Emergence of collective dynamics in active biological systems -- Swimming micro-organisms --

Transcription

1 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

2 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

3 Particles moving through fluids G Gravity: Sedimentation in colloidal disp. Gravity: A falling object at high Re=10 3 3

4 Basic equations for DNS Navier-Stokes (Fluid) Newton-Euler (Particles) ξ a t 1 2 u u p u f u p, 0 exchange momentum d R i d i d i Vi, m V i Fi, I Ω i N dt dt dt FEM: sharp solid/fluid interface on irregular lattice extremely slow FPD/SPM: smeared out interface on fixed square lattice much faster!! 4 i

5 FPD and SPM u n ( x, t) S u n P 1 ( x, t) FPD (2000) Tanaka, Araki P S S S SPM (2005) Nakayama, RY u*( x, t) u body force n 1 ( x, t) Define body force to enforce fluid/particle boundary conditions (colloid, swimmer, etc.) 5

6 PRE 2005 Implementation of no-slip b.c. Step 1 R n, V n, n, u n r i i i Step 2 Step 3 n1 n Momentum conservation 6

7 Mobility coefficient EPJE 2008 Numerical test: Drag force (1) Mobility coefficient of spheres at Re=1 This choice can reproduce the collect Stokes drag force within 5% error. 7

8 JCP 2013 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

9 RSC Advances 2014 Numerical test: Drag force (3) Drag coefficient of a sphere C D at Re<200 Re=10 (D=8Δ) 9

10 EPJE 2008 Numerical test: Lubrication force Approaching velocity of a pair of spheres at Re=0 under a constant F h V 1 V 2 F Lubrication (2-body) Two particles are approaching with velocity V under a constant force F. V tends to decrease with decreasing the separation h due to the lubrication force. RPY SPM Stokesian Dynamics (Brady) SPM can reproduce lubrication force very correctly until the particle separation becomes comparable to x (= grid size) 10

11 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

12 SM 2013 Implementation of surface flow x 2 tangential surface flow a a propulsion Total momentum is conserved 12

13 A model micro-swimmer: Squirmer Propulsion J. R. Blake (1971) z ê ˆr Polynomial expansion of surface slip velocity. Only ˆθ component is treated here. r ˆθ ˆφ y ( s) u neglecting n>2 x u ( s) B sin B sin θˆ 13

14 A spherical model: Squirmer J. R. Blake (1971) Ishikawa & Pedley (2006-) Surface flow velocity u ( s) B sin sin 2 1 θˆ down up propelling velocity stress against shear flow 14

15 A spherical model: Squirmer Microorganism extension contraction Bacteria chlamydomonas Squirmer Pusher 0 0 Puller 0 15

16 16 Sim. methods for squirmers SD Ishikawa, Pedley, (2006-) Swan, Brady, (2011-)... DNS LBM: Llopis, Pagonabarraga, (2006-) MPC / SRD: Dowton, Stark (2009-) Götze, Gompper (2010-) Navier-Stokes: Molina, Yamamoto, (2013-)... 16

17 SM 2013 A single swimmer Externally driven colloid (gravity, tweezers, etc ) Neutral swimmer 0 u( r) r 1 u( r) r 3 Box: 64 x 64 x 64 with PBC, Particle radius: a=6, φ=0.002 Re=0.01, Pe=, Ma=0 17

18 SM 2013 A single swimmer Pusher 2 Neutral 0 Puller 2 u( r) r 2 u( r) r 3 u( r) r 2 Box: 64 x 64 x 64 with PBC, Particle radius: a=6, φ=0.002 Re=0.01, Pe=, Ma=0 18

19 SM 2013 A single swimmer ur () Stream lines Puller 2 19

20 Swimmer dispersion, SM pusher 2 neutral 0 puller

21 SM 2013 Velocity auto correlation t 2 t C( t) U exp U exp s, l, Short-time Long-time l ( U ) 1 D 2 (, ) ~ U l ~ U r ( ) 2 c s weak dependency on, collision radius Analogous to low density gas (mean-free-path) 21

22 SM 2013 Collision radius of swimmers r c 2 2U 1/2 l rc a( 5) c r r c increases with increasing Nearly symmetric for puller (0) and pusher (0) Pusher Puller 22

23 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

24 Collective motion: flock of birds Interactions: Hydrodynamic Communication Re ~ 10 3~5 Ex. Vicsek model 24

25 Collective motion: E-coli bacteria Interactions: Hydrodynamic Steric (rod-rod) Re ~ 10-3~-5 Ex. Active LC model 25

26 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

27 unpublished Collective motion of squirmers confined between hard walls (at a volume fraction = 0.13) 27 puller with = +0.5 pusher with = -0.5

28 Dynamic structure factor Summary for bulk liquids 2D k T 2 Rayleigh mode (thermal diffusion) c s ad 1 T T b ω c dispersion relation with speed of sound: c s s k 2k 2 Brillouin mode (phonon) 28

29 unpublished Dynamic structure factor of bulk squirmers (puller with =+0.5) Brillouin mode (phonon-like?) ω c dispersion relation with speed of wave: c s s k ω 29

30 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. ω 30

31 unpublished Dynamic structure factor Dispersion relation ω c k s 31

32 unpublished Open questions Dependencies of the phenomena on,,l Mechanism of density wave naive guess for pullers Corresponding experiments contraction 32