Phase transition in peristaltic transport of granular particles - - PowerPoint PPT Presentation

Phase transition in peristaltic transport

• f granular particles

Naoki Yoshioka Hisao Hayakawa

Yukawa Institute for Theoretical Physics, Kyoto University

Physics of Granular Flows

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Outline

1 Intdocution

Peristaltic transport Objectives

2 Model (1)

Peristaltic flow of frictionless granular particles

3 Results (1)

Time evolution of mass flux Transition time Phase transition of peristaltic flow

4 Model (2)

Peristaltic flow of frictional granular particles Implementation of peristaltic motion

5 Results (2)

Time evolution of flow rate Stationary flow rate

6 Summary

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic transport

Progressive wave of area contraction/expansion. Biological systems

esophagus small intensine ureters

Peristaltic Pump

blood, corrosive fluids, foods, ... preventing the transported fluid from their mechanical parts.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic transport

Progressive wave of area contraction/expansion. Biological systems

esophagus small intensine ureters

Peristaltic Pump

blood, corrosive fluids, foods, ... preventing the transported fluid from their mechanical parts.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic transport

Progressive wave of area contraction/expansion. Biological systems

esophagus small intensine ureters

Peristaltic Pump

blood, corrosive fluids, foods, ... preventing the transported fluid from their mechanical parts.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous studies

Zien and Ostrach, J. Biomech. 3, 63 (1970) Shapiro et al., JFM 37, 799 (1969)

Newtonian fluids

Stokes approximation

assuming some of parameters are zero or small

reflux and trapping w/ pressure difference width at bottlenecks v.s. flow rate

Non-Newtonian fluids

many studies, e.g., Maxwell fluids, third-order fluids, power-law fluids, ...

Particles

• ne particle in fluids

dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous studies

Shapiro et al., JFM 37, 799 (1969)

Newtonian fluids

Stokes approximation

assuming some of parameters are zero or small

reflux and trapping w/ pressure difference width at bottlenecks v.s. flow rate

Non-Newtonian fluids

many studies, e.g., Maxwell fluids, third-order fluids, power-law fluids, ...

Particles

• ne particle in fluids

dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous studies

Shapiro et al., JFM 37, 799 (1969)

Newtonian fluids

Stokes approximation

assuming some of parameters are zero or small

reflux and trapping w/ pressure difference width at bottlenecks v.s. flow rate

Non-Newtonian fluids

many studies, e.g., Maxwell fluids, third-order fluids, power-law fluids, ...

Particles

• ne particle in fluids

dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous studies

Fauci, Computers Fluids 21, 583 (1992) Jim´ enez-Lozano et al., PRE 79, 041901

Newtonian fluids

Stokes approximation

assuming some of parameters are zero or small

reflux and trapping w/ pressure difference width at bottlenecks v.s. flow rate

Non-Newtonian fluids

many studies, e.g., Maxwell fluids, third-order fluids, power-law fluids, ...

Particles

• ne particle in fluids

dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Objectives

Hou et al., PRL 91, 204301 (2003).

Peristaltic transport of many particles. For example,

boluses/chymes in esophagus/intensine blood cells in blood vessel pumping corrosive sands, foods

Efficiency of pumping? Particles might jam at bottleneck

granular flow in silo

Minimum width w v.s. flux

large w—slow unjammed flow small w—fast jammed flow what’s inbetween? phase transition?

Role of friction? strain- v.s. stress-controlled

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Objectives

Hou et al., PRL 91, 204301 (2003).

Peristaltic transport of many particles. For example,

boluses/chymes in esophagus/intensine blood cells in blood vessel pumping corrosive sands, foods

Efficiency of pumping? Particles might jam at bottleneck

granular flow in silo

Minimum width w v.s. flux

large w—slow unjammed flow small w—fast jammed flow what’s inbetween? phase transition?

Role of friction? strain- v.s. stress-controlled

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Objectives

Hou et al., PRL 91, 204301 (2003).

Peristaltic transport of many particles. For example,

boluses/chymes in esophagus/intensine blood cells in blood vessel pumping corrosive sands, foods

Efficiency of pumping? Particles might jam at bottleneck

granular flow in silo

Minimum width w v.s. flux

large w—slow unjammed flow small w—fast jammed flow what’s inbetween? phase transition?

Role of friction? strain- v.s. stress-controlled

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Objectives

Hou et al., PRL 91, 204301 (2003).

Peristaltic transport of many particles. For example,

boluses/chymes in esophagus/intensine blood cells in blood vessel pumping corrosive sands, foods

Efficiency of pumping? Particles might jam at bottleneck

granular flow in silo

Minimum width w v.s. flux

large w—slow unjammed flow small w—fast jammed flow what’s inbetween? phase transition?

Role of friction? strain- v.s. stress-controlled

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Objectives

Hou et al., PRL 91, 204301 (2003).

Peristaltic transport of many particles. For example,

boluses/chymes in esophagus/intensine blood cells in blood vessel pumping corrosive sands, foods

Efficiency of pumping? Particles might jam at bottleneck

granular flow in silo

Minimum width w v.s. flux

large w—slow unjammed flow small w—fast jammed flow what’s inbetween? phase transition?

Role of friction? strain- v.s. stress-controlled

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic flow of frictionless granular particles

a b λ c b w Monodisperse dissipative particles Π = Πp ∪ Πw, w/o gravity & fluid. Spring and viscous force at contact; f el

ij = kξijΘ(ξij)nij,

f vis

ij = −η(vij · nij)Θ(ξij)nij,

Particles in a tube, Πp; m d2ri dt2 =

• j∈Π\{i}

(f el

ij + f vis ij ).

