1. 2013/6/27 1 2013/6/27 Koshiro Suzuki (Canon Inc.) Hisao Hayakawa (YITP) A rheological study of sheared granular flows by the Mode-Coupling Theory

2. 2013/6/27 Introduction Steady-state of the sheared granular flow Steady-state of the sheared granular flow Setting Setting monodisperse hard inelastic smooth spheres (3D) monodisperse hard inelastic smooth spheres (3D) uniform steady shear flow (bulk shearing) uniform steady shear flow (bulk shearing) Energy balance equation (energy/volume/time) Energy balance equation (energy/volume/time) 2 Control parameters volume fraction shear rate restitution coefficient x y d Granular temperature T is determined m volume V

3. 2013/6/27 Introduction Kinetic theory Kinetic theory Shear stress Shear stress Energy dissipation rate Energy dissipation rate Consequence of the energy balance Consequence of the energy balance Bagnold scaling Bagnold scaling 3 [Garzo and Dufty, PRE 59, 5895 (1999)]

4. 2013/6/27 Introduction Limitation of the kinetic theory Limitation of the kinetic theory It is known that kinetic theory works well up to It is known that kinetic theory works well up to However, there is a large deviation from simulation for However, there is a large deviation from simulation for 4 Granular temperatureEnergy dissipation rateShear stress [Mitarai and Nakanishi, PRE 75, 031305 (2007)]

5. 2013/6/27 Our work We aim to construct a theory viable for dense sheared granular flows, We aim to construct a theory viable for dense sheared granular flows, We attempt to apply the Mode-Coupling Theory (MCT) : We attempt to apply the Mode-Coupling Theory (MCT) : It is known that MCT captures the glass transition, It is known that MCT captures the glass transition, It incorporates memory kernels, which is neglected in the kinetic theory. It incorporates memory kernels, which is neglected in the kinetic theory. We expect that the memory kernels play significant roles. We expect that the memory kernels play significant roles. 5

6. 2013/6/27 Previous related works Driven granular fluids Driven granular fluids Kranz, Sperl, Zippelius, PRL 104, 225701 (2010) Kranz, Sperl, Zippelius, PRL 104, 225701 (2010) Sperl, Kranz, Zippelius, EPL 98, 28001 (2012) Sperl, Kranz, Zippelius, EPL 98, 28001 (2012) Kranz, Sperl, Zippelius, PRE 87, 022207 (2013) Kranz, Sperl, Zippelius, PRE 87, 022207 (2013) Sheared granular fluids Sheared granular fluids Hayakawa, Otsuki, PTP 119, 381 (2008) Hayakawa, Otsuki, PTP 119, 381 (2008) Suzuki, Hayakawa, AIP Conf. Proc., to be published (2013) [arXiv:1301.0866] Suzuki, Hayakawa, AIP Conf. Proc., to be published (2013) [arXiv:1301.0866] No work has been done for the rheology of sheared granular fluids No work has been done for the rheology of sheared granular fluids 6

7. 2013/6/27 Results MCT shows better compatibility than to simulation than the kinetic theory for the shear stress. MCT shows better compatibility than to simulation than the kinetic theory for the shear stress. However, MCT fails to obtain the energy dissipation rate, as is the case for the kinetic theory. However, MCT fails to obtain the energy dissipation rate, as is the case for the kinetic theory. 7 Shear stress

8. 2013/6/27 A short review of MCT MCT describes (for thermal glassy systems) : MCT describes (for thermal glassy systems) : two-step relaxation of the density time-correlation function (“cage effect”). two-step relaxation of the density time-correlation function (“cage effect”). the nonlinear rheology (constitutive relation of shear stress and shear rate). the nonlinear rheology (constitutive relation of shear stress and shear rate). 8 Newtonian yield stress high density low density [Suzuki and Hayakawa, PRE 87, 012304 (2013)]

9. 2013/6/27 A short review of MCT 9 Schematic picture Schematic picture The red particle is “caged” by the surrounding particles above the critical density until the cage is broken by shearing. The red particle is “caged” by the surrounding particles above the critical density until the cage is broken by shearing. This picture does not hold for granular particles ! This picture does not hold for granular particles !

