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Oscillatory instability in a driven granular gas

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.

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Oscillatory instability in a driven granular gas

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

  2. Motivation • Granular Materials are ubiquitous: • sand, sugar, flour, … • GMs are important: • powder metallurgy, pharmacology, … • GMs are interesting Surface Waves Avalanches Size separation Brazil Nut Effect

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

  4. 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

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

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

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

  8. A. Phase-separation instability Aspect ratio: H Marginal stability: unstable 4 3.5 3 stable R*c 2.5 0 0.8 1.6 Livne et al. (2002), Khain and Meerson (2002)

  9. Meerson, Sasorov, Pöschel,  and Schwager (2002) MD simulations, hydro simulations: Two coexisting phases One phase 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 !

  10. B. Oscillatory instability http://huji-phys.phys.huji.ac.il/staff/Khain/index.html Linear stability analysis: instability threshold Khain and Meerson (2003) 1 2 Unstable region Stable region

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

  12. MD simulations: stable region unstable region large-amplitude oscillations small-amplitude noise

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

  14. 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.

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