Example: Numerical Simulation of Simple Shear Flow
A binary solid mixture (1 and 2) with the same density was sheared between two infinite parallel plates set to move with relative velocities ±V 0/2. The motion of the granular mixture was in only the x-direction and was considered fully developed so that all the flow parameters are only a function of y. In this study, x and y are the axes parallel and perpendicular to the plates, respectively, as shown in Fig. 2.1. In
Fig. 2.1 Simple shear flow of a binary mixture (This figure was originally published in AlChE Journal 51, 2005 and has been reused with permission)
this example, the steady-state regime was considered, where the thermal equilibrium may be reached when the viscous effect due to continuous shearing is balanced by the dissipation due to collisions.
Simple shear flow is characterized by a linear profile of the velocity field (the shear rate, dv/dy, is constant). In this situation, the external forces are neglected, and the particles in the mixture move with the same center of mass average velocity, v. This means that the granular temperatures, solid volume fractions, and gradient of the velocities are all uniform throughout the flow zone. Therefore, fluctuating energy equations for phases 1 and 2 reduce to the following nonlinear algebraic equations
The system of Eqs. (2.92) and (2.93) was solved numerically for Q1 and 62, for the same flow parameters used by Galvin et al. (2005). The large particle mass, m1 is 1. The ratio of the plate spacing to the large particle diameter H/d is 4.45 or 9.8, depending on the total solid volume fraction and the solid volume fraction ratio. The ratio of H/d1 was chosen in the MD simulation to avoid cluster formation (Hopkins and Louge 1991; Liss and Glasser 2001; Clelland and Hrenya 2002; Alam et al. 2002). Furthermore, the shear rate was set to a constant value of V0/H = у = 1.
The calculated granular temperatures for different diameter ratios using the Iddir and Arastoopour (2005) model were compared with the Jenkins and Mancini (1987) theoretical results and the MD simulation results of Galvin et al. (2005).
Figure 2.2 shows the variation of the fluctuating temperature ratio with the particle size ratio at different restitution coefficients for e1/e2 = 0.5. The Iddir and Arastoopour model predicts well the non-equipartition of energy of the two interacting particles for inelastic collisions. As observed by several investigations in the literature (e.g., Clelland and Hrenya 2002), the fluctuating granular temperature of the large particles increases relative to that of the small particles with an increase in large to small diameter ratio.
In the range of parameters investigated, equipartition was observed in two cases: first, when the restitution coefficient is higher than 0.99 and, second, when the two particles have the same mechanical properties (p1 = p2 and d1 = d2). We noticed that the restitution coefficient is the most important parameter responsible for non-equipartition. The effect of the restitution coefficient on the deviation of the granular temperature of two particles from each other is enhanced by the size disparity. For example, for e = 0.99, the ratio 01/02 increases very slowly with the size ratio; however, for e = 0.8, a strong increase was observed for higher particle diameter ratios. Figure 2.3 shows the comparison between the calculated fluctuating energy ratio as a function of the diameter ratio based on the Iddir and Arastoopour (2005) model with the MD simulation and the theoretical results of
Fig. 2.2 Variation of the granular energy ratio with the diameter ratio for different restitution coefficients. pi/p2 = 1, sT = 0.5, and ?j/e2 = 0.5 . The subscripts 1 and 2 stand for large and small particles, respectively (This figure was originally published in AIChE Journal 51, 2005 and has been reused with permission)
Fig. 2.3 Variation of the granular energy with the diameter ratio for e = 0.95, eT = 0.5, and e1/e2 = 0.5. Comparison of Iddir and Arastoopour model (2005) (solid line) with the theory of Jenkins and Mancini (1987) (dashed line) and the MD simulation results of Galvin et al. (2005) (squares) (This figure was originally published in AIChE Journal 51 (2005) and has been reused with permission)
Jenkins and Mancini (1987) at eT = 0.5, e1/ e2 = 0.5, and a restitution coefficient of
0.95. Results given by the Iddir and Arastoopour (2005) model compared very well with the MD simulation results compared to those obtained by Jenkins and Mancini (1987). The deviation of the Jenkins and Mancini theory from the MD simulation and the Iddir and Arastoopour results is more pronounced for low restitution coefficients and high diameter ratios. At a diameter ratio less than 1.5, both theories exhibit a good agreement with the MD simulation results.