10. 2013/6/27 Microscopic dynamics of grains SLLOD equation (Newtonian eq. for uniform shear) SLLOD equation (Newtonian eq. for uniform shear) No external noises No external noises Viscous dissipation of soft, smooth grains Viscous dissipation of soft, smooth grains 10 10 ζ : viscous coefficient (mass/time) t=0 Equilibrium Relaxation to a steady state steady shear & dissipation hard-core limit is taken later k

11. 2013/6/27 Liouville equation (1) Time evolution of physical quantities A(q(t),p(t)) 11 11 Formal solution

12. 2013/6/27 Liouville equation (1) Physical quantities of interest Physical quantities of interest Shear stress Shear stress Energy dissipation rate Energy dissipation rate 12 12

13. 2013/6/27 Liouville equation (2) Time evolution of distribution function 13 13 Formal solution : Phase volume contraction (non-Hermitian)

14. 2013/6/27 Formula (MCT) : shear stress It is written in terms of time/wavenumber integration of time-correlation functions, It is written in terms of time/wavenumber integration of time-correlation functions, Dissipative vertex function Dissipative vertex function 14 14 dominant term (functions of equilibrium RDF at contact) (hard-core limit)

15. 2013/6/27 Formula (MCT) : energy dissipation rate The first term coincides with the kinetic theory ⇒ Maxwellian contribution The first term coincides with the kinetic theory ⇒ Maxwellian contribution 15 15 n.b. kinetic theory result non-Maxwellian contribution

16. 2013/6/27 Time correlation functions 16 16 Slow modes Slow modes Time correlation functions Time correlation functions Isotropic approximation Isotropic approximation current density fluctuation density fluctuation scalar functions,

17. 2013/6/27 MCT equations Isotropic, weak-shear approximation Isotropic, weak-shear approximation Effective friction coefficients Effective friction coefficients 17 17 g(d) : equilibrium RDF at contact

18. 2013/6/27 Memory kernels for Density-Density correlation function Φ for Density-Density correlation function Φ Dissipative kernels are negative Dissipative kernels are negative ⇒ They can cancel the glassy kernel 18 18 glassy kernel dissipative

19. 2013/6/27 Disappearance of the plateau Parametrical study (varying e ) Parametrical study (varying e ) Glassy plateau of the density-density correlation function disappears due to dissipative memory kernels. Glassy plateau of the density-density correlation function disappears due to dissipative memory kernels. Plateau appears complementarily in the density-current correlation function. Plateau appears complementarily in the density-current correlation function. 19 19 Density-Density correlation = 0.521 = 1.0e-2 T = 1.0 qd = 7.0 conditions Density-Current correlation

20. 2013/6/27 Disappearance of the plateau 20 20 Schematic picture Schematic picture The red particle loses its kinetic energy due to inelastic collisions ⇒ density correlations disappear. Eventually the cage is destructed by the shear.

21. 2013/6/27 Scheme of the analysis Temperature is not well predicted neither for MCT nor kinetic theory ⇒ adopt the result of the simulation for T ( energy balance equation is not solved ). Temperature is not well predicted neither for MCT nor kinetic theory ⇒ adopt the result of the simulation for T ( energy balance equation is not solved ). Compare the values of and separately for given sets of ( T,, e ); Compare the values of and separately for given sets of ( T,, e ); is chosen as the unit of time is chosen as the unit of time 5 conditions for (0.55, 0.57, 0.58, 0.59, 0.60) 5 conditions for (0.55, 0.57, 0.58, 0.59, 0.60) 3 conditions for e (0.98, 0.92, 0.70) 3 conditions for e (0.98, 0.92, 0.70) 21 21

22. 2013/6/27 Inputs Static structure factor (equilibrium) Static structure factor (equilibrium) Fact : MCT = 0.516 corresponds to g ～ 0.6 Fact : MCT = 0.516 corresponds to g ～ 0.6 ⇒ we adopt the following *, * = – g + MCT, g =0.595, MCT = 0.516. Radial distribution function (equilibrium, at contact) Radial distribution function (equilibrium, at contact) 22 22 Interpolation formula (Torquato) ν: volume fraction

23. 2013/6/27 Result : shear stress e=0.98/0.92/0.70 e=0.98/0.92/0.70 a Kinetic theory under-(over-)estimates for e=0.98(0.70) Kinetic theory under-(over-)estimates for e=0.98(0.70) MCT works well for e=0.98, 0.92, 0.70 MCT works well for e=0.98, 0.92, 0.70 23 23 Why does MCT work well ? ⇒ time-correlation function

24. 2013/6/27 Result : time-correlation function e=0.98 e=0.98 Dissipation is weak; glassy plateau appears in the density-density correlation. The plateau is crucial for the evaluation of the stress. The plateau is crucial for the evaluation of the stress. 24 24 Density-Density correlationDensity-Current correlation ・ ・ ・ qd = 7.0

25. 2013/6/27 Result : time-correlation function e=0.92 e=0.92 No clear plateau appears in the density-density correlation. 25 25 Density-Density correlationDensity-Current correlation ・ ・ ・ qd = 7.0