Particles embedded on a tube, Πw; ri =

• ri(t) cos φi, ri(t) sin φi, ζi
• ,

ri(t) = a + b sin 2π λ (ct + ζi)

• .

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic flow of frictionless granular particles

a b λ c b w Monodisperse dissipative particles Π = Πp ∪ Πw, w/o gravity & fluid. Spring and viscous force at contact; f el

ij = kξijΘ(ξij)nij,

f vis

ij = −η(vij · nij)Θ(ξij)nij,

Particles in a tube, Πp; m d2ri dt2 =

• j∈Π\{i}

(f el

ij + f vis ij ).

Particles embedded on a tube, Πw; ri =

• ri(t) cos φi, ri(t) sin φi, ζi
• ,

ri(t) = a + b sin 2π λ (ct + ζi)

• .

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic flow of frictionless granular particles

a b λ c b w Monodisperse dissipative particles Π = Πp ∪ Πw, w/o gravity & fluid. Spring and viscous force at contact; f el

ij = kξijΘ(ξij)nij,

f vis

ij = −η(vij · nij)Θ(ξij)nij,

Particles in a tube, Πp; m d2ri dt2 =

• j∈Π\{i}

(f el

ij + f vis ij ).

Particles embedded on a tube, Πw; ri =

• ri(t) cos φi, ri(t) sin φi, ζi
• ,

ri(t) = a + b sin 2π λ (ct + ζi)

• .

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic flow of frictionless granular particles

a b λ c b w Monodisperse dissipative particles Π = Πp ∪ Πw, w/o gravity & fluid. Spring and viscous force at contact; f el

ij = kξijΘ(ξij)nij,

f vis

ij = −η(vij · nij)Θ(ξij)nij,

Particles in a tube, Πp; m d2ri dt2 =

• j∈Π\{i}

(f el

ij + f vis ij ).

Particles embedded on a tube, Πw; ri =

• ri(t) cos φi, ri(t) sin φi, ζi
• ,

ri(t) = a + b sin 2π λ (ct + ζi)

• .

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Parameters

a b λ c b w Scaled by

mass m, diameter d,

• k/m

a = 1.5, λ = 10, η = 5.48 × 10−3 restitution coefficient e = exp

• −πη/
• 2 − η2

≃ 9.88 × 10−1

particles are almost elastic

Control parameters

width at a bottleneck w ≡ 2(a − b) strain rate ˙ ǫ ≡ c/λ volume fraction at b = 0, ¯ ρ ≡ N/6a2L

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Parameters

a b λ c b w Scaled by

mass m, diameter d,

• k/m

a = 1.5, λ = 10, η = 5.48 × 10−3 restitution coefficient e = exp

• −πη/
• 2 − η2

≃ 9.88 × 10−1

particles are almost elastic

Control parameters

width at a bottleneck w ≡ 2(a − b) strain rate ˙ ǫ ≡ c/λ volume fraction at b = 0, ¯ ρ ≡ N/6a2L

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Parameters

a b λ c b w Scaled by

mass m, diameter d,

• k/m

a = 1.5, λ = 10, η = 5.48 × 10−3 restitution coefficient e = exp

• −πη/
• 2 − η2

≃ 9.88 × 10−1

particles are almost elastic

Control parameters

width at a bottleneck w ≡ 2(a − b) strain rate ˙ ǫ ≡ c/λ volume fraction at b = 0, ¯ ρ ≡ N/6a2L

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Snapshots

t t+∆t t+2∆t

unjammed flow → jammed flow

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Typical time evolution of mass flux

0.2 0.4 0.6 0.8 1 2 4 6 8 10

J/Jmax t/103

ρ=0.167 a=2.0 ε=1.83×10-1 ·

τ

w=3.2 3.24 3.242 3.244 3.26 3.28 3.32

Initial condition: J = 0. Jmax ≡ Nc/L. Large w

steady slow unjammed flow

Small w

Transition from unsteady unjammed flow to steady fast jammed flow

Transition at w = wc.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Typical time evolution of mass flux

0.2 0.4 0.6 0.8 1 2 4 6 8 10

J/Jmax t/103

ρ=0.167 a=2.0 ε=1.83×10-1 ·

τ

w=3.2 3.24 3.242 3.244 3.26 3.28 3.32

Initial condition: J = 0. Jmax ≡ Nc/L. Large w

steady slow unjammed flow

Small w

Transition from unsteady unjammed flow to steady fast jammed flow

Transition at w = wc.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Typical time evolution of mass flux

0.2 0.4 0.6 0.8 1 2 4 6 8 10

J/Jmax t/103

ρ=0.167 a=2.0 ε=1.83×10-1 ·

τ

w=3.2 3.24 3.242 3.244 3.26 3.28 3.32

Initial condition: J = 0. Jmax ≡ Nc/L. Large w

steady slow unjammed flow

Small w

Transition from unsteady unjammed flow to steady fast jammed flow

Transition at w = wc.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Typical time evolution of mass flux

0.2 0.4 0.6 0.8 1 2 4 6 8 10

J/Jmax t/103

ρ=0.167 a=2.0 ε=1.83×10-1 ·

τ

w=3.2 3.24 3.242 3.244 3.26 3.28 3.32

Initial condition: J = 0. Jmax ≡ Nc/L. Large w

steady slow unjammed flow

Small w

Transition from unsteady unjammed flow to steady fast jammed flow

Transition at w = wc.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Transition time and its fluctuation

0.2 0.4 0.6 0.8 1 2 4 6 8 10

J/Jmax t/103

ρ=0.167 a=2.0 ε=1.83×10-1 ·

τ

w=3.2 3.24 3.242 3.244 3.26 3.28 3.32 101 102 103 104 105 106 107 10-810-710-610-510-410-310-210-1100

τ wc

• w

˙ ¯ ρ=2.96×10-1 ε*=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Time t = τ at which the transition

• ccurs.

