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Dynamics of Granular Materials: Flows, Relaxation and Convection

  • Daniel C. Hong (a1), Su Yue (a1) and Douglas A. Kurtze (a2)

Abstract

Using the diffusing void model of granular flows, we study the dynamic response of granular materials, in particular, the relaxation of granular pile, pipe flow of granular materials, and the convection of granular media in a two dimensional box subjected to vibrations. We first study the relaxation of a one dimensional granular pile of height L in a confined geometry under repeated tapping within the context of the diffusing void model. The reduction of height as a function of the number of taps is proportional to the accumulated void density at the top layer. The relaxation process is characterized by the two dynamic exponents z and z' which describe the time dependence of the height reduction, Δh(t) ≈ tz and the total relaxation time, T(L) ≈ Lz'. While the governing equation is nonlinear, we find numerically that z=z'=l, which is robust against perturbations and independent of the initial void distributions. We then show that the existence of a steady state traveling wave solution is responsible for such a linear behavior. Next, we examine the case where each void is able to maintain its overall topology as a round object that can subject itself to compression. In this regime, the governing equations for voids reduce to traffic equations and numerical solutions reveal that a cluster of voids arrives at the top periodically, which is manifested by the appearance of periodic solutions in the density at the top. In this case, the relaxation proceeds via a stick-slip process and the reduction of the height is sudden and discontinuous. We then carry out the long wave analysis to demonstrate the existence of the KdV solitons in the traffic equations. Finally, we show our preliminary studies on granular convection in a box. We observe the appearance of two rolls when the control parameter exceeds the critical value, which then undergoes bifurcations to four rolls.

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