Post on 04-Jul-2020
Shock Propagation in Loosely Packed Granular Media
Sudhir N. Pathak(Institute of Mathematical Sciences, Chennai)Zahera Jabeen (Univ. Michigan, USA)
Purusattam Ray (Institute of Mathematical Sciences, Chennai)R. Rajesh (Institute of Mathematical Sciences, Chennai)
Outline of the talk• The problem
★ Nuclear Explosion
★ Granular Explosion
• Motivation
• Analysis
• Comparison with experiments
• Modified models
• Conclusions
Nuclear Explosion
How does the radius increase with time?
Dimensional Analysis
[E0] = ML2T!2
[!] = ML!d
[t] = T
d = 3 =! R(t) " t2/5
R(t) = c
!E0t2
!
" 1d+2
R(t) = f(E0, t, !, T0)
Comparison with data
A computer model
• Particles at rest
• One particle given an impulse
• Interaction only on contact★ Energy conserving
★ Momentum conserving
A computer model
Radius vs timeR(t) = c
!E0t2
!
" 1d+2
1
2
3
4
5
6
7
2 3 4 5 6 7 8 9 10 11
log[
R(t)]
log(t)
t1/2
simulation
1
2
3
4
5
2 3 4 5 6 7 8 9 10
log[
R(t)]
log(t)
t2/5
simulation
2 dimensions 3 dimensions
Question
Take the above model and make the collisions inelastic.
Do the results change?
Granular systemsSand, steel balls, talcum powder
Size ! 1µm to 1mm
Mass ! 1 mg
Velocity ! 1cm/s
KE
kT=
10!610!4
kT! 1010
PE
kT=
10!61010!2
kT! 1013
Temperature plays no role
Different Cases
Jaeger et al, 1996
Blair et al, 2003 Goldhirsch et al, 1993
Aranson et al, 2006
Key ingredient
u
u t
un
Collisions are inelastic
vt = ut
vn = !run
r < 1
Boudet et al, PRL 2009
Experiments
Experiments
Walsh et al, PRL 2003
Crater formation
Experiments
Cheng et al, Nature Phys, 2008
Freely cooling granular gas
• Give initial energy to particles
• Isolate system
• Energy loss through collisions
• Why study?
★ Isolates effects of inelastic collisions
★ Direct experiments
★ As parts of larger driven systems
★ Interacting particle systems
Homogeneous Cooling
dE
dt! (1" r2)E
a/#
E
E ! 1(1" r2)t2 + c2
Haff’s law Haff, 1982
dE
dt= !!E
!
Assumption: particles are homogeneously distributed
Clustering
Clustering
• Breakdown of Haff ’s law (kinetic theory)
• New regime: inhomogeneous clustered regime
Experiments
• friction
• boundary effects
Experiments
Maaβ et al, 2008
Levitation
ExperimentsMicrogravity
Tatsumi et al, 2009
Boudet et al, PRL 2009
Clustering
A constant coefficient of restitution?
Brass Aluminium
Marble C. V. Raman, 1918
A constant coefficient of restitution?
Brass Aluminium
Marble
r ! 1 when v ! 0
r ! r0 when v !"
C. V. Raman, 1918
Coefficient of Restitution
r
v
1
δ
r ! 1 when v ! 0
r ! r0 when v !"
Do we need δ?
103
104
105
106
107
108
0 100 200 300 400 500 600 700 800 900 1000
# of
col
lisio
ns
t
!=0.00!=0.10
Inelastic collapse
Computer simulation
Elastic vs Inelastic
Scaling analysis
Let Rt ! t!
Length scales
R1
R2
δR
Do all these lengths scale as tα ?
Length scales
10-4
10-3
10-2
10-1
100
0 5000 10000 15000 20000 25000
A(t)
t
r=0.1r=0.5r=0.8r=1.0
R1
R2
A =!
!1 ! !2
!1 + !2
"2
!!R2
x" !RxRy"!RxRy" !R2
y"
"
-1
0
1
2
3
4
5
6
7
2 3 4 5 6 7 8 9 10 11 12
log[!(
t)]
log(t)
r=0.1; 3-d
!1!2!3
t1/2
Length scales
δRP (R, t) = f(R, !R, t)
P (R, t) = f(R, t)
P (R, t) =1R
g
!R
t!
