Granular flows under the shear

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Granular flows under the shear

• Hisao Hayakawa Kuniyasu Saitoh
• Dept. Phys. Kyoto Univ., JAPAN
• e-mail hisao_at_yuragi.jinkan.kyoto-u.ac.jp
• at Recent progress in glassy dynamics on
  September 29

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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 state
• Conclusion and discussion

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

• Strong fluctuations
• Concentration of stresses in small number of
  particles

• collapse of a silo by stress concentration!

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The jamming of granular particles

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

load
Inverse density
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II. Characteristics of granular fluid

• They have different properties from conventional
  flows
• Non-Newtonian constitutive equation
• Flow 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.

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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
  (Pouliquens talk)

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Purpose of this research

• To 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.

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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.

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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 theorem
• The existence of the long-range correlation (in
  freely cooling states)
• Homogeneous state cannot be maintained.

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III. Our system of the shear flow

• We 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 .
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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.

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The time evolution of area fraction ( )
Total energyKinetic (translational) Energy
Rotational Energy

• 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.

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Transient Dynamics
Granular temperature
X-component of velocity
Velocity (x-component)

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

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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.

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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.

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Theoretical treatment of the steady problem

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

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The equation of in the steady solution
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Details
radial distribution function
shear viscosity
thermal conductivity
Coefficient of density gradient in heat current

• From we can obtain the velocity and granular
  temperature.

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Steady state(without rotation)
Granular Temperature

• Agreement between the theory and the simulation
  is good.

Velocity (x-component)
We obtain the semi-quantitative results.
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Steady state(with rotation)
Granular Temperature

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

Velocity (x component)
The effects of rotation of particles can be
absorbed in the effective restitution constant.
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Discussion about high shear and elastic limit

• Kinetic Theory
• in strong shear stress
• in the elastic limit of e1
• No steady solution
• DEM We cannot reach a steady state
• (The energy increases with time.)

leads to break down of the steady solution.
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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.gt We may not need the
  liquid phase.

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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
  states
• 3-dimension The comparison with
  microgravity experiments.