Numerical Analysis of Granular Jet Impacts Yukawa Institute for - - PowerPoint PPT Presentation

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Numerical Analysis of Granular Jet Impacts Yukawa Institute for Theoretical Physics Tomohiko Sano & Hisao Hayakawa 6/25 Physics of Granular Flows 2013 (YITP, Kyoto Univ.) T. G. Sano and H. Hayakawa, Phys. Rev. E 86, 041308 (2012). T. G.

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Numerical Analysis of Granular Jet Impacts

Yukawa Institute for Theoretical Physics Tomohiko Sano & Hisao Hayakawa

  • T. G. Sano and H. Hayakawa, Phys. Rev. E 86, 041308 (2012).
  • T. G. Sano and H. Hayakawa, Powders & Grains 2013 (in press), arXiv: 1211.3533
  • T. G. Sano and H. Hayakawa, arXiv:1302.6734

6/25 Physics of Granular Flows 2013 (YITP, Kyoto Univ.)

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Outline of my talk Introduction: “Impact Process” Model: Discrete Element Method (DEM) Rheology of Granular jets in 3D : Is granular flow “perfect fluid ?” Rheology of Granular jets in 2D : Jet-induced jamming Discussion & Summary

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Introduction: “Impact Process”

Inkjet

Wide Length Scale

Industrial Application & Natural Science

Crater formation

micro

  • H. Katsuragi,
  • Phys. Rev. Lett. 104, 218001 (2010)
  • H. Sirringhaus, et al.

Science, 290 (5499) 2123-2126 (2000).

Nuclei Reaction(heavy ion)

http://lhc.web.cern.ch/lhc/

Granular Jet Impact

“Macroscopic” Impact Process

Interest

Fluid state after the impact

macro

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SLIDE 4

Experimental movie from Chicago group

http:/ /nagelgroup.uchicago.edu/Nagel-Group/Granular.html

  • X. Cheng et al. Phys. Rev. Lett. 99,

188001 (2007)

Granular Jet Impact

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SLIDE 5

INTRODUCTION Perfect-fluidity in Granular Jet experiment

  • 1. From Experimental Study
  • 2. From Numerical Study in 2D
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SLIDE 6

INTRODUCTION

  • 1. From experiments :

An analogy between Granular Flow & Quark Gluon Plasma(QGP)

→Perfect-Fluid like response Elliptic Flow Perfect-Fluid like response?

  • X. Cheng et al. Phys. Rev.
  • Lett. 99, 188001 (2007)

Impact of a rectangular jet

Anisotropic flow

Nuclei Reaction

Au Au

Jet Jet in in

QGP →Small shear viscosity

Perfect-fluidity in Granular Jet experiment

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SLIDE 7

INTRODUCTION

  • 2. From Two-dimensional simulation :

A correspondence between Granular Flow & Perfect Fluid

Profile of the velocity & pressure

  • J. Ellowits et al. arXiv:

1201.5562

Perfect-fluidity in Granular Jet experiment

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SLIDE 8
  • 1. Experiment :Similarity between QGP and granular flow
  • 2. Numerical study in 2D: Ellowtiz, et al. arXiv:1201.5562

Similar profile of pressure and velocity between perfect fluid and granular flow

Perfect-fluidity in Granular Jet experiment

・Why granular flow looks like a perfect fluid? ・Response to an impact in general and rheology

  • f flows under an impact should be investigated.

But, granular flow cannot be a perfect fluid.

Dense granular flow

Experimental data of viscosity of granular flow

  • J. Fluid. Mech 400 199 (1999)

σαβ = Pδαβ η = 0 Perfect fluid should be large density →large viscosity

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SLIDE 9

Outline of my talk Introduction: “Impact Process” Model: Discrete Element Method (DEM) Rheology of Granular jets in 3D : Is granular flow “perfect fluid ?” Rheology of Granular jets in 2D : Jet-induced jamming Discussion & Summary

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Model: Discrete Element Method (DEM)

e = 0.75:Restitution Coefficient

: Coulombic const. of spheres

Wall model:

µp = 0.2

p

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SLIDE 11

Outline of my talk Introduction: “Impact Process” Model: Discrete Element Method (DEM) Rheology of Granular jets in 3D : Is granular flow “perfect fluid ?” Rheology of Granular jets in 2D : Jet-induced jamming Discussion & Summary

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SLIDE 12

Simulation movie

initial value Tg = 0 volume fraction granular temperature (= fluctuation of velocity) φ0/φfcc = 0.90

z

z = 0

Calculation Region

Rheology of Granular jets in 3D

Rtar

Target ..... Jet

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SLIDE 13

Rheology of Granular jets in 3D : Is granular flow “perfect fluid ?”