τ depends on w. Diverges at w = wc(˙ ǫ);

τ ∼ (wc − w)−1

Transition time τ

τ ∼ ˙ ǫ−7/2f

• (wc − w)/˙

ǫ3/2 , f(x) ∼ x−1 for x ∼ 1.

χτ ≡ τ 2 − τ2

χτ ∼ (wc − w)−3 χτ ∼ ˙ ǫ−7g

• (wc − w)/˙

ǫ3/2 , g(x) ∼ x−3 for x ∼ 1.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Transition time and its fluctuation

0.2 0.4 0.6 0.8 1 2 4 6 8 10

J/Jmax t/103

ρ=0.167 a=2.0 ε=1.83×10-1 ·

τ

w=3.2 3.24 3.242 3.244 3.26 3.28 3.32 101 102 103 104 105 106 107 10-810-710-610-510-410-310-210-1100

τ wc

• w

˙ ¯ ρ=2.96×10-1 ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Time t = τ at which the transition

• ccurs.

τ depends on w. Diverges at w = wc(˙ ǫ);

τ ∼ (wc − w)−1

Transition time τ

τ ∼ ˙ ǫ−7/2f

• (wc − w)/˙

ǫ3/2 , f(x) ∼ x−1 for x ∼ 1.

χτ ≡ τ 2 − τ2

χτ ∼ (wc − w)−3 χτ ∼ ˙ ǫ−7g

• (wc − w)/˙

ǫ3/2 , g(x) ∼ x−3 for x ∼ 1.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Transition time and its fluctuation

10-4 10-3 10-2 10-1 100 101 102 103 10-3 10-2 10-1 100 101 102

τε7/2 (wc

• w)/ε3/2

˙ ¯ ρ=2.96×10-1

˙ ˙

ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2 101 102 103 104 105 106 107 10-810-710-610-510-410-310-210-1100

τ wc

• w

˙ ¯ ρ=2.96×10-1 ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Time t = τ at which the transition

• ccurs.

τ depends on w. Diverges at w = wc(˙ ǫ);

τ ∼ (wc − w)−1

Transition time τ

τ ∼ ˙ ǫ−7/2f

• (wc − w)/˙

ǫ3/2 , f(x) ∼ x−1 for x ∼ 1.

χτ ≡ τ 2 − τ2

χτ ∼ (wc − w)−3 χτ ∼ ˙ ǫ−7g

• (wc − w)/˙

ǫ3/2 , g(x) ∼ x−3 for x ∼ 1.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Transition time and its fluctuation

10-4 10-3 10-2 10-1 100 101 102 103 10-3 10-2 10-1 100 101 102

τε7/2 (wc

• w)/ε3/2

˙ ¯ ρ=2.96×10-1

˙ ˙

ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2 100 102 104 106 108 1010 1012 10-810-710-610-510-410-310-210-1100

χτ wc

• w

˙ ¯ ρ=2.96×10-1 ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Time t = τ at which the transition

• ccurs.

τ depends on w. Diverges at w = wc(˙ ǫ);

τ ∼ (wc − w)−1

Transition time τ

τ ∼ ˙ ǫ−7/2f

• (wc − w)/˙

ǫ3/2 , f(x) ∼ x−1 for x ∼ 1.

χτ ≡ τ 2 − τ2

χτ ∼ (wc − w)−3 χτ ∼ ˙ ǫ−7g

• (wc − w)/˙

ǫ3/2 , g(x) ∼ x−3 for x ∼ 1.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Transition time and its fluctuation

10-4 10-3 10-2 10-1 100 101 102 103 10-3 10-2 10-1 100 101 102

τε7/2 (wc

• w)/ε3/2

˙ ¯ ρ=2.96×10-1

˙ ˙

ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2 10-4 10-2 100 102 104 106 10-7 10-6 10-5 10-4 10-3 10-2 10-1100101

χτ ε7 (wc

• w)/ε3/2

¯ ρ=2.96×10-1

˙ ˙

˙ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Time t = τ at which the transition

• ccurs.

τ depends on w. Diverges at w = wc(˙ ǫ);

τ ∼ (wc − w)−1

Transition time τ

τ ∼ ˙ ǫ−7/2f

• (wc − w)/˙

ǫ3/2 , f(x) ∼ x−1 for x ∼ 1.

