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

• Analytical solution to the equation of motion of original (numerically solved) model.
• We explore effects of several parameters on evolution of the instability.
• Delayed fluidization time models forcing and granular response temporal coupling.
• Model now describes abrupt changes observed at large vibration accelerations.

The U-tube granular instability is understood as a tendency to increase of the height difference Δh between the granular material in both branches of a vertically vibrated U-tube. Δh 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 γ with the maximum dimensionless acceleration Γ 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 γ. The original model could not predict an abrupt decrease of γ observed at large Γ. In this work we show how considering a Γ dependent delayed coupling between the granular material and the vibrating tube allows the model to describe the rapid decrease of γ seen at large Γ and provides a new perspective to understand this behavior.