Microswimmers in Flow
Salima Rafaï
Laboratoire Interdisciplinaire de Physique, CNRS Grenoble
1180
1181
indeed, defining feature of active systems compared to more
familiar nonequilibrium systems is the fact that the energy
input that drives the system out of equilibrium is local, for
example, at the level of each particle, rather than at the
Active Matter
ns
1181
1182
1183
1184
1185
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er to a
he meand of
es from
m conner, Tu,
ny and
known
hey are
Surrey et al., 2001.
Motor proteins
FIG. 2. Patterns organized in vitro by the action of multimeric
kinesin complexes on microtubules, imaged by dark-field microscopy. The concentration of motor proteins increases from left to
right. (a) A disordered array of microtubules. The other two images
display motor-induced organization in (b) spiral and (c) aster
patterns. The bright spots in the images correspond to the minus
end of microtubules. These remarkable experiments from Surrey
et al. (2001) led the way to the study of pattern formation in active
systems. Adapted from Surrey et al., 2001.
Bacterial biofilm
Plankton bloom
Fish school
Bioconvection
Starling flock
2
Living microswimmers
E. Coli
Gonyaulax
Sperm cell
Paramecia
❖
Full name: Chlamydomonas Reinhardtii!
❖
Microalga!
❖
Size: 10 µm!
❖
Shape : Quasi-spherical!
❖
Gender : Puller-type!
❖
Velocity ~100µm/s!
❖
favorite drinking: freshwater + nutrients
3
The complex random walk of Chlamy
Power Stroke
Recovery Stroke
Re = 10-4
!F/2%
+F%!F/2%
20 µm
800 µm
400 Hz movie slowed down 20 times
4
Y (µm)
Particle tracking and
classical analysis
A persistent random walk
X (µm)
20 ms
2s
M. Garcia, S. Berti, P.Peyla & S.Rafaï
Phys. Rev. E (R) (Z011)
5
Y (µm)
Y (µm)
Chlamy & phototaxis
X (µm)
VX (µm/s)
VX (µm/s)
X (µm)
Time (s)
6
Time (s)
Phototaxis & Poiseuille flow
r(t) =
Z
t
V(r)dt0 ,
0
with
V(r) = Vswim sin ✓(r) er + [Vswim cos ✓(r) + u(r)
r z
u(r) = Umax (1
✓(r) =
8
>
< dr = Vswim sin ✓(r)
dt
dz
>
:
= Vswim cos ✓(r) + u(r)
dt
7
dr
Z
t
0
r2
)
2
R
!(r)dt0
2R
and
1mm
Phototaxis & Poiseuille flow: Photofocusing
Light Control of the Flow of Phototactic Microswimmer Suspensions
X. Garcia, S.Rafaï & P.Peyla (Z013) Phys. Rev. Lett. 110 , 138106
8
Half band width (µm)
Half band width (µm)
Photofocusing: a reversible effect
Flow rate (mL/min)
Time (s)
Microswimmer velocity controls the dynamics
A range of flow rate where focusing occurs
9
and
V(r) = Vswim sin ✓(r) er + [Vswim cos ✓(r) + u(r)] k
Z
A
non linear model: swimming
0
!(r)dt , u(r) = Umax r2 /R
speed Vswim+ flow speed u
Swimming in the flow
and
✓(r) =
0
u(r) = Umax (1
✓(r) =
Z
t
!(r)dt
0
r2
)
2
R r
z
(4
2R
8
>
< dr = Vswim sin ✓(r)
dt
dz
>
:
= Vswim cos ✓(r) + u(r)
dt
t
0
8
Z t
dr
<
steady but arbitrary>
swimming
angle
= Vswim sin
0
dr✓(r)
✓(r)
=
!(r)dt
dt
= Vswim sin ✓(r)
dz
>
0
dt ✓(r) + u(r)
:
= Vswim cos
8 dt
>
< dr = Vswim sin ✓(r)
dz
dr
dt
= Vswim cos ✓(r) + u(r)= Vswim sin ✓(r)
dz
>
:
dt
dt cos ✓(r) + u(r)
= Vswim
fluctuating arbitrary swimming
angle
dt
dz
= dr
Vswim cos ✓(r) + u(r)
= Vswim sin ✓(r)
dt
dt
[1] Viscous Fluid Flow, FM White, (1991), McGraw-Hill NY
dz
upstream reorienting swimming angle
= Vswim cos ✓(r) + u(r)
[2] Introduction `a la microfluidique, P.Tabeling
dt
[1] Viscous Fluid Flow, FM White, (1991), McGraw-Hill NY
⇤
[2] Introduction `a la microfluidique, P.Tabeling
Electronic address: [email protected]
⇤
Electronic address: [email protected]
[1] Viscous Fluid Flow, FM White, (1991), McGraw-Hill NY
z
[2] Introduction `a la microfluidique,
P.Tabeling
10
r
bonded to the spherical particle.