χτ ≡ τ 2 − τ2

χτ ∼ (wc − w)−3 χτ ∼ ˙ ǫ−7g

• (wc − w)/˙

ǫ3/2 , g(x) ∼ x−3 for x ∼ 1.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

w-dependence of flux

101 102 103 104 105 106 107 10-810-710-610-510-410-310-210-1100

τ wc

• w

˙ ¯ ρ=2.96×10-1 ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3

Jst/Jmax w

¯ ρ=2.96×10-1 ˙ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Estimating wc, using the relation τ ∼ (wc − w)−α. Mass flux J/Jmax, where Jmax ≡ Nc/L.

fast jammed flow for w < wc(˙ ǫ). slow unjammed flow for w > wc(˙ ǫ). jumps at w = wc. No such discontinuity has

• bserved in previous studies

(φ = 1 − w/2a)

wc linearly decreases as ˙ ǫ, wc ≃ −3.75˙ ǫ + wmax.

but behavior for ˙ ǫ 1.0 × 10−2 is not well understood

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

w-dependence of flux

101 102 103 104 105 106 107 10-810-710-610-510-410-310-210-1100

τ wc

• w

˙ ¯ ρ=2.96×10-1 ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3

Jst/Jmax w

¯ ρ=2.96×10-1 ˙ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Estimating wc, using the relation τ ∼ (wc − w)−α. Mass flux J/Jmax, where Jmax ≡ Nc/L.

fast jammed flow for w < wc(˙ ǫ). slow unjammed flow for w > wc(˙ ǫ). jumps at w = wc. No such discontinuity has

• bserved in previous studies

(φ = 1 − w/2a)

wc linearly decreases as ˙ ǫ, wc ≃ −3.75˙ ǫ + wmax.

but behavior for ˙ ǫ 1.0 × 10−2 is not well understood

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

w-dependence of flux

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3

Jst/Jmax w

¯ ρ=2.96×10-1 ˙ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Shapiro et al., JFM 37, 799 (1969)

Estimating wc, using the relation τ ∼ (wc − w)−α. Mass flux J/Jmax, where Jmax ≡ Nc/L.

fast jammed flow for w < wc(˙ ǫ). slow unjammed flow for w > wc(˙ ǫ). jumps at w = wc. No such discontinuity has

• bserved in previous studies

(φ = 1 − w/2a)

wc linearly decreases as ˙ ǫ, wc ≃ −3.75˙ ǫ + wmax.

but behavior for ˙ ǫ 1.0 × 10−2 is not well understood

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

w-dependence of flux

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3

Jst/Jmax w

¯ ρ=2.96×10-1 ˙ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2 1.5 2 2.5 3 0.1 0.2 0.3 0.4

wc ε ˙

¯ ρ=2.96×10-1

Estimating wc, using the relation τ ∼ (wc − w)−α. Mass flux J/Jmax, where Jmax ≡ Nc/L.

fast jammed flow for w < wc(˙ ǫ). slow unjammed flow for w > wc(˙ ǫ). jumps at w = wc. No such discontinuity has

• bserved in previous studies

(φ = 1 − w/2a)

wc linearly decreases as ˙ ǫ, wc ≃ −3.75˙ ǫ + wmax.

but behavior for ˙ ǫ 1.0 × 10−2 is not well understood

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Density dependence

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3

Jst/Jmax w

˙ ε=1.83×10-1 ¯ ρ=1.48×10-1 2.96×10-1 4.44×10-1 5.92×10-1 7.40×10-1 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1

wc ρ ¯

˙ε=1.83×10-1 2.1 2.2 2.3 0.5 1

Fixing ˙ ǫ and changing ¯ ρ Normalised flux J/Jmax decreases as ρ. wc(˙ ǫ) is almost constant for ρ. α ≃ 1 [τ ∼ (wc − w)−α] for 0.15 ρ 0.60. Changing density ρ affects only J/Jmax.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Density dependence

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3

Jst/Jmax w

˙ ε=1.83×10-1 ¯ ρ=1.48×10-1 2.96×10-1 4.44×10-1 5.92×10-1 7.40×10-1 0.5 1 1.5 2 2.5 0.2 0.4 0.6 0.8 1

wc ρ ¯

˙ε=1.83×10-1 2.1 2.2 2.3 0.5 1

Fixing ˙ ǫ and changing ¯ ρ Normalised flux J/Jmax decreases as ρ. wc(˙ ǫ) is almost constant for ρ. α ≃ 1 [τ ∼ (wc − w)−α] for 0.15 ρ 0.60. Changing density ρ affects only J/Jmax.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Density dependence

0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3

Jst/Jmax w

˙ ε=1.83×10-1 ¯ ρ=1.48×10-1 2.96×10-1 4.44×10-1 5.92×10-1 7.40×10-1 0.5 1 1.5 2 2.5 3 3.5 0.2 0.4 0.6 0.8 1

α ρ ¯

˙ε=1.83×10-1

Fixing ˙ ǫ and changing ¯ ρ Normalised flux J/Jmax decreases as ρ. wc(˙ ǫ) is almost constant for ρ. α ≃ 1 [τ ∼ (wc − w)−α] for 0.15 ρ 0.60. Changing density ρ affects only J/Jmax.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Hysteresis

0.2 0.4 0.6 0.8 1 2000 4000 6000 8000 10000

J/Jmax t

ρ=0.167 a=2.0 γ=1.83×10-1 · w=3.6 3.52 3.48 3.4 0.2 0.4 0.6 0.8 1 3 3.2 3.4 3.6 3.8 4

J/Jmax w

ρ=1.67×10-1 γ=1.83×10-1 · J(0)/Jmax=0 J(0)/Jmax=1

Initial condition: J = Jmax. Small w

steady jammed flow

Large w

Transition from unsteady jammed flow to steady unjammed flow

Transition at w = wc′ = wc.

First-order transition

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Hysteresis

0.2 0.4 0.6 0.8 1 2000 4000 6000 8000 10000

J/Jmax t

ρ=0.167 a=2.0 γ=1.83×10-1 · w=3.6 3.52 3.48 3.4 0.2 0.4 0.6 0.8 1 3 3.2 3.4 3.6 3.8 4

J/Jmax w

ρ=1.67×10-1 γ=1.83×10-1 · J(0)/Jmax=0 J(0)/Jmax=1

Initial condition: J = Jmax. Small w

steady jammed flow

Large w

Transition from unsteady jammed flow to steady unjammed flow

Transition at w = wc′ = wc.

