conserved. However, for a flow that the symmetric motion has not been formed, the bulk motion of the layer may be toward some directions. Hence, subtracting the bulk motion is very important to estimate the really quantity of vertical convection flow, especially for the flow that the symmetric convection has not been form yet. Assuming that the solid fraction is constant across each horizontal layer, by dividing the layer into (12°Ñ12) cubic bins, the dimensionless vertical flow rate J of each layer can be expressed by

where
 and
 are the length of each bin, L_x and L_z are the length of the bed in x and z directions, respectively. To measure the magnitude of the convection flow, the dimensionless convection flow rate, J_conv, is evaluated from the maximum J in the bed.

In granular flows, the velocity fluctuations of particles induce the diffusive motion. The self-diffusion coefficient is used to measure the strength of the diffusive motion. From the concept for analyzing the diffusive phenomena of suspended particles undergoing Brownian motion in a liquid, the self-diffusion coefficient tensor D_ij is defined as

where
and
 are the diffusive displacements in directions i and j, respectively. The brackets, <>, denote an ensemble average for the bed. Note that the displacements of particles result from mean velocity of the bulk motions have been subtracted so that the diffusive displacements are induced by the particles' fluctuations. For a vertical vibrated granular bed, the self-diffusion coefficient in vertical direction, D_yy, is the most significant index to describe the diffusion motion in the bed. Thus, D_yy is used to regard as the magnitude of the diffusive motion.

Figure 1. The Variations of dimensionless convective flow rate, J_conv, with the dimensionless vibration acceleration, G, under different vibration frequencies.

Figure 2. The Variations of self-diffusion coefficient, D_yy, with the dimensionless vibration acceleration, G, under different vibration frequencies.

Figure 1 shows the dimensionless convection flow rate, J_conv, plotted against the dimensionless vibration acceleration, G. The dimensionless convection flow rates increase with the increasing dimensionless vibration acceleration in all cases with different vibration frequencies. For the same acceleration, the dimensionless convection flow rate is greater for the case with lower vibration frequency (with greater vibration amplitude). Figure 2 shows D_yy vary with the dimensionless vibration acceleration, G, under different vibration frequencies. D_yy increases with the increasing G, since the greater G induces the greater energy to promote the velocity fluctuations of particles. Similar to the relation of the convection flow rate with the acceleration (Fig. 1), at the same vibration acceleration, D_yy is greater for a case with the lower vibration frequency. However, comparing Fig.2 with Fig. 1 carefully, it is observed that the increasing trend of D_yy with G is a little different from that of J_conv with G. In Fig. 2, D_yy as a function of G is increasing concave upward no matter at larger G or smaller G (see the inset of Fig. 2). In Fig. 1, J_conv as a function of G is increasing concave upward at larger G, while that is increasing concave downward at smaller G (see the inset of Fig. 1). The increasing rate of J_conv with G is different from that of D_yy with G, especially at smaller G.

Figure 3. The Variations of Péclet number, Pe, with the dimensionless vibration acceleration, G, under different vibration frequencies.

In a granular flow, the Péclet number, Pe, can be used to describe the convective and diffusive motions very well. For a vertical vibrated granular bed, Peis the advection in vertical flow of speed V_yto diffusion with the vertical diffusion coefficient D_yy at the scale of a particle diameter d. The Péclet number is expressed by

Figure 3 shows the variations of the Péclet number with the dimensionless vibration acceleration. It is interesting to note that Pe increase firstly and then decrease with the increasing G in all cases with different vibration frequencies. There is an inflection point of Pe under each frequency. Also, the corresponding acceleration of the inflection point, G_inf, is larger when the frequency is higher. For f = 12 Hz, 20 Hz, 30 Hz and 40 Hz, G_inf are 1.4, 1.6, 2.0 and 2.6, respectively. Before and after the inflection point, what is the difference of the flow pattern between them? The central section's velocity fields as the acceleration near G_inf under different vibration frequencies are plotted, and the detailed physical meaning is discussed in this study.