Small off-diagonal part of stress tensor

  • 0.1

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.4 0.6 0.8 1 1.2

stress

Profile of the stress tensor

σαβ ≡ σk

αβ + σc αβ

σk

αβ ≡ 1

V X

i

muiαuiβ σc

αβ ≡ 1

V X

i<j

F ij

α rij β

Large normal stress difference !! : σzz, σrr, σθθ → Origin of Perfect-fluidity σαβ = Pαδαβ ηDαβ ' Pαδαβ uiα ≡ viα − ¯ vα

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SLIDE 14

How about shear viscosity ?

Is granular flow “perfect fluid ?”

Shear viscosity: consistent with kinetic theory

Frictionless Frictional

5 10 15 20 25 30 35 0.1 0.2 0.3 0.4 0.5

5 10 15 20 25 30 35 0.1 0.2 0.3 0.4 0.5

→ Small strain rate →

Note.

η∗ = η∗(φ, e)

In general,

σrz = σY − ηDrz

However, we assume σY = 0 Deviation: the effect of the source point:

r ∼ 0

The kinetic theory is not valid here.

D∗

rz ≡

Drzd p Tg/m = O(0.01) ∼ 0.4

σαβ = Pαδαβ ηDαβ ' Pαδαβ σrz = −ηDrz η∗ ≡ η/η0 η0 ≡ 5 p mTg/π 16d2

Tg = 1 N X

i

mu2

i

3

Dαβ ≡ 1 2 ✓∂¯ vα ∂xβ + ∂¯ vβ ∂xα ◆

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SLIDE 15

Results for Rheology of Granular jets in 3D Granular flow cannot be a perfect fluid. Granular flow looks like a perfect fluid. Why?

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Results for Rheology of Granular jets in 3D

  • T. G. Sano and H. Hayakawa, Phys. Rev. E 86, 041308 (2012).
  • T. G. Sano and H. Hayakawa, Proceedings Powders & Grains 2013 (accepted), arXiv: 1211.3533

Shear viscosity: consistent with kinetic theory Large normal stress difference

σαβ 6= Pδαβ η 6= 0

Granular flow cannot be a perfect fluid. Granular flow looks like a perfect fluid. Why? Profile of the stress tensor

Shear stress looks very small in this setup.

σαβ = Pαδαβ ηDαβ ' Pαδαβ

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SLIDE 17

Outline of my talk Introduction: “Impact Process” Model: Discrete Element Method (DEM) Rheology of Granular jets in 3D : Is granular flow “perfect fluid ?” Rheology of Granular jets in 2D : Jet-induced jamming Discussion & Summary

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However....

Are the rheological properties in 2D granular jets qualitatively the same as those in 3D ?? The aim of 2D rheological studies: To clarify the qualitative difference between 2D and 3D granular jets Previous Granular Jet studies investigate 2D numerical studies to reproduce 3D experiments.

  • N. Guttenberg, Pys. Rev. E 85 051303 (2012).
  • J. Ellowitz, N. Guttenberg and W

. W . Zhang, arXiv:1201.5562 (2012).

  • J. Ellowitz, H. Turlier, N. Guttenberg, W

. W . Zhang, S. R. Nagel, arXiv:1304.4671 (2013).

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SLIDE 19

Rheology of Granular jets in 2D

Bi-disperse case mass , Frictionless grains

Visualization of contact forces

Simulation movie

d1/d2 = 0.8

m

(i) (ii)

Snapshot

Z ' 0.526

Coordination number : 71.5% of particles are NOT in contact.

Grains are well packed dense flow with contact-force network Jet-induced Jammed state

(a) layer (b) layer

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SLIDE 20

Rheology of Granular jets in 2D

The asymptotic divergence of the pressure : Frictionless case

Is ≡ Dxy p m/P

P mD2

xy

∼ (φs − φ)−αs φs = 0.834 ± 0.001 αs = 1.36 ± 0.05

P ≡ σxx + σyy 2

0.68 0.72 0.76 0.8 0.84 0.02 0.04 0.06 0.08 0.1 0.12

(a) (b)

αs

Jamming under shear

Hatano(2008) Otsuki & Hayakawa(2009) 4.0

2.7 1.0

Kinetic Theoretical regime Pd2 Tg ∼ P mD2

xy

∼ φg(φ) ∼ (φc − φ)−1

{

Critical φ

φJ = 0.8425 ' φs

Mean field picture of jamming

Exponent is smaller than those of the sheared granular systems, and are close to the extrapolation from the kinetic theoretical regime. Results

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The asymptotic divergence of shear stress

: Frictionless case

10 100 1000 10000 100000 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84

(a) (b)

1e+06

Hatano(2008) Otsuki et al.(2010) Garcia-Rojo et al. (2006) : Kinetic Theoretical

βs

Results

∝ mD2

xy(φs − φ)−(1−β/2)αs

(1 β/2)αs ' 0.96

−σxy = µ∗P

βs ' 0.96 The asymptotic divergence is similar to the extrapolation from kinetic theoretical regime.