First-order transition

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Hysteresis

0.2 0.4 0.6 0.8 1 2000 4000 6000 8000 10000

J/Jmax t

ρ=0.167 a=2.0 γ=1.83×10-1 · w=3.6 3.52 3.48 3.4 0.2 0.4 0.6 0.8 1 3 3.2 3.4 3.6 3.8 4

J/Jmax w

ρ=1.67×10-1 γ=1.83×10-1 · J(0)/Jmax=0 J(0)/Jmax=1

Initial condition: J = Jmax. Small w

steady jammed flow

Large w

Transition from unsteady jammed flow to steady unjammed flow

Transition at w = wc′ = wc.

First-order transition

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Hysteresis

0.2 0.4 0.6 0.8 1 2000 4000 6000 8000 10000

J/Jmax t

ρ=0.167 a=2.0 γ=1.83×10-1 · w=3.6 3.52 3.48 3.4 0.2 0.4 0.6 0.8 1 3 3.2 3.4 3.6 3.8 4

J/Jmax w

ρ=1.67×10-1 γ=1.83×10-1 · J(0)/Jmax=0 J(0)/Jmax=1

Initial condition: J = Jmax. Small w

steady jammed flow

Large w

Transition from unsteady jammed flow to steady unjammed flow

Transition at w = wc′ = wc.

First-order transition

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic flow of frictional granular particles

Polydisperse granular particles

diameter di, 0.8 ≤ di/d∗ ≤ 1.0 mass mi = m∗(di/d∗)3 no gravity, no ambient fluid

f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

f n

ij: Hertz force w/ damping term

f n

ij = 2Y

• Rij

3(1 − ν2)

• ξ3/2

ij

− A

• ξijvn

ij

• f t

ij: tangential force

f t

ij =

• ˜

f

t ij

if

• ˜

f

t ij

• < µsf n

ij

µkf n

ijtij

• therwise

˜ f

t ij = −ktut ij − ηtvt ij

Linear spring and no tangential force in our previous model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic flow of frictional granular particles

vi vj vij rij

fij

n

fij

t

Polydisperse granular particles

diameter di, 0.8 ≤ di/d∗ ≤ 1.0 mass mi = m∗(di/d∗)3 no gravity, no ambient fluid

f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

f n

ij: Hertz force w/ damping term

f n

ij = 2Y

• Rij

3(1 − ν2)

• ξ3/2

ij

− A

• ξijvn

ij

• f t

ij: tangential force

f t

ij =

• ˜

f

t ij

if

• ˜

f

t ij

• < µsf n

ij

µkf n

ijtij

• therwise

˜ f

t ij = −ktut ij − ηtvt ij

Linear spring and no tangential force in our previous model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic flow of frictional granular particles

vi vj vij rij

fij

n

fij

t

Polydisperse granular particles

diameter di, 0.8 ≤ di/d∗ ≤ 1.0 mass mi = m∗(di/d∗)3 no gravity, no ambient fluid

f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

f n

ij: Hertz force w/ damping term

f n

ij = 2Y

• Rij

3(1 − ν2)

• ξ3/2

ij

− A

• ξijvn

ij

• f t

ij: tangential force

f t

ij =

• ˜

f

t ij

if

• ˜

f

t ij

• < µsf n

ij

µkf n

ijtij

• therwise

˜ f

t ij = −ktut ij − ηtvt ij

Linear spring and no tangential force in our previous model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic flow of frictional granular particles

vi vj vij rij

fij

n

fij

t

Polydisperse granular particles

diameter di, 0.8 ≤ di/d∗ ≤ 1.0 mass mi = m∗(di/d∗)3 no gravity, no ambient fluid

f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

f n

ij: Hertz force w/ damping term

f n

ij = 2Y

• Rij

3(1 − ν2)

• ξ3/2

ij

− A

• ξijvn

ij

• f t

ij: tangential force

f t

ij =

• ˜

f

t ij

if

• ˜

f

t ij

• < µsf n

ij

µkf n

ijtij

• therwise

˜ f

t ij = −ktut ij − ηtvt ij

Linear spring and no tangential force in our previous model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

Monodisperse particles embedded in a tube’s wall “Particle-Wall” interactions

f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

neglecting rotation

dw = d∗, mw = 0.1m∗

“Wall-Wall” interactions

Linear spring force w/ natural length l

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

• 2π(zi − ct)/λ
• ri = (ri cos φi, ri sin φi, 0), ri = a + b sin
• 2π(zi − ct)/λ
• in our previous “strain-controlled” model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

Monodisperse particles embedded in a tube’s wall “Particle-Wall” interactions

f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

neglecting rotation

dw = d∗, mw = 0.1m∗

“Wall-Wall” interactions

Linear spring force w/ natural length l

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

• 2π(zi − ct)/λ
• ri = (ri cos φi, ri sin φi, 0), ri = a + b sin
• 2π(zi − ct)/λ
• in our previous “strain-controlled” model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

Monodisperse particles embedded in a tube’s wall “Particle-Wall” interactions

f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

neglecting rotation

dw = d∗, mw = 0.1m∗

“Wall-Wall” interactions

Linear spring force w/ natural length l

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

• 2π(zi − ct)/λ
• ri = (ri cos φi, ri sin φi, 0), ri = a + b sin
• 2π(zi − ct)/λ
• in our previous “strain-controlled” model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

