1. Symmetry breaking instabilities in granular gases. Lectures I and II. Baruch Meerson Hebrew University of Jerusalem, Israel Plan 1.Examples of symmetry breaking instabilities + motivation. 2.Granular hydrodynamics (a brief overview). 3.Thermal granular convection. 4.Phase separation. 5.Giant fluctuations at onset of phase separation: is hydrodynamics sufficient? 6.Summary ESCHOOL/Advanced Study Institute - PHYSBIO3 Benasque 2003

2. A few pictures of the Hebrew University of Jerusalem

3. A few pictures of Jerusalem

4. Example 1. Granular Maxwell’s Demon: phase separation by symmetry breaking Schlichting and Nordmeier (1996), Eggers (1999), Lohse et al. (2001), Brey et al. (2002),… original experiment more experiment + theory theory

5. Granular Maxwell’s Demon (continued): coarsening Lohse et al. (2002) Coarsening will be considered in Lecture 3.

6. Example 2. Phase separation (and coarsening) by symmetry breaking in a granular monolayer driven by vertical vibrations Experiment: Olafsen and Urbach (1998) MD simulations: Nie et al. (2000), Cafiero et al. (2000) static (crystalline) cluster “gas”

7. Before going further: Gas of Inelastic Hard Spheres inelastic binary collisions (simplest model) coefficient of normal restitution: coefficient of normal restitution  : particle diameter m: particle mass

8. In most cases, this model cannot describe granular solids: It does describe many aspects of granular liquids: and, of course, granular gas.

9. Driving wall g Inelastic particle collisions cause a negative temperature gradient Experiment: Wildman et al. (2001,2002). Theory: He, M. and Doolen (2002), Khain and M. (2003) Example 3. “Thermal” granular convection MD simulations: Ramìrez et al. (2000), Sunthar and Kumaran (2001).

10. density map Luding (2002) Goldhirsch and Zanetti (1993) Example 4. Clustering in a freely cooling granular gas Nearly elastic collisions, 1-r<<1 much wider cluster bands: Granular Hydrodynamics has better chances to be valid Strongly inelastic collisions, narrow cluster bands (few particle diameters)

11. Motivation: why should we care about symmetry breaking instabilities (SBIs)? SBIs provide sensitive tests to models of granular flows They improve our understanding of pattern formation far from equilibrium They are cool

12. II. Navier-Stokes granular hydrodynamics (GH): an overview GH operates with three coarse-grained fields:  (r,t): granular density v(r,t): granular velocity T(r,t): granular temperature Formal definitions: Averaging over cells that are small compared to the system size, but still include many particles n(r,t): local number density a compressible fluid!

13. Equations of Navier-Stokes GH P: stress tensor Q: heat flux  : rate of energy loss by collisions per unit volume f: external force More fundamental kinetic theory validates these equations and supplies constitutive relations: Jenkins and Richman, Tan and Goldhirsch, Brey et al. mass conservation momentum conservation energy equation d=2 or 3: dimension of space

14. “Standard” constitutive relations of GH Valid for nearly elastic collisions, 1-r<<1, and at small or moderate densities rate of deformation tensor deviatoric part of D Jenkins and Richman (1985,1988) G, F, J and M: functions of n  2 (d=2), or n  3 (d=3) a 1, a 2, a 3, a 4 and a 5,: numbers Conveniently presented by Babić (1993)

15. The energy sink term  a  simple estimate Energy lost in each collision: A particle with speed v T ~ (T/m) 1/2 loses, on the average, energy ~ m (1-r 2 ) v T 2. Collision rate ~ v T / l, where l is the mean free path. n particles in unit volume. d=2, dilute limit

16. When is the Navier-Stokes GH valid? Not a simple question. Important necessary conditions are: 1.Mean free path and particle diameter are small compared to any length scale described hydrodynamically. 2.Mean time between two consecutive collisions are small compared to any time scale described hydrodynamically. 3.Only single, liquid-like phase. 4.Velocity distribution function f(v,r,t) is “close to Maxwellian” at all r and t of interest. 5.Inelastic collapse doesn’t happen (or removed by a regularization). 1,2,4 and 5 are usually satisfied for nearly elastic collisions, 1-r<<1 (quite a restrictive condition! OK for glass balls, steel balls.) 3 is expected to break down at high densities. In general, this requires introduction of an order parameter into theory, cf. Aranson and Tsimring (2002), even for the hard sphere model. Sometimes, condition 3 can be relaxed: Grossman et al. (1997), M., Pöschel and Bromberg (2003), Gollub et al… Not always clear why.

