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

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

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

4. Introduction – shear viscosity
Dilute region
Einstein (1905)
Batchelor, Green (1972)
Dense region
Chong et al. (1971), Quemada (1977)
volume V
suspending liquid
(viscosity 　)
: empirical
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5. Introduction – pressure viscosity
Morris, Boulay (1999)
Zarraga et al. (2000)
These models are all empirical
(Deboeuf et al., 2009)
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6. Introduction - recent results
Pressure-controlled experiment
(Boyer et al., 2011)
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7. Introduction - theoretical approach
Brownian suspensions
Thermal noise & diffusion
Contribution of Brownian & particle-contact stresses
(Brady, 1993)
(Smoluchowski eq.)
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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
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9. Contents Introduction Outline of the theory MD simulation Discussions
Microscopic model
Equation of continuity
Steady-state averages
Pressure, shear stress, normal stress differences
MD simulation
Discussions
Summary 9

10. Microscopic model Equation of motion (Lees-Edwards b.c.)
No thermal noises (non-Brownian)
No fluid interactions (e.g. lubrication forces)
Overdamped approximation
mass y m
diameter d
volume V
suspension
(viscosity ) x
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11. Equation of continuity
Stress tensor
8　　　 11

12. Equation of continuity
Averaged stress
9　　　 12

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

14. Steady-state averages
Inter-particle force y x
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15. Steady-state averages
Steady-state distribution function
Grad’s expansion
Kinetic theory of gases
Extension to dense suspensions ?
(Santos et al., 2004)
: peculiar velocity
: pressure (ideal gas)
: kinetic stress
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16. Steady-state averages
Steady-state distribution function
T : temperature of the solvent
: contact stress
11　　　 16

17. Pressure, shear stress, normal stress differences
Coupled equations
Pressure P
Shear stress σxy
Normal stresses σxx , σyy
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18. Pressure, shear stress, normal stress difference
Final result
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19. Contents Introduction Outline of the theory MD simulation Discussions
Microscopic model
Equation of continuity
Steady-state averages
Pressure, shear stress, normal stress differences
MD simulation
Discussions
Summary 19

20. MD simulation Event-driven MD (hard spheres)
Uniform shear (Lees-Edwards b.c.)
Parameters
N = 1000, samples S = 100
Sampling time tm = 1000 collisions
Procedure
sampling time tm
initial
configuration
start
sampling
equilibrate
sample1
sample2
sample3
sampleS
2015/09/09
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21. Pressure & shear viscosities
Density dependence
Both exhibit ~ δφ-2
Slight difference → stress ratio
16　　　 21

22. Stress ratio
Approaches as φ→φJ
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23. Contents Introduction Outline of the theory MD simulation Discussions
Microscopic model
Equation of continuity
Steady-state averages
Pressure, shear stress, normal stress differences
MD simulation
Discussions
Summary 23

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

25. Diffusivity Brownian suspensions Non-Brownian suspensions
Divergence is not related to the diffusion constant
(Brady, 1993)
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26. Contents Introduction Outline of the theory MD simulation Discussions
Microscopic model
Equation of continuity
Steady-state averages
Pressure, shear stress, normal stress differences
MD simulation
Discussions
Summary 26

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.
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