Monodisperse particles embedded in a tube’s wall “Particle-Wall” interactions

f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

neglecting rotation

dw = d∗, mw = 0.1m∗

“Wall-Wall” interactions

Linear spring force w/ natural length l

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

• 2π(zi − ct)/λ
• ri = (ri cos φi, ri sin φi, 0), ri = a + b sin
• 2π(zi − ct)/λ
• in our previous “strain-controlled” model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

Monodisperse particles embedded in a tube’s wall “Particle-Wall” interactions

f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

neglecting rotation

dw = d∗, mw = 0.1m∗

“Wall-Wall” interactions

Linear spring force w/ natural length l

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

• 2π(zi − ct)/λ
• ri = (ri cos φi, ri sin φi, 0), ri = a + b sin
• 2π(zi − ct)/λ
• in our previous “strain-controlled” model

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Parameters, etc.

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0.2 0.4 0.6 0.8 1

e vt*/d* t∗ ≡

• m∗/Y d∗

Parameters

a = 3.5d∗, λ ≃ 20.0d∗ A = 0.1t∗, ν = 0.5, kt = 1.0Y d∗, ηt = 0.1Y d∗t∗, µs = 0.5, µk = 0.4

Restitution coeff. (di = d∗, mi = m∗) e ≃ 0.85 for v ≃ d∗/t∗

M¨ uller and P¨

• schel, PRE (2011)

Control parameters

amplitude of peristaltic force f p peristaltic speed c number of particles N

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Parameters, etc.

0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1 0.2 0.4 0.6 0.8 1

e vt*/d* t∗ ≡

• m∗/Y d∗

Parameters

a = 3.5d∗, λ ≃ 20.0d∗ A = 0.1t∗, ν = 0.5, kt = 1.0Y d∗, ηt = 0.1Y d∗t∗, µs = 0.5, µk = 0.4

Restitution coeff. (di = d∗, mi = m∗) e ≃ 0.85 for v ≃ d∗/t∗

M¨ uller and P¨

• schel, PRE (2011)

Control parameters

amplitude of peristaltic force f p peristaltic speed c number of particles N

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Snapshots

N/V0 = 7.10 × 10−1/d∗3, c/λ = 4.01 × 10−3/t∗ fp = 0.005Y d∗2 Blue: ⇐, Red: ⇒

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Snapshots

N/V0 = 7.10 × 10−1/d∗3, c/λ = 4.01 × 10−3/t∗ fp = 0.004Y d∗2 Blue: ⇐, Red: ⇒

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Time evolution of averaged flow rate

J/t∗ =

i vzi/L, J∗/t∗ = Nc/L

• 0.2

0.2 0.4 0.6 0.8 1 1.2 5000 10000

J/J* t

Nd*3/V0=1.18 ct*/λ=4.01×10-3 fp/Yd*2=0.001 0.005 0.007 0.008 0.01 0.015 0.02

Transitions exist for certain fp’s

from a jammed flow to a unjammed flow

because of stress-contrlled walls

different transition which is found in the previous models

J can be negative for small fp’s

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Time evolution of averaged flow rate

J/t∗ =

i vzi/L, J∗/t∗ = Nc/L

• 0.2

0.2 0.4 0.6 0.8 1 1.2 5000 10000

J/J* t

Nd*3/V0=1.18 ct*/λ=4.01×10-3 fp/Yd*2=0.001 0.005 0.007 0.008 0.01 0.015 0.02

Transitions exist for certain fp’s

from a jammed flow to a unjammed flow

because of stress-contrlled walls

different transition which is found in the previous models

J can be negative for small fp’s

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Stationary flow rate

n ≡ Nd∗3/V0, ˙ ǫ ≡ ct∗/λ

0.2 0.4 0.6 0.8 1 0.005 0.01

J/J* fp

n=7.10×10-1 ˙ ε=5.02×10-4 1.00×10-3 2.01×10-3 4.01×10-3

• 0.2

0.2 0.4 0.6 0.8 1 0.005 0.01

J/J* fp

n=1.18 ˙ ε=5.02×10-4 1.00×10-3 2.01×10-3 4.01×10-3

Discontinuous transition for large c’s No transition? or continuous transition? for small c’s Negative J’s for small fp’s

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Stationary flow rate

n ≡ Nd∗3/V0, ˙ ǫ ≡ ct∗/λ

0.2 0.4 0.6 0.8 1 0.005 0.01

J/J* fp

n=7.10×10-1 ˙ ε=5.02×10-4 1.00×10-3 2.01×10-3 4.01×10-3

• 0.2

0.2 0.4 0.6 0.8 1 0.005 0.01

J/J* fp

n=1.18 ˙ ε=5.02×10-4 1.00×10-3 2.01×10-3 4.01×10-3

Discontinuous transition for large c’s No transition? or continuous transition? for small c’s Negative J’s for small fp’s

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Stationary flow rate

n ≡ Nd∗3/V0, ˙ ǫ ≡ ct∗/λ

0.2 0.4 0.6 0.8 1 0.005 0.01

J/J* fp

n=7.10×10-1 ˙ ε=5.02×10-4 1.00×10-3 2.01×10-3 4.01×10-3

• 0.2

0.2 0.4 0.6 0.8 1 0.005 0.01

J/J* fp

n=1.18 ˙ ε=5.02×10-4 1.00×10-3 2.01×10-3 4.01×10-3

Discontinuous transition for large c’s No transition? or continuous transition? for small c’s Negative J’s for small fp’s

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Negative J—rough v.s. smooth

n ≡ Nd∗3/V0, ˙ ǫ ≡ ct∗/λ

• 0.2

0.2 0.4 0.6 0.8 1 0.005 0.01

J/J* fp

n=1.18 ε=4.01×10-3 ˙ rough smooth

No negative J’s for smooth granular particles?

because of friction?