  • B. Dapeng, et al. Nature 480, 355‒358

(15 December 2011)

Dxy Dxy

σxy

Shear stress:

−σxy ∼ mD2

xy(φs − φ)−βs

Jamming under shear

1.0

2.6 4.0

Origin of the difference between our case and systems under shear : (i) Our system cannot reach the true jamming transition (ii) Uncontrollability of shear rate → Bagnold’s scaling regime → We do nothing after the impact.

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SLIDE 22

Outline of my talk Introduction: “Impact Process” Model: Discrete Element Method (DEM) Rheology of Granular jets in 3D : Is granular flow “perfect fluid ?” Rheology of Granular jets in 2D : Jet-induced jamming Discussion & Summary

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SLIDE 23

Discussion Rheology of Granular jets in 3D

①

Rheology of Granular jets in 2D :Jet-induced jamming

②

Small: Geometrical constraint

Grains: consistent with kinetic theory

Shear stress looks small as a whole.

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

  • 1
  • 0.8 -0.6 -0.4 -0.2

0.2 0.4 0.6 0.8 1

max

3D

2D(a) 2D(b)

In 2D, grains are well packed, compared with those in 3D.

Note. Critical phenomena of jamming under shear do not depend on spatial dimensions.

Jet-induced jammed state

~ Response to an impact ~

Shear viscosity would be different if we use different particles.

σαβ = Pαδαβ ηDαβ ' Pαδαβ

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SLIDE 24

Summary

(ii) ：Large normal stress difference (i) ：Small shear stress

σαβ = Pαδαβ ηDαβ ' Pαδαβ

σαβ σαβ

Rheology of Granular jets in 3D : Is granular flow “perfect fluid ?”

①

Shear viscosity consistent with kinetic theory ＋ Small strain rate

Asymptotic divergence of pressure and shear stress

② Rheology of Granular jets in 2D :Jet-induced jamming

Dense flow with contact-force network Thank you !

→ Extrapolation from kinetic theoretical regime →Jet-induced “jammed” state

  • T. G. Sano and H. Hayakawa, Phys. Rev. E 86, 041308 (2012).
  • T. G. Sano and H. Hayakawa, Powders & Grains 2013 (in press), arXiv: 1211.3533
  • T. G. Sano and H. Hayakawa, arXiv:1302.6734
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SLIDE 25

Effect of friction of grains

0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 0.02 0.04 0.06 0.08 0.1 0.12

(a) (b)

µp = 1.0

Two branches on vs plane

φ Is

P/mD2

xy ∼ (φs − φ)−αs

large → decreases

φs

µp

0.81 0.815 0.82 0.825 0.83 0.835 0.84 0.845 0.2 0.4 0.6 0.8 1

(a) (b) ,

Bi-disperse vs Mono-disperse

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.02 0.04 0.06 0.08 0.1 0.12

(a) (b)

µ∗ ≡ −σxy/P

Effective friction const.

Two branches on vs plane

Is

µ∗

φL

:jamming density of sheared frictional grains

  • M. Otsuki & H. Hayakawa, Phys. Rev. E 83, 051301 (2011).

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.02 0.04 0.06 0.08 0.1 0.12 (a) (b)

d1/d2 = 0.8

d1/d2 = 1.0

Bi-disperse Mono-disperse

(i)

(ii)

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SLIDE 26

Jamming Transition

A granular system has rigidity above a critical value of density , and has no rigidity below .

φJ φJ

Grains

Tanaka Lab., Univ. Tokyo

Jamming of foams

  • M. Le Merrer, et al. Phys. Rev. Lett. 108 188301 (2012).

Jamming gripper

  • Univ. Chicago, Univ. Cornell,

iRobot and DARPA

Rigidity Divergence of Pressure and shear stress

Jamming of ... : Grains, Foams, etc... Characterization of Jamming

  • B. Dapeng, et al. Nature 480, 355‒358

(15 December 2011)

Jamming under shear Phenomenology of jamming Ex.) Kinetic theoretical divergence

g(φ) ∼ (φc − φ)−1

Radial distribution function Ex.) Mean field picture

  • M. Otsuki and H. Hayakawa
  • Prog. Theor. Phys. 121 647 (2009).

{