Granular flows under the shear

Hisao Hayakawa* & Kuniyasu SaitohDept. Phys. Kyoto Univ., JAPAN*e-mail : hisao@yuragi.jinkan.kyoto-u.ac.jp

at Recent progress in glassy dynamics on September 29

Outline of this talk

Introduction: What is granular material?Characteristics of granular flows: Is there a liquid phase?Simulation of granular flow

Metastable dynamics and the plug flow

The description based on the kinetic theory for the steady stateConclusion and discussion

Strong fluctuationsConcentration of stresses in small number of particles

collapse of a silo by stress concentration!

I. Introduction: What is granular material? – force chains under the shear-

The jamming of granular particles

It is known that there is an analogy between glass transition and the jamming.

load

Inverse density

II. Characteristics of granular fluid

They have different properties from conventional flowsNon-Newtonian constitutive equationFlow is heterogeneous and it strongly depends on boundary conditions.There are many cases which coexist both flow regions and glassy (solid-like) regions. Theoretical treatments are mainly based on the kinetic theory for gases 　　　　 There are not so many phenomena those can be explained by the theory.There are a lot of phenomenology but the range of applications is limited.

A fundamental question

Is there a liquid phase separately from the gas phase?

No definite answer

No: there is only the dense gas phase because no attractive interaction is included (my talk).Yes: the behavior of dense granular flow has common properties different from the dense gases (Pouliquen’s talk)

Purpose of this researchTo extract the essence of granular flows, we focus on the simple shear flow for relatively dilute granular gases without the influence of gravity.

We do not introduce any particular liquid phase.

We are interested in the relaxation dynamics and the steady state. We examine the validity of the kinetic theory in the heterogeneous system.We also investigate the effect of the tangential contact force and the rotation of particles in granular flows.

Previous Studies on Granular Gases

Flows on slopes (or inclined planes) 　　　　　There are many experiments and theories. 　The system is anisotropic under the influence of the gravity. 　

Freely Cooling Processes 　　　 There are many theories and simulations but no experiments.Most of simulations do not take into account the rotations of particles.

Simple Shear Flows (Couette flows) 　　　 Experiments are limited to high density case . Systems are strongly influenced by the gravity.Theories （ Jenkins, Alam etc ） are based on the kinetic theory and applied to dilute case.

Additional characteristics of granular gases

Absence of the standard Green-Kubo formula

The transport coefficient is given by the complicated correlation function

Absence of the fluctuation theoremThe existence of the long-range correlation (in freely cooling states)Homogeneous state cannot be maintained.

III. Our system of the shear flowWe apply the shear to a system of 2-dimensional granular gas.No. of particles = 5000, Average Area fraction = 0.12 Initial condition : The configuration is uniform and velocity distribution obeys Gaussian. 　　　　　　 　　　　　　　　　　

Shear speed : U . Shear rate : . ( 　　　 : diameter 、 　 : gravitational

acceleration, 　　　　　　　 )Bumpy boundary condition at sheared wall ( ) 　Periodic boundary condition at .

dg /1089.8 2gd0.8

d g

2/y2/x

d90U

Discrete element method ( DEM )

Elastic force Linear spring （　　　　　　　　　 ）Viscous force （ viscous constant ）Coulomb friction in the tangential force （　　　　 ）Contact force

　　　　　　　 represent the relative displacements in the normal and the tangential direction, respectively.The tangential force causes the rotation of particles.

dt

dk n

nnnn

uuF

n

tttt

t dt

dk

F

uuF

)|(| nt FF

)|(| nt FF

tunu

4100.5 k0.12

2.0

]/[ dg

])/[( 2/1dg

The time evolution of area fraction ( )

Normal （ Without rotation) 　　： One peak exists through the time evolution to form a band like cluster.Tangential （ with rotation ）

　　： The are two peaks in the transient dynamics.Steady states in both systems are similar.

Total energy=Kinetic (translational) Energy+ Rotational Energy

Transient Dynamics

Transient 　　 According to rotational effects of particles, velocity and granular temperature become almost zero in the central region.

X-component of velocityVelocity (x-component) Granular temperature

Suggestion from the simulation

Even when the average density is not high, there appear dense clusters.In the dense clusters, the motion of particles are frozen like a glassy state.

The coexistence of the dense region and the dilute region is a typical characteristics of granular flows.

However, as will be shown, it is surprised that we can use the kinetic theory.

IV. The steady solution of fluid equations

With the aid of fluid equations derived from the kinetic theory by Jenkins & Richman (1985) we have obtained the steady solution of Couette flow for the case without the rotation.We also obtain the steady solution for the case with the particles’ rotation based on the idea by Yoon & Jenkins (2005) .

　　　　　 The effects of rotations （ friction constant ） can be absorbed with the introduction of the effective restitution constant.

　　　　　　　　2)1(2 eeeeff

Theoretical treatment of the steady problem

We can derive a set of fluid equations based on the dense gas kinetic theory. (Enskog+dissipation)Equations include the conservations of the mass, the linear momentum and the energy.

TquPDn

TuT

Puuu

nun

t

t

t

):(2

1

0)(

The equation of in the steady solution

)]()1(1[2

)(

)()()1(2

1)()(2)(

)}(]))((

)1(1[2

)()(){()(

)(})({

2/32212/1

212/3

ge

geRG

d

gdeF

Gdy

dF

dy

d

Details

　 From 　　 we can obtain the velocity and granular temperature.

)(g )()( )(: radial distribution function

: thermal conductivity

: shear viscosity:Coefficient of density gradient

in heat current

2)(p

R

boundaryxyP |

boundaryyyPp |

Steady state （ without rotation ）

Agreement between the theory and the simulation is good. 　　　　　　　

We obtain the semi-quantitative results.

Velocity (x-component)

Granular Temperature

Steady state （ with rotation ）

The disagreement of the area fraction between the simulation and the theory is enlarged. But not bad!

Granular Temperature

Velocity (x component)

The effects of rotation of particles can be absorbed in the effective restitution constant.

Discussion about high shear and elastic limit

Kinetic Theory　　 in strong shear stress in the elastic limit of e=1

　 No steady solution

DEM We cannot reach a steady state

　（ The energy increases with time. ）

leads to break down of the steady solution.

0)( G

V. Summary

According to the rotation, we find that a characteristic behavior appears in the transient dynamics.

The motion of particles are frozen in the region between two dense clusters.

In the steady states, qualitative behaviors are common with regardless to the existence of the rotation of particles. The hydrodynamic variables in the steady state can be described by the kinetic theory.

This is astonishing because the motion of some particles are frozen.=> We may not need the liquid phase.

Perspective

We have to construct a theory to describe dynamics of particles in the metastable state. We need to improve the theory to describe high density region which may be correction to the kinetic theory. We need to investigate the effects of system size, because there is a paper to indicate that there are two dense clusters in the transient region if the system size is enough large. Extension to our theory

High shear case or elastic limit 　 Unsteady states3-dimension The comparison with microgravity experiments.