articles with an inner viscosity ⌘p much higher than the
particle, we need to know both it
uter fluid viscosity ⌘0 (⌘p /⌘0 = 100). The stokes equa< v >i to determine its position
on is then solved on the entire domain: inside and outity < ⌦ >i to determine its orien
de the particles, thus avoiding particle tracking. The
firsttomodel
reposition the set of forc
ranslational velocity of the ith particle, < v >i is ob- Aorder
i.
ained by averaging the velocity v on the volume of parA single swimmer, described
cle number i. The symbol < . >i means that avermoves in a fluid at rest with a v
ging is done over the volume of the particle number
M
!F/2%
the force : v0 = M f . The swimm
It is also convenient to calculate the angular velocity
1
that
M
=
(6⇣⇡⌘R)
where ⇣ is
< ⌦ >i = 1/2 < r ⇥ v >i . A time step t is defined
+F%!F/2%
eter which depends on the geome
nd at each iteration, the out-of-lattice sphere number i
around the swimmer. For exampl
moved by the quantity ri =< v >iModel:
t. Then, the field
Chlamy
plied on the spherical body, we w
f viscosity
is recalculated
from the new
positions of
Phantom ⌘(r)
flagella
model
=
with
setParticle
of three
forces bon
he sphere
centers.
Time t is then simplySphere
t = n t where
3Dthe
Fluid
Dynamics
Mehandia
& Nott
found
⇣⇠
1.14
in our
simulations
J. ofnumber
Fluid Mech.
Peyla
et al.
J. Fluid
Mech.
2012
+
is the
of 2008
iterations.
3D
Fluid
particle
dynamics
consistent
with
multipole of forces
is -very
close
to the
analytical
valu
Despite the smallness of the
Re
number,
we
solve
the
Tanaka
&
Araki,
PRL,
2000
Experimental hydrodynamic field rived with the Green’s function th
- Peyla, EPL 2007,Davit & Peyla, EPL 2
Navier-Stokes equation (andDrescher
not the Stokes
equation)
et al. Phys
Rev Lett on
2010
fluid. Penalization method
he whole domain with a field of viscosity ⌘(r):
WeTanaka
initially
begin2000),
the simulat
2D (PRL
of swimmers
as describ
Peyla 3D modeled
(EPL 2007)"
Dv
t
⇢
= rP + r.[⌘(r)(rv + rv )] + f ,
(1)
neously distributed across a chan
Dt
Modeling with full hydrodynamics
M
M
M
Navier Stokes + Active forces
11
Clustering
Full hydrodynamics numerical simulations (FPD method)
Self-focusing and jet instability of a microswimmer suspension
L. Jibuti et al. (2014) Phys. Rev. E, 90, 063019
12
Perspectives
together with Philippe Peyla (Prof. UJF)
Experimental evidence of clustering?
Effect of hydrodynamic interactions on the dispersion of µswimmers
numerics
Johannes Greber PhD
(Bayreuth)
13
experiments
Matthieu Martin PhD
Thank you
People: "
Philippe Peyla, Prof. "
Michaël Garcia, PhD (2009-‐2013) Levan JibuC, PhD (2008 – 2011) Stefano BerC, Post-‐Doc 2008 Xabel Garcia, Master 2011 MaKhieu MarCn, PhD (2014-‐ ) "
Prof. W. Zimmermmann Bayreuth Johannes Greber, PhD "
"
Fundings:
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