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

Phase transition in peristaltic transport of 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 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, Zien and Ostrach, J. Biomech. 3 , 63 (1970) third-order fluids, power-law fluids, ... Particles Shapiro et al. , JFM 37 , 799 (1969) one particle in fluids dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Previous studies 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 one particle in fluids dilute particles in fluids Shapiro et al. , JFM 37 , 799 (1969)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Previous studies 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 one particle in fluids dilute particles in fluids Shapiro et al. , JFM 37 , 799 (1969)

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Previous studies 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 Fauci, Computers Fluids 21 , 583 (1992) many studies, e.g. , Maxwell fluids, third-order fluids, power-law fluids, ... Particles one particle in fluids Jim´ enez-Lozano et al. , PRE 79 , 041901 dilute particles in fluids

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Objectives 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 Hou et al. , PRL 91 , 204301 (2003).

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Objectives 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 Hou et al. , PRL 91 , 204301 (2003).

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Objectives 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 Hou et al. , PRL 91 , 204301 (2003).

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Objectives 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 Hou et al. , PRL 91 , 204301 (2003).

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Objectives 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 Hou et al. , PRL 91 , 204301 (2003).

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Peristaltic flow of frictionless granular particles Monodisperse dissipative particles Π = Π p ∪ Π w , w/o gravity & fluid. Spring and viscous force at contact; f el ij = kξ ij Θ( ξ ij ) n ij , a f vis ij = − η ( v ij · n ij )Θ( ξ ij ) n ij , Particles in a tube, Π p ; c m d 2 r i � ( f el ij + f vis d t 2 = ij ) . j ∈ Π \{ i } Particles embedded on a tube, Π w ; λ � � r i = r i ( t ) cos φ i , r i ( t ) sin φ i , ζ i , � 2 π � r i ( t ) = a + b sin λ ( ct + ζ i ) . w b b

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Peristaltic flow of frictionless granular particles Monodisperse dissipative particles Π = Π p ∪ Π w , w/o gravity & fluid. Spring and viscous force at contact; f el ij = kξ ij Θ( ξ ij ) n ij , a f vis ij = − η ( v ij · n ij )Θ( ξ ij ) n ij , Particles in a tube, Π p ; c m d 2 r i � ( f el ij + f vis d t 2 = ij ) . j ∈ Π \{ i } Particles embedded on a tube, Π w ; λ � � r i = r i ( t ) cos φ i , r i ( t ) sin φ i , ζ i , � 2 π � r i ( t ) = a + b sin λ ( ct + ζ i ) . w b b

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Peristaltic flow of frictionless granular particles Monodisperse dissipative particles Π = Π p ∪ Π w , w/o gravity & fluid. Spring and viscous force at contact; f el ij = kξ ij Θ( ξ ij ) n ij , a f vis ij = − η ( v ij · n ij )Θ( ξ ij ) n ij , Particles in a tube, Π p ; c m d 2 r i � ( f el ij + f vis d t 2 = ij ) . j ∈ Π \{ i } Particles embedded on a tube, Π w ; λ � � r i = r i ( t ) cos φ i , r i ( t ) sin φ i , ζ i , � 2 π � r i ( t ) = a + b sin λ ( ct + ζ i ) . w b b

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Peristaltic flow of frictionless granular particles Monodisperse dissipative particles Π = Π p ∪ Π w , w/o gravity & fluid. Spring and viscous force at contact; f el ij = kξ ij Θ( ξ ij ) n ij , a f vis ij = − η ( v ij · n ij )Θ( ξ ij ) n ij , Particles in a tube, Π p ; c m d 2 r i � ( f el ij + f vis d t 2 = ij ) . j ∈ Π \{ i } Particles embedded on a tube, Π w ; λ � � r i = r i ( t ) cos φ i , r i ( t ) sin φ i , ζ i , � 2 π � r i ( t ) = a + b sin λ ( ct + ζ i ) . w b b

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Parameters Scaled by mass m , diameter d , � k/m a = 1 . 5 , λ = 10 , η = 5 . 48 × 10 − 3 a restitution coefficient � � 2 − η 2 � e = exp − πη/ ≃ 9 . 88 × 10 − 1 c particles are almost elastic Control parameters width at a bottleneck λ w ≡ 2( a − b ) strain rate ˙ ǫ ≡ c/λ volume fraction at b = 0 , ρ ≡ N/ 6 a 2 L ¯ w b b

Intdocution Model (1) Results (1) Model (2) Results (2) Summary Parameters Scaled by mass m , diameter d , � k/m a = 1 . 5 , λ = 10 , η = 5 . 48 × 10 − 3 a restitution coefficient � � 2 − η 2 � e = exp − πη/ ≃ 9 . 88 × 10 − 1 c particles are almost elastic Control parameters width at a bottleneck λ w ≡ 2( a − b ) strain rate ˙ ǫ ≡ c/λ volume fraction at b = 0 , ρ ≡ N/ 6 a 2 L ¯ w b b