201 7 /03/0 9 1 5 : 0 0 – 1 5 :3 0 Koshiro Suzuki (Canon Inc.)

1. 2017/03/09 15:00 – 15:30 Koshiro Suzuki (Canon Inc.) in collaboration with Hisao Hayakawa (YITP) Non-equilibrium theory of rheology for non-Brownian dense suspensions 2017/03/09 Non-Gaussian fluctuation and rheology in jammed matter

2. Contents • Introduction • Outline of the theory • Microscopic model • Equation of continuity • Steady-state averages • Pressure, shear stress, normal stress differences • MD simulation • Discussions • Summary

3. Contents • Introduction • Outline of the theory • Microscopic model • Equation of continuity • Steady-state averages • Pressure, shear stress, normal stress differences • MD simulation • Discussions • Summary

4. Introduction – shear viscosity volume V suspending liquid (viscosity 　) : empirical • Shear viscosity • Dilute region • Einstein (1905) • Batchelor, Green (1972) • Dense region • Chong et al. (1971), Quemada (1977) 1

5. Introduction – pressure viscosity (Deboeuf et al., 2009) • Pressure viscosity • Morris, Boulay (1999) • Zarraga et al. (2000) • These models are all empirical 2

6. Introduction - recent results (Boyer et al., 2011) • Pressure-controlled experiment 3

7. Introduction - theoretical approach (Brady, 1993) (Smoluchowski eq.) • Brownian suspensions • Thermal noise & diffusion • Contribution of Brownian & particle-contact stresses 5

8. Introduction - summary • Study of the rheology of suspensions • Well studied since Einstein • Established by experiments and simulations • Theoretical framework for dense suspensions • Shear viscosity: only for Brownian suspensions • Pressure: absent • Aim of this work • Construct a unified theoretical framework for the pressure and viscosity of dense non-Brownian suspensions 6

9. Contents • Introduction • Outline of the theory • Microscopic model • Equation of continuity • Steady-state averages • Pressure, shear stress, normal stress differences • MD simulation • Discussions • Summary

10. Microscopic model mass y m diameter d volume V suspension (viscosity ) x • Equation of motion (Lees-Edwards b.c.) • No thermal noises (non-Brownian) • No fluid interactions (e.g. lubrication forces) • Overdamped approximation 7

11. Equation of continuity • Stress tensor 8

12. Equation of continuity • Averaged stress 9

13. Steady-state averages ss ss =0 ss ss • Averaged stress • Inter-particle force and steady-state distribution is necessary 9

14. Steady-state averages y x • Inter-particle force 10

15. Steady-state averages (Santos et al., 2004) : peculiar velocity : pressure (ideal gas) : kinetic stress • Steady-state distribution function • Grad’s expansion • Kinetic theory of gases • Extension to dense suspensions ? 18

16. Steady-state averages : contact stress • Steady-state distribution function • T : temperature of the solvent 11

17. Pressure, shear stress, normal stress differences • Coupled equations • Pressure P • Shear stress σxy • Normal stresses σxx , σyy 12

18. Pressure, shear stress, normal stress difference • Final result 13

19. Contents • Introduction • Outline of the theory • Microscopic model • Equation of continuity • Steady-state averages • Pressure, shear stress, normal stress differences • MD simulation • Discussions • Summary

20. MD simulation sampling time tm start sampling initial configuration equilibrate sample1 sample2 sample3 sampleS • Event-driven MD (hard spheres) • Uniform shear (Lees-Edwards b.c.) • Parameters • N = 1000, samples S = 100 • Sampling time tm = 1000 collisions • Procedure 2015/09/09 14

21. Pressure & shear viscosities • Density dependence • Both exhibit ~ δφ-2 • Slight difference→ stress ratio 16

22. Stress ratio • Approaches 0.1-0.2 as φ→φJ 17

23. Contents • Introduction • Outline of the theory • Microscopic model • Equation of continuity • Steady-state averages • Pressure, shear stress, normal stress differences • MD simulation • Discussions • Summary

24. Grad’s expansion (Santos et al., 2004) : peculiar velocity : pressure (ideal gas) : kinetic stress (collisional stress) : contact stress • Original • Extension 18

25. Diffusivity (Brady, 1993) • Brownian suspensions • Non-Brownian suspensions • Divergence is not related to the diffusion constant 19

26. Contents • Introduction • Outline of the theory • Microscopic model • Equation of continuity • Steady-state averages • Pressure, shear stress, normal stress differences • MD simulation • Discussions • Summary

27. Summary Theory of rheology for dense non-Brownian suspensions is proposed. It relies on the extension of the Grad’s expansion. It successfully describes the correct divergence for the pressure, shear stress, and the normal stress differences. The distinction between Brownian and non-Brownian suspensions is shown. Quantitative validity of the theory is now under investigation. 24