Instabilities in Garnular Shear Flows · 2. Instabilities in granular shear flows 3. Aim & strategy...
Transcript of Instabilities in Garnular Shear Flows · 2. Instabilities in granular shear flows 3. Aim & strategy...
Kuniyasu Saitoh
Faculty of Engineering Technology,
University of Twente,
The Netherlands
*Instabilities in
Garnular Shear Flows
Contents I. Introduction
1. Instabilities in freely cooling state
2. Instabilities in granular shear flows
3. Aim & strategy
II. Molecular dynamics simulations
1. Setup & results
2. Kinetic theory of granular gases
III. Linear stability analysis
1. Basic equations & scaling units
2. Layering mode
3. Non-layering mode
IV. Weakly nonlinear analysis
1. 1-dimensional TDGL equation
2. Higher order amplitude equation
3. Bifurcation analysis
4. Hybrid approach & 2-dimensional TDGL equaiton
V. Discussion
1. Finite-size system
2. Hysteresis loop
VI. Conclusion
Instabilities in freely cooling state
Linear stability analysis
Hydrodynamic mode
kkkkk uuu| | , ,
Growth rate
)()()( kikk
Shear mode
Heat mode
Sound mode
0)( ),( kk
0)( ),( HH kk
0)( ,0)( SS kk
2
H 1 , ekk
2~ k
ku
| |
ku
ku
k
vortex
clustering
Instabilities in granular shear flows
Finite-size systems
Hydrodynamic limit
Linear stability analysis
Weakly nonlinear analysis
M. Alam & P. R. Nott, J. Fluid Mech. 377 (1998) 99
P. Shukla & M. Alam, Phys. Rev. Lett. 103 (2009) 068001
P. Shukla & M. Alam, J. Fluid Mech. 666 (2011) 204
Weakly nonlinear analysis (Ginzburg-Landau equation)
K. Saitoh & H. Hayakawa, Granular Matter 13 (2011) 697
Numerical solution of the Ginzburg-Landau equation
K. Saitoh & H. Hayakawa, AIP. Conf. Proc. 1501 (2012) 1001
K. Saitoh & H. Hayakawa, Phys. Fluids (2013) in press
Molecular dynamics simulation
K. Saitoh & H. Hayakawa, Phys. Rev. E 75 (2007) 021302
Strategy
Molecular dynamics
simulations
Observation
Kinetic theory
Linear & non-linear
analysis
Theory
Numerical
simulations of
amplitude eq.
Modeling
Comparison
Molecular dynamics simulations
Model
2-dimensional frictional granular particles
85.0e 2.0System size
5000N
Mean area fraction
ddLL 180180
12.00
Molecular dynamics simulations
Dense plug formation
Time development of the area fraction
(Saitoh & Hayakawa, 2007)
y
Bumpy wall
We also observed a similar plug formation of frictionless granular particles
under the Lees-Edwards boundary condition (Saitoh & Hayakawa, 2013)
a. Area fraction, velocity fields, and granular temperature .
b. Heat flux
c. Energy-sink term
d. Hydrostatic pressure & transport coefficients are the functions of .
Continuum equation
Equation of motion
Equation of energy
Granular hydrodynamic equations
Kinetic theory of granular gases
)( )( 1 2
2/3
1
2
jjue
Jenkins & Richman (1985)
Linear stability analysis
Granular hydrodynamic equations
Jenkins & Richman (1985)
2-dimensional frictionless disks
The Lees-Edwards boundary conditions
Scaling units
Mass
Length
Time
md
Udt 0
Shear rate 1
0
1
0 LdLU
: the ratio of particle’s diameter to gap
Hydrodynamic limit
1
Linear stability analysis
Homogeneous state
Finite temperature approximation
Hydrodynamic fields
)1(~0 O i.e. 221 e
),(),( 0 tt rr
T , , , wu
T000 ,0 , , y
T , , , wu
yxy
y
y
k
ti
t
kk
yik
k eAeAt rk
kr)(NL
)(
0
NLLL),(
Layering mode Non-layering mode
0 xk 0 xk
Kelvin mode xyx ktkkt , )( k
)1( ,0, , 22
00 ey uy : non-dimensionalized coordinate
Linear stability analysis
Linearized granular hydrodynamic equation
II )( tLdt
d
yxyxxyx kLtkkkLtkkkLtL 2
2
00 ,,)(
Layering mode
ykLtL ,0)( 0 : independent of time Matrix
Growth rate t
k ey
L
Eigenvalue problem
LL
0 ,0yy kkykL
0 xk
Perturbative calculations
0 LEigenvalue problem
Linear stability analysis
Wave vector qk y cf.) Clustering instabilities
, H
kk
2
2
10 MML
2
2
1
10
10~ ~ ~
Matrix
Eigenvalue
Right eigenvector
Left eigenvector
1*) We omit the superscript “L” and subscript “ky”.
