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

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. Sano and H. Hayakawa, Powders & Grains 2013 (in press), arXiv: 1211.3533 T. G. Sano and H. Hayakawa, arXiv:1302.6734

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

Introduction: “Impact Process” Phys. Rev. Lett. 104, 218001 (2010) Fluid state after the impact Interest “Macroscopic” Impact Process (2000). Science, 290 (5499) 2123-2126 Inkjet H. Sirringhaus, et al. H. Katsuragi, micro Crater formation Natural Science & Industrial Application Wide Length Scale macro Nuclei Reaction(heavy ion) Granular Jet Impact http://lhc.web.cern.ch/lhc/

X. Cheng et al. Phys. Rev. Lett. 99 , Granular Jet Impact 188001 (2007) Experimental movie from Chicago group http:/ /nagelgroup.uchicago.edu/Nagel-Group/Granular.html

INTRODUCTION Perfect-fluidity in Granular Jet experiment 1. From Experimental Study 2. From Numerical Study in 2D

INTRODUCTION Nuclei Reaction →Small shear viscosity QGP in in Jet Jet Anisotropic flow 1. From experiments : Impact of a rectangular jet Lett. 99, 188001 (2007) X. Cheng et al. Phys. Rev. Perfect-Fluid like response? Elliptic Flow →Perfect-Fluid like response An analogy between Granular Flow & Quark Gluon Plasma(QGP) Perfect-fluidity in Granular Jet experiment Au Au

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

1. Experiment :Similarity between QGP and granular flow 2. Numerical study in 2D: Ellowtiz, et al. arXiv:1201.5562 large density Perfect fluid should be J. Fluid. Mech 400 199 (1999) granular flow Experimental data of viscosity of Dense granular flow But, granular flow cannot be a perfect fluid. of flows under an impact should be investigated. ・Response to an impact in general and rheology ・Why granular flow looks like a perfect fluid? Perfect-fluidity in Granular Jet experiment perfect fluid and granular flow Similar profile of pressure and velocity between →large viscosity σ αβ = P δ αβ η = 0

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

Model: Discrete Element Method (DEM) : Coulombic const. of spheres Wall model: p e = 0 . 75 :Restitution Coefficient µ p = 0 . 2

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

Simulation movie initial value volume fraction granular temperature (= fluctuation of velocity) Calculation Region Rheology of Granular jets in 3D z = 0 Target z Jet R tar ..... φ 0 / φ fcc = 0 . 90 T g = 0

Small off-diagonal part of stress tensor → Origin of Perfect-fluidity Profile of the stress tensor Rheology of Granular jets in 3D : Is granular flow “perfect fluid ?” 0.7 σ αβ ≡ σ k αβ + σ c αβ 0.6 αβ ≡ 1 0.5 X σ k mu i α u i β stress V 0.4 i αβ ≡ 1 0.3 X α r ij σ c F ij β V 0.2 i<j 0.1 u i α ≡ v i α − ¯ v α 0 -0.1 0 0.2 0.4 0.6 0.8 1 1.2 Large normal stress difference !! : σ zz , σ rr , σ θθ σ αβ = P α δ αβ � η D αβ ' P α δ αβ

How about shear viscosity ? Note. Shear viscosity: consistent with kinetic theory In general, Is granular flow “perfect fluid ?” σ rz = − η D rz p mT g / π η 0 ≡ 5 16 d 2 35 35 η ∗ ≡ η / η 0 30 30 Frictionless m u 2 T g = 1 25 X i 20 25 N 3 i 15 ✓ ∂ ¯ ◆ + ∂ ¯ D αβ ≡ 1 v α v β 20 10 ∂ x β ∂ x α 2 5 15 0 0 0.1 0.2 0.3 0.4 0.5 σ rz = σ Y − η D rz 10 Frictional However, we assume σ Y = 0 5 0 Deviation: the e ff ect of 0 0.1 0.2 0.3 0.4 0.5 r ∼ 0 the source point: η ∗ = η ∗ ( φ , e ) The kinetic theory is not valid here. D rz d D ∗ = O (0 . 01) ∼ 0 . 4 rz ≡ p T g /m σ αβ = P α δ αβ � η D αβ ' P α δ αβ → Small strain rate →

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

Shear viscosity: consistent with kinetic theory Large normal stress difference 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. Results for Rheology of Granular jets in 3D σ αβ = P α δ αβ � η D αβ ' P α δ αβ η 6 = 0 σ αβ 6 = P δ αβ 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

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

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). 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

Rheology of Granular jets in 2D Coordination number : (a) layer Jet-induced Jammed state contact-force network dense flow with Grains are well packed 71.5% of particles are NOT in contact. (b) layer Bi-disperse case mass , Frictionless grains Snapshot Simulation movie Visualization of contact forces Z ' 0 . 526 d 1 /d 2 = 0 . 8 m (ii) (i)

kinetic theoretical regime. Rheology of Granular jets in 2D systems, and are close to the extrapolation from the Exponent is smaller than those of the sheared granular Mean field picture of jamming The asymptotic divergence of the pressure : Frictionless case 0.84 0.8 P ≡ σ xx + σ yy p I s ≡ D xy m/P 2 0.76 Results (a) (b) 0.72 P ∼ ( φ s − φ ) − α s 0.68 mD 2 φ s = 0 . 834 ± 0 . 001 xy 0 0.02 0.04 0.06 0.08 0.1 0.12 α s = 1 . 36 ± 0 . 05 Jamming under shear Hatano(2008) 2 . 7 { Otsuki & Hayakawa(2009) 4 . 0 Pd 2 P α s Kinetic Theoretical regime 1 . 0 ∼ φ g ( φ ) ∼ ( φ c − φ ) − 1 ∼ mD 2 T g Critical φ φ J = 0 . 8425 ' φ s xy

→ We do nothing after the impact. The asymptotic divergence is → Bagnold’s scaling regime (ii) Uncontrollability of shear rate jamming transition (i) Our system cannot reach the true case and systems under shear : Origin of the difference between our Shear stress: (15 December 2011) B. Dapeng, et al. Nature 480, 355‒358 from kinetic theoretical regime. similar to the extrapolation The asymptotic divergence of shear stress : Frictionless case 1e+06 (a) (b) σ xy ∝ mD 2 xy ( φ s − φ ) − (1 − β / 2) α s 100000 − σ xy = µ ∗ P (1 � β / 2) α s ' 0 . 96 10000 Results 1000 xy ( φ s − φ ) − β s − σ xy ∼ mD 2 100 β s ' 0 . 96 10 0.68 0.7 0.72 0.74 0.76 0.78 0.8 0.82 0.84 Jamming under shear β s Garcia-Rojo et al. (2006) 1 . 0 : Kinetic Theoretical Hatano(2008) 2 . 6 Otsuki et al.(2010) 4 . 0 D xy D xy

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

Discussion ① we use different particles. ② Small: Geometrical constraint Grains: consistent with kinetic theory Shear stress looks small as a whole. Shear viscosity would be different if ~ Response to an impact ~ Jet-induced jammed state not depend on spatial dimensions. Critical phenomena of jamming under shear do Note. compared with those in 3D. In 2D, grains are well packed, Rheology of Granular jets in 3D σ αβ = P α δ αβ � η D αβ ' P α δ αβ Rheology of Granular jets in 2D :Jet-induced jamming 1 0.9 max 0.8 0.7 0.6 0.5 3D 0.4 2D � (a) 2D � (b) 0.3 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1