shock propagation in loosely packed granular mediafracmeet/ffpm2012/rajesh_porous.pdf · r. rajesh...
TRANSCRIPT
![Page 1: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/1.jpg)
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)
![Page 2: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/2.jpg)
Outline of the talk• The problem
★ Nuclear Explosion
★ Granular Explosion
• Motivation
• Analysis
• Comparison with experiments
• Modified models
• Conclusions
![Page 3: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/3.jpg)
Nuclear Explosion
How does the radius increase with time?
![Page 4: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/4.jpg)
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)
![Page 5: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/5.jpg)
Comparison with data
![Page 6: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/6.jpg)
A computer model
• Particles at rest
• One particle given an impulse
• Interaction only on contact★ Energy conserving
★ Momentum conserving
![Page 7: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/7.jpg)
A computer model
![Page 8: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/8.jpg)
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
![Page 9: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/9.jpg)
Question
Take the above model and make the collisions inelastic.
Do the results change?
![Page 10: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/10.jpg)
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
![Page 11: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/11.jpg)
Different Cases
Jaeger et al, 1996
Blair et al, 2003 Goldhirsch et al, 1993
Aranson et al, 2006
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Key ingredient
u
u t
un
Collisions are inelastic
vt = ut
vn = !run
r < 1
![Page 13: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/13.jpg)
Boudet et al, PRL 2009
Experiments
![Page 14: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/14.jpg)
Experiments
Walsh et al, PRL 2003
Crater formation
![Page 15: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/15.jpg)
Experiments
Cheng et al, Nature Phys, 2008
![Page 16: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/16.jpg)
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
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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
![Page 18: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/18.jpg)
Clustering
![Page 19: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/19.jpg)
Clustering
• Breakdown of Haff ’s law (kinetic theory)
• New regime: inhomogeneous clustered regime
![Page 20: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/20.jpg)
Experiments
• friction
• boundary effects
![Page 21: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/21.jpg)
Experiments
Maaβ et al, 2008
Levitation
![Page 22: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/22.jpg)
ExperimentsMicrogravity
Tatsumi et al, 2009
![Page 23: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/23.jpg)
Boudet et al, PRL 2009
Clustering
![Page 24: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/24.jpg)
A constant coefficient of restitution?
Brass Aluminium
Marble C. V. Raman, 1918
![Page 25: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/25.jpg)
A constant coefficient of restitution?
Brass Aluminium
Marble
r ! 1 when v ! 0
r ! r0 when v !"
C. V. Raman, 1918
![Page 26: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/26.jpg)
Coefficient of Restitution
r
v
1
δ
r ! 1 when v ! 0
r ! r0 when v !"
![Page 27: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/27.jpg)
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
![Page 28: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/28.jpg)
Computer simulation
![Page 29: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/29.jpg)
Elastic vs Inelastic
![Page 30: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/30.jpg)
Scaling analysis
Let Rt ! t!
![Page 31: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/31.jpg)
Length scales
R1
R2
δR
Do all these lengths scale as tα ?
![Page 32: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/32.jpg)
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
![Page 33: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/33.jpg)
Length scales
δRP (R, t) = f(R, !R, t)
P (R, t) = f(R, t)
P (R, t) =1R
g
!R
t!
"
![Page 34: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/34.jpg)
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
![Page 35: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/35.jpg)
Scaling analysis
vt =dRt
dt! t!!1
Let Rt ! t!
Et ! Ntv2t ! t!d+2!!2
Nt ! Rdt ! t!d
![Page 36: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/36.jpg)
Scaling (elastic limit)
Et ! Ntv2t ! t!d+2!!2
But energy is a constant
!d + 2!! 2 = 0
! =2
d + 2
![Page 37: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/37.jpg)
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
![Page 38: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/38.jpg)
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
![Page 39: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/39.jpg)
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
![Page 40: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/40.jpg)
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
![Page 41: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/41.jpg)
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
![Page 42: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/42.jpg)
Comparison with kinetic theory
100
101
102
103
100 101 102 103 104 105 106
R(t)
t
Eq. (6)SimulationSlope = 1/3
![Page 43: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/43.jpg)
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
![Page 44: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/44.jpg)
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
![Page 45: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/45.jpg)
Experiments
Boudet et al, PRL 2009
![Page 46: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/46.jpg)
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
![Page 47: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/47.jpg)
Non-zero ambient temperature
![Page 48: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/48.jpg)
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
![Page 49: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/49.jpg)
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
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Experiment
Walsh et al, PRL 2003
Crater formation
![Page 51: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/51.jpg)
Data (Crater)
100
101
101 102
R(t)
t
t0.18t0.19t0.24
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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
![Page 53: Shock Propagation in Loosely Packed Granular Mediafracmeet/FFPM2012/rajesh_porous.pdf · R. Rajesh (Institute of Mathematical Sciences, Chennai) Outline of the talk ... C. V. Raman,](https://reader035.vdocuments.us/reader035/viewer/2022081407/5f1caf6195cecc084529976b/html5/thumbnails/53.jpg)
Summary & Outlook
• Understanding crossovers
• Can the freely cooling gas be understood?
• Related experiments
Cheng et al, Nature Phys, 2008