kinetic theory of shear thickening for inertial suspensionsjam2018/file/hayakawa1.pdfkinetic theory...
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Kinetic theory of shear thickening for inertial suspensions
Hisao Hayakawa (YITP, Kyoto Univ.) collaboration with Satoshi Takada (Univ. Tokyo),
& Vicente Gárzo (Univ. Estremadura)
Refs: HH and S. Takada, arXiv:1611.07295, HH et al, PRE96, 042903 (2017).
Rheolgy of disordered particles- suspensions, glassy and granular particles, June 18th-29th, 2018 (talk is held on June 21st).
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Contents
• Introduction of shear thickening
• Kinetic model
• Kinetic theory of discontinuous shear thickening (DST) for dilute suspensions
• From DST to continuous shear thickening (CST) at finite density
• Application to binary mixtures
• Discussions
• Conclusions
June 21st, 208 Kinetic theory of shear thickening 2
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Contents
• Introduction of shear thickening
• Kinetic model
• Kinetic theory of discontinuous shear thickening (DST) for dilute suspensions
• From DST to continuous shear thickening (CST) at finite density
• Application to binary mixtures
• Discussions
• Conclusions
June 21st, 208 Kinetic theory of shear thickening 3
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Shear thickening
• Shear thickening is the increment of the viscosity
against shear rate 𝛾 .
• In particular, the discontinuous shear thickening (DST) has a drastic increase of the viscosity.
• Industrial applications to protective vest and traction control (break suspension)
June 21st, 208
http://youtube-video-download.info/video/EhdgkziFhrY
Kinetic theory of shear thickening 4
solvent (viscosity 𝜂0 )
𝛾 =𝜕𝑣𝑥/𝜕𝑦
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Discontinuous shear
thickening (DST)
DST in experiments • Normal stress difference
is important. C. D. Cwalina & N.J. Wagner, J. Rheol. 58, 949 (2014)
For DST in dense systems • Mutual friction to stabilize
percolaition is important M.Otsuki & H. Hayakawa, PRE 83, 051301 (2011)
R. Seto et al., PRL 111, 218301 (2013).
DST attracts much attention from physicists.
Seto et al., PRL 111, 218301 (2013).
June 21st, 208
𝛾 Kinetic theory of shear thickening 5
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Gas-solid suspensions
June 21st, 208
• Inertial suspensions • Homogenous state is relialized by the
balance between injecting gas flow & gravity.
• Relatively dilute system (theoretical treatment is available) without percolation picture
Ignited state
Quenched state
Tsao and Koch, JFM 296, 211 (1995)
analyzed dilute gas-solid suspensions without thermal noise. • Quenched-Ignited transition
(DST-like transition for temperature but not for viscosity)
Previous (theoretical) study
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Contents
• Introduction of shear thickening
• Kinetic model
• Kinetic theory of discontinuous shear thickening (DST) for dilute suspensions
• From DST to continuous shear thickening (CST) at finite density
• Application to binary mixtures
• Discussions
• Conclusions
June 21st, 208 Kinetic theory of shear thickening 7
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Langevin model
• Suspensions are influenced by collisions and dag as well as thermal activation:
• where the noise satisfies
• Here, we have introduced the peculiar momentum
June 21st, 208
= 𝑚 𝑽 𝑖
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Assumptions behind the Langevin model
• We assume that the drag force from the fluid is proportional to the fluid velocity. – Stokesian flow is assumed for relatively small
grains.
• The drag coefficient is only determined by the average density of suspensions. – This is a mean field model 𝜁 = 𝜁0𝑅(𝜑).
• Noise plays an important role for low shear regime to recover a thermal equilibrium state.
June 21st, 208 Kinetic theory of shear thickening 9
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Simulation of Langevin equation
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Boltzmann-Enskog equation
• Langevin model is equivalent to the kinetic equation (moderately dense case)
• We adopt Boltzmann-Enskog equation
June 21st, 208
friction from back ground
collision integral shear
Decoupling (Enskog equation) The coefficient of restitution
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Contents
• Introduction of shear thickening
• Kinetic model
• Kinetic theory of discontinuous shear thickening (DST) for dilute suspensions
• From DST to continuous shear thickening (CST) at finite density
• Application to binary mixtures
• Discussions
• Conclusions
June 21st, 208 Kinetic theory of shear thickening 12
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Stress equation for dilute suspensions
June 21st, 208
To get a closure we adopt Grad’s 13 moment method
Kinetic theory of shear thickening 13
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A set of dynamic equations: dilute gases
June 21st, 208
Kinetic temperature
Anisotropy of the temperature
𝑃𝑘𝑥𝑦
=𝑃𝑥𝑦
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Some relations in the steady state (dilute case)
• Temperature ratio
• Stress
• Shear rate
• Viscosity
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(normal stress difference)
ex ex Kinetic theory of shear thickening 15
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DST in dilute suspensions
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Flow curves for various restitution constant =>Perfect agreement between the theory & simulation
The lines are the analytic expressions of steady equations and the data are obtained from the Langevin simulation.