2*) We have 4 eigenvalues and 4 eigenvectors.
Linear stability analysis
Dispersion relation
The eigenvalue with the maximum real part
4
4
2
221 ,0 qaqa
4
4
2
2
2)( qaqaq
42 2aaqc
The most unstable mode
2~ q
4~ q
Open circles
Numerical solution
Solid line
4
4
2
2)( qaqaq
Linear stability analysis
Eigenvectors (layering mode)
Open circles
Numerical solutions
Solid line
10)( q
1*) We also confirm good agreements of ~
Linear stability analysis
Eigenvectors (non-layering mode)
)(cos)()( 30
2
0
0NL
)( ttEJ
tEJ
pt
q
)(sin)()(1
1)(
)(13
21
2
NL
)( ttEt
tEt
tu t
q
)(sin)()(1
)()(1
13
21
2
NL
)( ttEt
ttE
tw t
q
)(cos)(2
)('
30
20NL
)( ttEJ
ptE
J
pt q
tetEtEtE2
~)( ),( ),( 321
NLNL )( tLdt
d
Weakly nonlinear analysis
Scaling of the eigenvalue and wave number
Long length & long time scales
2 qky
Note:
Long time scale
Long length scale
t 2
y
2
t
y
The most unstable solution
c.c. , LL
m c
c
iq
q eA
1*) Other modes can be suppressed.
t qyky
Weakly nonlinear analysis
Perturbative expansions
2
2
10 MML
2
2
1 AAA
Perturbative calculations
OO ,1 : absent
2O 0L
1 cqM 0~
1
L
1
L cc qq M
consistent
3O2L
1
L
13
L
1
2L
1
L
2
L
1
L || AANADAMAcc qq
2L
1
L
1
L
1
2L
12
L
1 || AAAdAA
3
LL
2
L
2
LLL ~ ,~ ,~ ,1~ NDdMcccccc qqqqqq
1D TDGL eq.
Weakly nonlinear analysis
Higher order calculations
4O
5O
3O2L
1
L
1
L
1
2L
12
L
1 || AAAdAA
L
2
2L
1
L
2
2L
1
L
2
2L
22
L
2 ||2 AAAAAdAA
4L
1
L
1
2L
2
L
1
L
3
2L
32
L
3 || AAAAAdAA
2
4LL2LLL2L
2
L |~
|~
|~
|~
~
~~
AAAAAdAA
L
3
2L
2
L
1
L ~
AAAA
Envelop function
Sum up
2
Weakly nonlinear analysis
Bifurcation analysis
Supercritical
Subcritical
|| A
super sub
0
Elastic & non-shear limit
i.e. Equilibrium gas 0
Weakly nonlinear analysis
Hybrid approach
22
2
2
12 || )( )( AAAdAdAdAA
2D TDGL eq.
New terms
cq )(q
)(q
c.c. ,, )(NL
)(
L
h zq
q
i
q eAc
Small deviation around the most unstable mode
)()( qqq c
The most unstable solution
c.c. )(L
m zq i
q eAc
c.c. )(NL
)(d zq
q
ieA
“Deviated” solution
Weakly nonlinear analysis
Time dependent diffusion coefficients
22
2
2
12 || )( )( AAAdAdAdAA
2D TDGL eq.
22
2 || AAAdAA 1D TDGL eq.
Zero
e~NL
)(q
Decay to zero
before 1)(1 d
)(2 d
Numerical solutions
Scheme
The 4th order Runge-Kutta method for the time-integration
The central difference method for the diffusion terms
Periodic boundary conditions in the sheared frame
Small perturbations with randomly chosen wave numbers
22
2
2
12 || )( )( AAAdAdAdAA
Numerical solutions
2D
TDGL
MD
(CG)
MD
Area fraction
Scaling function
Numerical solutions
Discussion
Finite-size systems
Hysteresis loop
0
|| A
sub
0
Does not exist! By definition
We couldn’t discuss hysteresis!
The ratio of a particle’s diameter to gap does not need to be small. Ld
2
2
01 e
The mean granular temperature
diverges in the elastic limit (e=1) Problem 1
is independent of and becomes another parameter.
What is a small parameter for the nonlinear analysis?
eProblem 2
The Fourier transformations could be questionable. Problem 3
Conclusion
Observation
Dense plug formations in 2-dimensional granular shear flows
are observed in both the bumpy & Lees-Edwards boundaries.
Theory
Granular hydrodynamic equations derived by the kinetic theory
can describe the dynamics of dense plug formations.
We perturbatively solved the linearized granular hydrodynamic
equations and found the hydrodynamic modes and eigenvalues.
From the weakly nonlinear analysis, we derived the 1D TDGL
equation and discussed the bifurcations of steady amplitude.
By taking the “hybrid” approach, we also introduced the 2D
TDGL equation with the time dependent diffusion coefficients.
Modeling
The 2D TDGL equation “qualitatively” describes the plug formations.