12/08/2015, YITP, Kyoto Emergence of collective dynamics in active biological systems -- Swimming micro-organisms -- Norihiro Oyama John J. Molina Ryoichi Yamamoto* Department of Chemical Engineering, Kyoto University 1
Outline 1. Introduction: – DNS for particles moving through Fluids Stokes friction, Oseen (RPY), … are not the end of the story -> Need DNS to go beyond 2. DNS of swimming (active) particles: – Self motions of swimming particles – Collective motions of swimming particles 2
Particles moving through fluids G Gravity: Gravity: Sedimentation A falling object in colloidal disp. at high Re=10 3 3
Basic equations for DNS Navier-Stokes 1 2 , 0 u u p u f u (Fluid) p t exchange momentum Newton-Euler Ω R V d d d i i i , , V m F I N (Particles) i i i i i dt dt dt • FEM : sharp solid/fluid interface on irregular lattice → extremely slow… a ξ • FPD/SPM : smeared out interface on fixed square lattice → much faster!! 4
FPD and SPM FPD (2000) S Tanaka, Araki n ( , ) u x t P P S 1 ( , ) n u x t SPM (2005) Nakayama, RY S S body Define body force to force enforce fluid/particle boundary conditions 1 ( , ) n *( , ) u x t u x t (colloid, swimmer, etc.) 5
PRE 2005 Implementation of no-slip b.c. n n n n , , , R V u r i i i Step 1 Step 2 Step 3 Momentum conservation 1 n n 6
EPJE 2008 Numerical test: Drag force (1) Mobility coefficient of spheres at Re=1 Mobility coefficient This choice can reproduce the collect Stokes drag force within 5% error. 7
JCP 2013 13 Numerical test: Drag force (2) Drag coefficient of non-spherical rigid bodies at Re=1 Any shaped rigid bodies can be formed by assembling spheres Simulation vs. Stokes theory 8
RSC Advanc ances 2014 Numerical test: Drag force (3) Drag coefficient of a sphere C D at Re<200 Re=10 (D=8Δ) 9
EPJE 2008 Numerical test: Lubrication force Approaching velocity of a pair of spheres at Re=0 under a constant F h F Lubrication (2-body) V 1 V 2 SPM Two particles are approaching with velocity V under a constant force F. RPY V tends to decrease with decreasing the separation h due to the lubrication force. Stokesian Dynamics (Brady) SPM can reproduce lubrication force very correctly until the particle separation becomes 10 comparable to x (= grid size)
Outline 1. Introduction: – DNS for particles moving through Fluids 2. DNS of swimming (active) particles: – Self motions of swimming particles – Collective motions of swimming particles 11
SM 2013 Implementation of surface flow x 2 a a tangential propulsion surface flow Total momentum is conserved 12
A model micro-swimmer: Squirmer Propulsion J. R. Blake (1971) ˆ e Polynomial expansion of surface slip ˆ z velocity. Only component is treated θ ˆ here. r ˆ φ ˆ r θ y ( ) s u neglecting n>2 ˆ x ( ) θ s sin sin 2 u B B 1 2 13
A spherical model: Squirmer J. R. Blake (1971) Ishikawa & Pedley (2006-) ˆ ( ) θ s sin sin2 u B Surface flow 1 velocity down up stress against shear flow propelling velocity 14
A spherical model: Squirmer extension contraction Micro- organism Bacteria chlamydomonas Puller Pusher 0 0 0 Squirmer 15
16 Sim. methods for squirmers SD DNS LBM: Llopis, Pagonabarraga , … (2006 -) Ishikawa, Pedley , … (2006 -) Swan, Brady, … (2011 -) MPC / SRD: Dowton, Stark (2009-) . Götze, Gompper (2010-) . . Navier-Stokes: Molina, Yamamoto, … (2013 -) . . . 16
SM 2013 A single swimmer Externally driven colloid Neutral swimmer (gravity, tweezers, etc…) 0 ( ) | | ( ) | | 1 3 u r r u r r Box: 64 x 64 x 64 with PBC, Particle radius: a=6, φ=0.002 Re=0.01, Pe= ∞ , Ma=0 17
SM 2013 A single swimmer Pusher Neutral Puller 2 0 2 ( ) | | ( ) | | ( ) | | 2 3 2 u r r u r r u r r Box: 64 x 64 x 64 with PBC, Particle radius: a=6, φ=0.002 Re=0.01, Pe= ∞ , Ma=0 18
SM 2013 A single swimmer Stream lines Puller 2 ( ) u r 19
SM 2013 Swimmer dispersion , 0.05 0.01 0.10 0.124 pusher 2 neutral 0 puller 2 20 20
SM 2013 Velocity auto correlation t t 2 ( ) exp ex p C t U U , , s l Short-time Long-time 1 ( ) U U l 2 ( , ) ~ ~ D U l 2 ( ) r c ↑ collision radius weak dependency on , s Analogous to low density gas (mean-free-path) 21
SM 2013 Collision radius of swimmers 1/2 2 2 r U c l a ( 5) r c r increases with c increasing | | r c Nearly symmetric for puller ( 0 ) and pusher ( 0 ) Pusher Puller 22
Outline 1. Introduction: – DNS for particles moving through Fluids 2. DNS of swimming (active) particles: – Self motions of swimming particles – Collective motions of swimming particles 23
Collective motion: flock of birds Interactions: • Hydrodynamic • Communication Re ～ 10 3 ～ 5 Ex. Vicsek model 24
Collective motion: E-coli bacteria Interactions: • Hydrodynamic • Steric (rod-rod) Re ～ 10 -3 ～ -5 Ex. Active LC model 25
Question Can any non-trivial collective motions take place in a system composed of spherical swimming particles which only hydrodynamically interacting to each other? ↓ DNS is an ideal tool to answer this question. 26
unpublished Collective motion of squirmers confined between hard walls (at a volume fraction = 0.13) puller with ＝ +0.5 pusher with ＝ -0.5 27
Dynamic structure factor Summary for bulk liquids 2 Rayleigh mode 2D k T (thermal diffusion) 2 2 k Brillouin mode 1 (phonon) c s T ω c k s dispersion relation with a D b speed of sound: c s T 28
unpublished Dynamic structure factor of bulk squirmers (puller with ＝ +0.5) Brillouin mode (phonon-like?) ω c k s dispersion relation with speed of wave: c s 29 ω
unpublished Dynamic structure factor of bulk squirmers (pusher with ＝ -0.5) Similar to the previous puller case ( ＝ +0.5), but the intensity of the wave is much suppressed. 30 ω
unpublished Dynamic structure factor Dispersion relation ω c k s 31
unpublished Open questions • Dependencies of the phenomena on , , L • Mechanism of density wave n aive guess … for pullers contraction • Corresponding experiments 32
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