𝛾 ∗ = 𝛾 /𝜁0
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High and low shear limits
(e=1)
(e<1)
June 21st, 208 Kinetic theory of shear thickening 17
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Normal stress difference, and shear ratio
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𝜇 = −𝑃𝑥𝑦/𝑃 ∆𝜃 = ∆𝑇/𝑇𝑒𝑥
Kinetic theory of shear thickening 18
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Contents
• Introduction of shear thickening
• Kinetic model
• Kinetic theory of discontinuous shear thickening (DST) for dilute suspensions
• From DST to continuous shear thickening (CST) at finite density
• Application to binary mixtures
• Discussions
• Conclusions
June 21st, 208 Kinetic theory of shear thickening 19
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Extension to finite density
• We can apply Grad’s method to moderately dense suspensions.
• The collisional stress involves an additional contribution.
– It is known that Enskog theory gives precise results for 𝜑 < 0.5.
• The treatment of the shear rate in the collisional stress is difficult.
=> This does not appear in Chapman-Enskog theory.
June 21st, 208 Kinetic theory of shear thickening 20
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Stress equation for moderately dense gas
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Stress equations under Grad’s approximation
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𝑃𝑥𝑦 = 𝑃𝑥𝑦𝑘 + 𝑃𝑥𝑦
𝑐 Kinetic theory of shear thickening 22
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From DST to CST
• DST becomes CST if the density becomes finite.
• We plot the line of 𝜕𝛾/ 𝜕𝜃 = 0, where the red one is 𝜕2𝛾/ 𝜕𝜃 2>0.
• It is analogous to the first order transition.
June 21st, 208 𝝋 = 𝟎. 𝟎𝟓
Kinetic theory of shear thickening 23
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Temperature & Viscosity ⇒ good agreement
𝜑 = 0.30
Moderately dense cases
June 21st, 208
𝜑 = 0.50
Temperature ⇒ good Viscosity ⇒ fare Kinetic theory of shear thickening 24
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Stress ratio
June 21st, 208 Kinetic theory of shear thickening 25
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Normal stress differences
June 21st, 208 Kinetic theory of shear thickening 26
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Contents
• Introduction of shear thickening
• Kinetic model
• Kinetic theory of discontinuous shear thickening (DST) for dilute suspensions
• From DST to continuous shear thickening (CST) at finite density
• Application to binary mixtures
• Discussions
• Conclusions
June 21st, 208 Kinetic theory of shear thickening 27
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Motivation for study of binary mixtures
• To avoid crystallization for dense systems
• To discuss shear thinning for low shear regime
• To clarify the role of mutual diffusion between two spieces
• To extend the simple kinetic theory for mono-disperse systems to the complicated(?) kinetic theory for binary mixtures
June 21st, 208 Kinetic theory of shear thickening 28
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Some examples of simulation for moderately dense systems
• Bidisperse system • Number of particles:
𝑁 = 𝑁1 + 𝑁2, 𝑁1 = 𝑁2 = 500 • Diameter:
𝑑1: 𝑑2 = 1: 1.4, 𝑑1 = 𝑑, 𝑑2 = 1.4𝑑 • Restitution coefficient:
𝑒11 = 𝑒22 = 𝑒12 ≡ 𝑒
The notations are same as those in our previous paper. (All quantities are nondimensionalized in terms of 𝑚, 𝑑, 𝜁.)
June 21st, 208 Kinetic theory of shear thickening 29
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Simulation movie (𝝋 = 𝟎. 𝟒, 𝒆 = 𝟎. 𝟗)
Simulation works! June 21st, 208 30
Small particles: gray Large particles: blue Kinetic theory of shear thickening
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Results: moderately dense case 𝜑 = 0.40
Shear thinning appears for 𝑒 = 1! But plateau for 𝑒 < 1.
