Oscillatory instability in a driven granular gas Evgeniy Khain Baruch Meerson Racah Institute of Physics Hebrew University of Jerusalem Granular gas: a simple model of a fluidized granular medium Granular hydrodynamics Phase-separation instability Oscillatory instability Summary

Granular Materials are ubiquitous: sand, sugar, flour, … GMs are important: powder metallurgy, pharmacology, … GMs are interesting Surface WavesAvalanches Brazil Nut Effect Size separation Motivation

The simplest model of granular gas: Inelastic Hard Spheres inelastic binary collisions coefficient of normal restitution: elastic collisions The energy loss in each collision

Hydrodynamics of gases with inelastic collisions Continuous approach: coarse-grained variables Granular temperature T Granular density ρ Granular pressure P Works well for nearly elastic collisions Kinetic theory Constitutive relations

Eqs. of Granular Hydrodynamics These equations and constitutive relations can be derived from kinetic theory (for nearly elastic collisions) Jenkins and Richman (1985), … P - stress tensor q - heat flux rate of energy losses by collisions f - external force

ρ grows, T decreases 1-D static cluster can become unstable! 1-D static cluster state Simplest setting of driven granular gas P = g(ρ)T =const Grossman, Zhou and Ben-Naim (1997) – MD simulations + hydrodynamic model, Kudrolli, Wolpert and Gollub (1997) - experiment Thermal wall Thermal wall Thermal wall Tobochnik (1999), Brey and Cubero (1999) Khain and Meerson (2003)

Governing parameters Governing equations stress tensor Relative heat loss parameter Transport parameter Khain and Meerson (2003) General scenario for instabilities: R exceeds a critical value Area fraction

A. Phase-separation instability Livne et al. (2002), Khain and Meerson (2002) R*cR*c Marginal stability: unstable stable H Aspect ratio:

Two coexisting phasesOne phase Meerson, Sasorov, Pöschel, and Schwager (2002) MD simulations, hydro simulations: Explanations and further exciting issues: wait for the lecture of Baruch Meerson tomorrow Let's consider a small aspect ratio. 1-D static cluster can become unstable even in this case !

Linear stability analysis: instability threshold B. Oscillatory instability Khain and Meerson (2003) Unstable region 1 2 Stable region

MD simulations: Cluster oscillates back and forth away from the thermal walls

MD simulations: stable regionunstable region small-amplitude noise large-amplitude oscillations

What happens for larger aspect ratios? The two instabilities coexist Small isolated cluster with broken symmetry oscillates back and forth

Summary We found a novel oscillatory instability in a simple driven granular system Hydrodynamic linear stability analysis performed, instability threshold determined Predictions of linear theory verified in MD simulations. Next step should be nonlinear theory Hydrodynamics is instrumental in analysis of rapid granular flow.