26. 2013/6/27 Result : time-correlation function e=0.70 e=0.70 Dependence on density decreases as e ↓ Dependence on density decreases as e ↓ The origin of the density dependence resides in the memory kernel, which dissapears as e ↓ The origin of the density dependence resides in the memory kernel, which dissapears as e ↓ 26 26 Density-Density correlationDensity-Current correlation ・ ・ ・ qd = 7.0

27. 2013/6/27 Significance of the plateau e=0.98 e=0.98 The result of the kinetic theory is close to the MCT without the memory kernel. The result of the kinetic theory is close to the MCT without the memory kernel. Precise evaluation of the relaxation of the time- correlation function is crucial for the shear stress. Precise evaluation of the relaxation of the time- correlation function is crucial for the shear stress. 27 27 ・ Density-Density correlation qd = 7.0 Shear stress

28. e=0.70 e=0.70 The result of MCT is coincident to MCT without memory kernels. The result of MCT is coincident to MCT without memory kernels. 2013/6/27 Significance of the plateau 28 28 ・ Density-Density correlation qd = 7.0 Shear stress

29. 2013/6/27 Summary of the results MCT shows better compatibility with the simulation than the kinetic theory for the shear stress. MCT shows better compatibility with the simulation than the kinetic theory for the shear stress. Precise evaluation of the relaxation time of the time- correlation function is crucial. Precise evaluation of the relaxation time of the time- correlation function is crucial. The relaxation of the time-correlation function is determined by the dissipative memory kernel. The relaxation of the time-correlation function is determined by the dissipative memory kernel. However, MCT fails to explain the energy dissipation rate ⇒ we will retry ! However, MCT fails to explain the energy dissipation rate ⇒ we will retry ! 29 29

30. 2013/6/27 Discussions Nonlinear rheology Nonlinear rheology In MCT, deviation from the Bagnold scaling is not observed. In MCT, deviation from the Bagnold scaling is not observed. This is due to the hard-core limit. This is due to the hard-core limit. Proper treatment of the soft spheres is necessary. Proper treatment of the soft spheres is necessary. 30 30 [Hatano, Otsuki, Sasa, JPSJ 76, 023001 (2007)]

31. 2013/6/27 Discussions Jamming transition Jamming transition Precise evaluation of the vibrational frequency of the contact networks (instead of the collision frequency) is necessary. Precise evaluation of the vibrational frequency of the contact networks (instead of the collision frequency) is necessary. The replica theory might be of help for this issue. The replica theory might be of help for this issue. 31 31

32. 32 32 Appendix 2013/6/27

33. 2013/6/27 Hard-core limit Relation of ζ and e Relation of ζ and e Collision frequency Collision frequency It is difficult to derive in MCT. It is difficult to derive in MCT. We adopt its expression for the kinetic theory. We adopt its expression for the kinetic theory. 33 33 (contact duration time) k : linear spring coefficient(1) hard-core limit effective collision frequency (1) [Otsuki, Hayakawa, Luding, PTP Suppl. 184, 110 (2010)]

34. 2013/6/27 Hard-core limit Collision frequency (kinetic theory) Collision frequency (kinetic theory) 34 34 g(φ, d) : equilibrium RDF at contact The step function is formally replaced by the delta function. viscous coefficient (hard-core limit)

35. 2013/6/27 Interpretation of the granular temperature Normalization of current projection Normalization of current projection Requirement 1 : T should be treated as constant Requirement 1 : T should be treated as constant T is not included as a basis of the projected space T is not included as a basis of the projected space Requirement 2 : T should satisfy the energy balance eq. Requirement 2 : T should satisfy the energy balance eq. 35 35 T : granular temperature We simply assume T = T SS in this work.

36. 2013/6/27 Projection operator formalism Basis of the projected space Basis of the projected space Projection operators Projection operators Mori equation Mori equation Mode-Coupling Approximation Mode-Coupling Approximation 36 36 density fluctuation current density fluctuation specific for sheared granular systems : static structure factor

37. 2013/6/27 Time correlation functions Additional functions are required from Additional functions are required from Initial conditions Initial conditions 37 37 additional correlation functions for granular systems translational invariance in the sheared frame

38. 2013/6/27 Mori equation Exact equation for the time correlation functions Exact equation for the time correlation functions Coefficients, are equilibrium quantities Coefficients, are equilibrium quantities (n.b. T is exceptional) 38 38

39. 2013/6/27 Approximations Isotropic approximation Dissipation is almost isotropic For reducing the load of calculation （ 3D ⇒ 1D ） Weak-shear approximation Weak-shear approximation Second (and higher) order terms in and/or are neglected. e.g.,,, etc. Second (and higher) order terms in and/or are neglected. e.g.,,, etc. 39 39 e.g. reduces to two scalar functions,