"
Prob dist (2-D)
0
0.01
0.02
0.03
0.04
0 20 40 60 80
P(R,
t) t!
R /t!
(a)
!=1/3
0
0.002
0.004
0.006
0.008
0.01
0 0.001 0.002 0.003
P(v,
t)/ t1-
!
v t1- !
(b)
1-!=2/3
Scaling analysis
vt =dRt
dt! t!!1
Let Rt ! t!
Et ! Ntv2t ! t!d+2!!2
Nt ! Rdt ! t!d
Scaling (elastic limit)
Et ! Ntv2t ! t!d+2!!2
But energy is a constant
!d + 2!! 2 = 0
! =2
d + 2
Scaling (inelastic limit)• clustering for all r < 1
• Particle direction remains constant
10-6
10-5
10-4
10-3
10-2
10-1
100 101 102 103 104 105 106
Ave
rage
scat
terin
g an
gle
time
! = 10"4
! = 10"5
! = 10"6
Radial Mometum• radial momentum is conserved
101
102
103
101 102 103 104 105 106
!Rad
ial m
omen
tum"
t
Elastic # = 10$2
# = 10$3
# = 10$4
# = 10$5
Scaling (inelastic limit)
vt =dRt
dt! t!!1
Let Rt ! t!
Et ! Ntv2t ! t!d+2!!2
Nt ! Rdt ! t!d
Ntvtd! = constant
!d + !! 1 = 0
! =1
d + 1
!el =2
d + 2
A calculation in one dimensiona av0
Special case: r=0 [sticky limit]
After m! 1 collisions, mass of particle is m
Momentum conservation =! vm!1 = v0m
tm = av0
+ av1
+ . . . + avm!1
= a!m
i=1i
v0! m2
m ! Nt ! Rt !"
t
! = 12 , d = 1 ! =
1d + 1
Recall
Simulation-2d
1 2 3 4 5 6 7 8 9
10 11
2 3 4 5 6 7 8 9 10 11 12
log[
N(t)
]
log(t)
Number - 2D
r=0.1r=0.5r=0.8
t2/3
-8-7-6-5-4-3-2-1 0 1
2 3 4 5 6 7 8 9 10 11 12
log[
E(t)]
log(t)
Energy - 2D
r=0.1r=0.5r=0.8
t-2/3
!N(t)" # t2/3 !E(t)" # t!2/3
Comparison with kinetic theory
100
101
102
103
100 101 102 103 104 105 106
R(t)
t
Eq. (6)SimulationSlope = 1/3
Simulation-3d
!E(t)" # t!3/4!N(t)" # t3/4
2
4
6
8
10
12
14
2 3 4 5 6 7 8 9 10 11 12
log[
N(t)
]
log(t)
Number - 3D
r=0.1r=0.5r=0.8
t3/4
-10-9-8-7-6-5-4-3-2-1
2 3 4 5 6 7 8 9 10 11 12
log[
E(t)]
log(t)
Energy - 3D
r=0.1r=0.5r=0.8
t-3/4
δ-dependence
10-9
10-8
10-7
10-6
101 102 103 104 105
E t
t
!=10-2!=10-3!=10-4!=10-5
Experiments
Boudet et al, PRL 2009
Data (Shock)
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
0 1 2 3 4 5
ln(R
/Rs)
ln(t/ts)
4mm8mm
16mm
Non-zero ambient temperature
Non-zero ambient temperature
101
102
103
101 102 103 104 105 106
R(t)
t
a b
c d
! = 1/1600! = 1/800! = 1/400
1
3
5
10-3 10-1 101
R(t)
t-1/3
t !3/2
!=1/1600!=1/800!=1/400
Model with escape rate
101
102
101 102 103 104 105 106 107
R(t)
t
v0 = 1v0 = 2v0 = 4v0 = 8
no hopping
t0.18
Experiment
Walsh et al, PRL 2003
Crater formation
Data (Crater)
100
101
101 102
R(t)
t
t0.18t0.19t0.24
Summary & Outlook
• A generalization of the Taylor-Sedov problem
• Inelastic ⇒ clustering and band formation
• Conservation of radial momentum
• New exponents independent of r
• Describes experimental data well
Summary & Outlook
• Understanding crossovers
• Can the freely cooling gas be understood?
• Related experiments
Cheng et al, Nature Phys, 2008