17. Checking GH in MD simulations of a simple steady state M., Pöschel, Sasorov and Schwager (2002) thermal wall Works well Many other systems checked. No major disagreements found when conditions 1-5 are met (one open question is still under investigation, wait for the end of Lecture II). Boundary conditions may introduce additional limitations, if one wants to have a full hydrodynamic problem.

18. Relaxing condition 3: close-packed “floating”cluster Lohse et al. (2003) MD simulations M., Pöschel and Bromberg (2003) experiment

19. A simple version of Granular Hydrostatics [Grossman, Zhou and Ben-Naim (1997)] works well at all densities, up to hexagonal close packing M., Pöschel and Bromberg (2003) T=const g Phase coexistence doesn’t seem to play a big role here

20. III. Back to thermal granular convection He, M. and Doolen (2002) Governing equations in the dilute limit stress tensor Governing parameters: Froude number (may go to infinity) Knudsen number Relative heat loss parameter g H T0T0

21. Basic state: no mean flow “Thermal” wall at y=0 T goes down with the height y:  T y  may show a density inversion Inelastic collisions cause a negative temperature gradient Solvable analytically

22. unstable 123 0.7 0.8 0.9 Linear stability analysis of the trivial state R > R * c (F,K) Fourier analysis in x direction Linearization of equations, numerical solution. Marginal (or neutral) stability condition:  =0 convection R*cR*c Khain and M. (2002) like in Rayleigh-Benard convection

23. Large-F limit F>>1 drops out: localization of granulate near base Dependence of the threshold on the Froude number 024 1 2 3 Convection cells at onset Phase separation instability

24. Schwarzschild’s criterion for classical convection in compressible fluid no convection S: entropy of fluid

25. Schwarzschild’s criterion works for granular convection: 01234 1 2 3 convection threshold Schwarzschild’s criterion Define entropy S( ,T) like in classic case (r=1)

26. Convection threshold versus Knudsen number K viscosity heat conduction suppress convection 00.020.04 0 0.5 1 1.5 Schwarzschild’s value

27. Hydrodynamic simulations of thermal granular convection: solving full hydrodynamic equations numerically He, M. and Doolen (2002) Supercritical bifurcation, in agreement with MD simulations by Ramìrez et al. (2000) Next (huge) step: developing a weakly nonlinear theory close to onset. Can Boussinesque approximation be validated?

28. IV. Phase-separation instability “hot” wall gravity = 0 elastic wall Steady state obeys: 00.20.40.60.81 0 0.1 0.2 0.3 0.4 X  “hot” wall elastic wall T Early work: Theory: Grossman et al. (1997). Experiment: Kudrolli et al. (1997) stripe state Model:

29. GH predicts instability of the stripe state, 2D steady states appear. Livne, M. and Sasorov (2000,2002), Khain and M.(2002) Steady-state describable by hydrostatic equations: heat conduction heat loss nonlinear problem multiple hydrostatic solutions possible, stripe can be unstable or metastable Stripe state: 1D solution of these eqns.

30. Governing parameters of the hydrostatic equations area fraction aspect ratio hydrodynamic heat loss parameter Livne, M. and Sasorov (2000,2002); Khain and M. (2002)

31. f p Negative compressibility!  C Physics of stripe state p(f,  ): steady-state pressure of stripe state Khain and M. (2002), Argentina et al. (2002)

32. 00.060.120.18 0 0.01 0.02 0.03 0.04 0.05 f p Infinite layer Stripe state unstable within spinodal interval spinodal interval Spinodal interval everywhere critical point  C C:C:  C Argentina et al. (2002), KMS  C

33. Analogy: P-V isotherms of van der Waals gas Coexistence region Spinodal region

34. What if  is finite? Negative lateral compressibility: destabilizing effect f P Lateral heat conduction: stabilizing effect  c Critical lateral length:  c  C for instability Negative compressibility