𝛾 ∗
𝜂∗
𝛾 ∗
𝜃
Temperature ratio Shear viscosity
June 21st, 208 Kinetic theory of shear thickening 31
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Some simulation for dilute cases
• Bidisperse system • Number of particles:
𝑁 = 𝑁1 +𝑁2, 𝑁1 = 𝑁2 = 500 • Diameter: 1:1.1, 1:1.4, 1:2, and 1:3
ex.) 𝑑1: 𝑑2 = 1: 1.4 ⟹ 𝑑1 = 𝑑, 𝑑2 = 1.4𝑑
mass ratio: 𝑚2
𝑚1=
𝑑2𝑑1
3
• Restitution coefficient: 𝑒11 = 𝑒22 = 𝑒12 ≡ 𝑒 = 0.9
• System size: 𝐿 = 39.4𝑑 𝑑2 = 1.1𝑑 , 𝐿 = 46.1𝑑 𝑑2 = 1.4𝑑 , 𝐿 = 61.8𝑑 𝑑2 = 2𝑑 , 𝐿 = 90.2𝑑 (𝑑2 = 3𝑑).
June 21st, 208
• Packing fraction
𝜑 =𝑁1
𝜋6 𝑑1
3 +𝑁2𝜋6 𝑑2
3
𝐿3= 0.01
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Dimensionless parameters
• All quantities are nondimensionalized by mass 𝑚 = 𝑚1, diameter 𝑑 = 𝑑1, and drag coefficient 𝜁
𝑇ex = 0.01𝑚𝑑2𝜁2
Shear rate: 𝛾 ∗ =𝛾
𝜁
Temperature: 𝜃𝑖 =𝑇𝑖
𝑇ex
Stress tensor: 𝑃𝑘∗ =𝑃
𝑛𝑇ex
Viscosity: 𝜂∗ =𝜂𝜁
𝑛𝑇ex
June 21st, 208 33
Same as H. Hayakawa and S. Takada, arXiv:1611.07295 and H. Hayakawa, S. Takada, and V. Garzó, Phys. Rev. E 96, 042903 (2017)
Kinetic theory of shear thickening
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Partial temperature ratio Partial temperature ratio 𝑇2/𝑇1
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Low shear limit: 𝑇2/𝑇1 → 1 High shear limit:
𝑇2/𝑇1 → 1.1 𝑑2/𝑑1 = 1.4 , 1.3 (𝑑2/𝑑1 = 2.0)
Existence of peak
Kinetic theory of shear thickening
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Current status of theory
• We are focusing on dilute situations.
• We have precisely evaluated collision integrals.
• The theoretical treatment for the temperature ratio seems to work for not-large size ratios, but has some troubles for large size ratios (V. Garzo).
• Flow curves are insensitive to the size dispersion.
• Perhaps, we will report the results in details in near future.
June 21st, 208 Kinetic theory of shear thickening 35
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Contents
• Introduction of shear thickening
• Kinetic model
• Kinetic theory of discontinuous shear thickening (DST) for dilute suspensions
• From DST to continuous shear thickening (CST) at finite density
• Application to binary mixtures
• Discussions
• Conclusions
June 21st, 208 Kinetic theory of shear thickening 36
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Effect of hydrodynamic interactions; shear thickening model
• The hydrodynamic interactions play crucial roles in moderately dense suspensions.
• It is difficult to analyze the serious model of hydrodynamically interacting suspensions.
• Nevertheless, it is easy to implement the hydrodynamic effect as a mean field by changing the drag coefficient.
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Effect of mutual friction in dense systems
• The mutual friction plays an important role for dense systems.
• Kuniyasu Saitoh & HH are developing the kinetic theory of dense frictional dry granular systems.
• Listen to Kuni’s talk.
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Conceptual phase diagram
June 21st, 208
density
friction
DST CST
CST
DST
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Shear jammed state and DST
• DST for high density system is related to the shear jammed (SJ) state. – SJ may be a protocol
dependent state for frictional grains.
• (Right) Bertland et al, PRL 114, 098301 (2015) but it is under debate.
• This is a current hot subject in this field.
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Contents
• Introduction of shear thickening
• Kinetic model
• Kinetic theory of discontinuous shear thickening (DST) for dilute suspensions
• From DST to continuous shear thickening (CST) at finite density
• Discussions
• Conclusions
June 21st, 208 Kinetic theory of shear thickening 41
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Conclusions for shear thickening
• We develop a simple kinetic theory for shear thickening.
• Discontinuous shear thickening (DST) can be reproduced for arbitrary 𝑇𝑒𝑥 and dilute situations.
• Normal stress difference is important. • All of the theoretical results perfectly agree with
simulation results for dilute cases. • We also develop Enskog theory for moderately
dense suspensions and find the existence a transition from DST to CST.
• See Hayakawa et al, PRE 96, 042903(2017).
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• Thank you for your attention.
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Configurations
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