Improved model for the U-tube granular instability: Analytical solution and delayed coupling

摘要

The U-tube granular instability is understood as a tendency to increase of the height difference Delta h between the granular material in both branches of a vertically vibrated U-tube. Delta h has been reported to increase exponentially with time for a certain range of experimental parameters. A simple model based on a cyclic fluid/solid response of the granular material has been used to explain the dependence of the exponential growth rate gamma with the maximum dimensionless acceleration Gamma of vibration, at least within a certain range of applicability. We introduce an analytic solution of the model and use it to discuss the effect of several parameters on gamma. The original model could not predict an abrupt decrease of gamma observed at large Gamma. In this work we show how considering a Gamma dependent delayed coupling between the granular material and the vibrating tube allows the model to describe the rapid decrease of gamma seen at large Gamma and provides a new perspective to understand this behavior. (C) 2015 Elsevier Ltd. All rights reserved.