35. How to find  c, the critical lateral length for instability? Marginal stability analysis scaled inverse density kk is smalllinearization of hydrostatic eqns. Schrödinger equation for  k (x) k: discrete energy level  Close to continuous bifurcation threshold

36. f k/  1/2 Stripe stable beyond curve  c at different   c =  /k c and f 1 <f<f 2 needed for instability k: wave number along stripe LMS (2000,2002); KM (2002), Brey et al. (2002)

37.  =12,500 f=0.0378 Δ=1.2 Δ=1.3 Δ=3  c = 1.28 LMS 2002 Bifurcated state (droplet) above critical value of  color density maps close-packed droplet

38. Coarsening dynamics: hydrodynamic simulations Λ=12,500 f=0.0378 Δ=6 Final state: one close-packed droplet LMS 2002 Gas-mediated coarsening Hydro simulations used Stokes friction instead of full viscosity

39. Direct mergers of clusters also happen Λ=12,500 f=0.0378 Δ=6 Same final state! LMS 2002

40. Supercritical bifurcation diagram: theory vs. (overdamped) hydrodynamic simulations Supercritical bifurcation diagram: theory vs. (overdamped) hydrodynamic simulations Y c : normalized y-coordinate of center of mass 0123 4 567  YcYc Δ c  1.28 theory  =12,500 f=0.0378 0.1 0.2 0.3 0.4 0.5 Excellent agreement with analytic theory not too far from threshold

41. MD simulations MD simulations (M, Pöschel, Sasorov and Schwager 2002)  c weakly fluctuating stripe state, hydrostatics (and overdamped hydrodynamics) are doing well Δ c = 0.5 Δ = 0.1 N=2·10 4

42. MD simulations hydro simulations  c Nucleation, coarsening of droplets, one droplet survives. Hydrostatics (and overdamped HD) are doing well. MPSS (2002)LMS (2002) N=2·10 4

43. Very large aspect ratio:  c MD simulations Argentina et al. (2002) fluctuations weak, hydrostatics OK Outside of spinodal interval, inside coexistence interval

44. What happens around  c ? MD simulations Transitions between the two states with broken symmetry? N=2·10 4

45. Hydrostatics fails in a wide region around  c Center-of-mass dynamics Giant fluctuations

46. Symmetry-breaking transition occurs somewhere at 0.3 <  < 1, hydrostatics predicts  c = 0.5. Probability distribution of Y c at different 

47. Probability distribution maxima vs.  hydro simulation Systematic disagreement with hydrostatics within fluctuation- dominated region. Change of bifurcation character? hydrostatic bifurcation curve maxima of PDF

48. What is so unusual here? Fluctuation region anomalously wide N=2,300 MD simulations: Ramirez et al. (2000), Hydrodynamics: HMD (2002), KM (2003) For comparison: transition to convection (when one adds gravity) doesn’t show any “noise anomaly”: Where do the giant fluctuations come from?

49. Y c vs. N at fixed , f and  c N=20000 N=15000 N=10000 N=5000 Low-frequency oscillations don’t seem to go down as N increases

50. Scenario I. Giant fluctuations are driven hydrodynamically: by a (yet unknown) instability (Ia), or by a long-lived transient (Ib) To check this scenario, one needs full hydrodynamic simulations (Livne, M. and Sasorov; work in progress).

51. Scenario II. Amplification of discrete-particle noise. Here “Fluctuating Granular Hydrodynamics” is needed Now P, q and  include delta- correlated gaussian random noise terms, in the spirit of Landau and Lifshitz (1959). Further simplifications needed: very small or large aspect ratio, work close to the critical point, … Once it is computed we can start working, but: the problem is difficult because of mode coupling: two acoustic modes, entropy mode and shear mode. Hopefully, Scenario I is correct! Nasty surprise:  noise part of  is not known!

52. Conclusions 1.Symmetry-breaking instabilities (SBIs) are sensitive probes of models of granular flow 2.SBIs are ubiquitous in granular gases 3.Two particular and novel SBIs discussed in Lectures 1 and 2: thermal granular convection and phase separation 4.A whole lot of work remains to be done: convection: Boussinesque (?), weakly-nonlinear theory close to onset, phase separation: giant fluctuations, role of oscillatory instability: yesterday’s talk by Evgeniy Khain Finally, a take-home message about Granular Hydrodynamics: 5. When it can be used, it must be used