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