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

• 2
• 1

1 2 16 18 20 22 24 y z

• 2
• 1

1 2 16 18 20 22 24 y z

• 2
• 1

1 2 16 18 20 22 24 y z

• 2
• 1

1 2 16 18 20 22 24 y z

• 2
• 1

1 2 16 18 20 22 24 y z

• 2
• 1

1 2 16 18 20 22 24 y z

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

• 6
• 4
• 2

2 4 6 10 15 20 25 30 35 40 y z

• 6
• 4
• 2

2 4 6 10 15 20 25 30 35 40 y z

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Summary

Peristalsis transport of granular particles Frictionless case

Discontinuous transition between jammed flow and unjammed flow Scaling relationships

N.Y. and H. Hayakawa, Phys. Rev. E 85, 031302 (2012). Frictional case

Discontinuous transition between jammed flow and “unjammed flow”

this unjammed flow is different from that in frictionless case

Back flow

Intdocution Model (1) Results (1) Model (2) Results (2) Summary 42 44 46 48 50 52 54 56 58 60 200 400 600 800 1000

z t

Intdocution Model (1) Results (1) Model (2) Results (2) Summary 30 35 40 45 50 55 60 65 70 2000 4000 6000 8000 10000

z t

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Negative J

N/V0 = 7.10 × 10−1/d∗3, c/λ = 4.01 × 10−3/t∗ fp = 0.002Y d∗2 rotation smooth Blue: ⇐, Red: ⇒

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic transport

Progressive wave of area contraction/expansion. Biological systems

esophagus small intensine ureters vasomotion (spontaneous oscillation)

• f small blood vessels

Peristaltic Pump

blood, corrosive fluids, foods, ... preventing the transported fluid from their mechanical parts.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic transport

Progressive wave of area contraction/expansion. Biological systems

esophagus small intensine ureters vasomotion (spontaneous oscillation)

• f small blood vessels

Peristaltic Pump

blood, corrosive fluids, foods, ... preventing the transported fluid from their mechanical parts.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic transport

Progressive wave of area contraction/expansion. Biological systems

esophagus small intensine ureters vasomotion (spontaneous oscillation)

• f small blood vessels

Peristaltic Pump

blood, corrosive fluids, foods, ... preventing the transported fluid from their mechanical parts.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic transport

Progressive wave of area contraction/expansion. Biological systems

esophagus small intensine ureters vasomotion (spontaneous oscillation)

• f small blood vessels

Peristaltic Pump

blood, corrosive fluids, foods, ... preventing the transported fluid from their mechanical parts.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous studies

Zien and Ostrach, J. Biomech. 3, 63 (1970) Shapiro et al., JFM 37, 799 (1969)

Newtonian fluids

Stokes approximation

assuming some of parameters are zero or small.

reflux, trapping.

Non-Newtonian fluids

many studies, e.g., Maxwell fluids, third-order fluids, power-law fluids, ...

Particles

• ne particle in fluids

dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous studies

Shapiro et al., JFM 37, 799 (1969)

Newtonian fluids

Stokes approximation

assuming some of parameters are zero or small.

reflux, trapping.

Non-Newtonian fluids

many studies, e.g., Maxwell fluids, third-order fluids, power-law fluids, ...

Particles

• ne particle in fluids

dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous studies

Shapiro et al., JFM 37, 799 (1969)

Newtonian fluids

Stokes approximation

assuming some of parameters are zero or small.

reflux, trapping.

Non-Newtonian fluids

many studies, e.g., Maxwell fluids, third-order fluids, power-law fluids, ...

Particles

• ne particle in fluids

dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous studies

Fauci, Computers Fluids 21, 583 (1992) Jim´ enez-Lozano et al., PRE 79, 041901

Newtonian fluids

Stokes approximation

assuming some of parameters are zero or small.

reflux, trapping.

Non-Newtonian fluids

many studies, e.g., Maxwell fluids, third-order fluids, power-law fluids, ...

Particles

• ne particle in fluids

dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous results—snapshots

• N. Y. and H. H., Phys. Rev. E 85, 031302 (2012).

a b λ c b w t t+∆t t+2∆t Peristaltic transport of smooth dissipative particles Strain-controlled peristaltic motion Unjammed flow → Jammed flow

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Previous results—flow rate

0.2 0.4 0.6 0.8 1 2 4 6 8 10

J/Jmax t/103

ρ=0.167 a=2.0 ε=1.83×10-1 ·

τ

w=3.2 3.24 3.242 3.244 3.26 3.28 3.32 0.2 0.4 0.6 0.8 1 1 1.5 2 2.5 3

Jst/Jmax w

¯ ρ=2.96×10-1 ˙ε=3.65×10-1 2.61×10-1 1.83×10-1 9.13×10-2 4.56×10-2 3.65×10-2

Large w ⇒ steady slow unjammed flow Small w ⇒ steady fast jammed flow Discontinuous transition at w = wc.

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Objectives

Peristaltic transport of frictional granular particles More realistic systems

rough v.s. smooth stress- v.s. strain-controlled

slow peristaltic speed

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Model—granular particles

a b λ c b w Polydisperse granular particles w/o gravity & fluid

diameter di, 0.8 ≤ di/d∗ ≤ 1.0 mass mi = m∗(di/d∗)3

f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

nij = rij/|rij|, rij = ri − rj, ξij = (di + dj)/2 − |rij|,

Hertzian contact force w/ damping term fn

ij = 2Y

• Rij

3(1 − ν2)

• ξ3/2

ij

− A

• ξijvn

ij

• vn

ij = vij · nij, vij = vi − vj,

Rij = didj/2(di + dj)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Model—granular particles

a b λ c b w Polydisperse granular particles w/o gravity & fluid

diameter di, 0.8 ≤ di/d∗ ≤ 1.0 mass mi = m∗(di/d∗)3

f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

nij = rij/|rij|, rij = ri − rj, ξij = (di + dj)/2 − |rij|,

Hertzian contact force w/ damping term fn

ij = 2Y

• Rij

3(1 − ν2)

• ξ3/2

ij

− A

• ξijvn

ij

• vn

ij = vij · nij, vij = vi − vj,

Rij = didj/2(di + dj)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Model—granular particles

a b λ c b w Polydisperse granular particles w/o gravity & fluid

diameter di, 0.8 ≤ di/d∗ ≤ 1.0 mass mi = m∗(di/d∗)3

f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

nij = rij/|rij|, rij = ri − rj, ξij = (di + dj)/2 − |rij|,

Hertzian contact force w/ damping term fn

ij = 2Y

• Rij

3(1 − ν2)

• ξ3/2

ij

− A

• ξijvn

ij

• vn

ij = vij · nij, vij = vi − vj,

Rij = didj/2(di + dj)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Model—granular particles

a b λ c b w f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

Cundall-Strack f t

ij =

˜ f

t ij

if

• ˜

f

t ij

• < µsfn

ij

µkfn

ijtij

• therwise

˜ f

t ij = −ktut ij − ηtvt ij

˙ ut

ij = vt ij − [(ut ij · vij)/|rij|]nij

vt

ij = (vij −vn ijnij)+ di − ξij

2 nij ×ωi −dj − ξij 2 nji × ωj tij = ˜ f

t ij/

• ˜

f

t ij

• Solving eqs. of motion

by Two-step Adams–Bashforth method

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Model—granular particles

a b λ c b w f ij = (fn

ijnij + f t ij)Θ(ξij)Θ(fn ij)

Cundall-Strack f t

ij =

˜ f

t ij

if

• ˜

f

t ij

• < µsfn

ij

µkfn

ijtij

• therwise

˜ f

t ij = −ktut ij − ηtvt ij

˙ ut

ij = vt ij − [(ut ij · vij)/|rij|]nij

vt

ij = (vij −vn ijnij)+ di − ξij

2 nij ×ωi −dj − ξij 2 nji × ωj tij = ˜ f

t ij/

• ˜

f

t ij

• Solving eqs. of motion

by Two-step Adams–Bashforth method

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

a b λ c b w Monodisperse particles embedded in a tube’s wall “Particle-Wall”

Hertzian force w/ damping term f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

no rotation

diameter of “wall” particle dw/d∗ = 1.0 mass of “wall” particle mw/m∗ = 0.1

“Wall-Wall”

Linear spring force w/ natural length l f ij = −k(|rij| − l)nij

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

2π λ (zi − ct)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

a b λ c b w Monodisperse particles embedded in a tube’s wall “Particle-Wall”

Hertzian force w/ damping term f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

no rotation

diameter of “wall” particle dw/d∗ = 1.0 mass of “wall” particle mw/m∗ = 0.1

“Wall-Wall”

Linear spring force w/ natural length l f ij = −k(|rij| − l)nij

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

2π λ (zi − ct)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

a b λ c b w Monodisperse particles embedded in a tube’s wall “Particle-Wall”

Hertzian force w/ damping term f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

no rotation

diameter of “wall” particle dw/d∗ = 1.0 mass of “wall” particle mw/m∗ = 0.1

“Wall-Wall”

Linear spring force w/ natural length l f ij = −k(|rij| − l)nij

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

2π λ (zi − ct)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Peristaltic tube

a b λ c b w Monodisperse particles embedded in a tube’s wall “Particle-Wall”

Hertzian force w/ damping term f ij = (f n

ijnij + f t ij)Θ(ξij)Θ(f n ij)

no rotation

diameter of “wall” particle dw/d∗ = 1.0 mass of “wall” particle mw/m∗ = 0.1

“Wall-Wall”

Linear spring force w/ natural length l f ij = −k(|rij| − l)nij

Peristaltic external force f i = (fp

i cos φi, fp i sin φi, 0) + f keep i

f p

i = f p sin

2π λ (zi − ct)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Parameters, etc.

a b λ c b w Scaled by

largest mass m∗, largest diameter d∗,

• m∗/Y d∗

Parameters

a = 3.5, λ ≃ 20.0 A = 0.1, ν = 0.5, kt = 1.0, ηt = .1, µs = 0.5, µk = 0.4

Control parameters

amplitude of peristaltic force f p strain rate ˙ ǫ ≡ c/λ initial number density n ≡ N/πa2L

Intdocution Model (1) Results (1) Model (2) Results (2) Summary

Parameters, etc.

a b λ c b w Scaled by

largest mass m∗, largest diameter d∗,

• m∗/Y d∗

Parameters

a = 3.5, λ ≃ 20.0 A = 0.1, ν = 0.5, kt = 1.0, ηt = .1, µs = 0.5, µk = 0.4

Control parameters

amplitude of peristaltic force f p strain rate ˙ ǫ ≡ c/λ initial number density n ≡ N/πa2L