COLLECTIVE HYDRODYNAMICS OF SWIMMING
MICROORGANISMS
Timothy J Pedley
DAMTP, University of Cambridge
APS-DFD, Salt Lake City, 2007
Copyright T. Pedley, 2007
OUTLINE
1 Bioconvection
•
Mechanisms
•
•
•
2
•
•
•
•
Model
Instability of a uniform suspension
Predictions for algae, bacteria, spermatozoa
Concentrated suspensions
Coherent structures
Simulations
Pairwise interactions of squirmers
Diffusion or aggregation?
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Particle Tracking (projection)
The result of measuring the velocities. Each cross corresponds to velocity
detected. Number of tracks analysed is 359, number of points displayed is
2168. Dense cloud below the origin corresponds to sedimenting particles.
Vladimirov, V.A. et al (2000) Marine & Freshwater Res. 51: 589-600.
Copyright T. Pedley, 2007
Cell Conservation
Dn
= −∇ ⋅ ( nVc + J r )
Dt
[+ birth, death, etc]
where
Vc = mean cell swimming velocity,
Jr
= flux due to random cell swimming (chemokinesis) [ =
− D ⋅ ∇n ?]
Both consequences of cell swimming, directed and random respectively
Copyright T. Pedley, 2007
Cell Conservation Equation
∂n
= −∇ ⋅ ⎡⎣ n ( u + Vc ) − D ⋅∇n ⎤⎦
∂t
Swimming
diffusion
The random swimming behaviour can be quantified in terms of a
probability density function f ( p) for the cell swimming direction p
[and, in principle, another one for swimming speed V ]. Then
s
ensemble averages are defined by
⋅⋅ = ∫
2
⋅
⋅
f
(
p
)
d
p
∫
where the integral is taken over the unit sphere in
p -space.
Thus Vc = Vs p = Vs p
But what is f ( p) ?
And how do we calculate
D?
Copyright T. Pedley, 2007
Rational model (?)
P&K (JFM, 212, 155-182, 1990)
Intrinsic randomizing influences are balanced by
the tendency to return to Pˆ ; f ( p) satisfies a
Fokker-Planck equation:
⎛ ∂f ⎞
2
&
⎜ + ⎟ ∇ p ⋅ ( pf ) = Dr ∇ p f
⎝ ∂t ⎠
(probability conservation in p − space)
(treats the suspension as analogous to a colloidal
suspension subject to Brownian rotation)
Copyright T. Pedley, 2007
Bottom heavy algae will rotate as a result of the torque balance
p
k
h
g
grav
But note that sedimenting cells
with uniform density but asymmetric
shape will also rotate
(e.g. spermatozoa)
visc
(Katz & Pedrotti 1977; Roberts & Deacon 2002)
Copyright T. Pedley, 2007
Torque balance for algae
.
p
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Algal cell
“Puller”
S = + Tl
Sperm
Bacterium
“Pushers”
S = - Tl
[Note that pushers tend to be head-heavy, not bottom-heavy]
Copyright T. Pedley, 2007
- 2 DR
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PARAMETER VALUES
C nivalis
_
β
B subtilis
(upswimming)
-3
B subtilis
(sedimenting)
?
-1.4 x 10
-3
Spermatozoa
(sedimenting)
0.2
5 x 10
-0.02
S
1.4
-3400
-3400
-1.2 x 10
g
11
20
-37
-6.9 x 10
ν
250
2750
2750
_
_
-5
0
-7
_
Note:
_
ν >> 1
Also
_
DR = DR D
___
n
n0
1200
= 1/6
2
VC
Hence
_
_
β + 2 DR > 0
in all cases
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For B subtilis this would be about
10 9 cells per ml
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Collective behaviour:
Bioconvection - bottom-heavy algae
(Chlamydomonas nivalis)
Bioconvection - oxytactic bacteria
(Bacillus subtilis)
“Whirls and jets” - bacteria, in 3D or 2D configurations
(Ray Goldstein’s movie)
Copyright T. Pedley, 2007
Question - can the observed behaviour of the bacteria
be explained in terms of hydrodynamic
interactions alone, not chemical or other
‘sensory’ signals?
Copyright T. Pedley, 2007
Recent modelling of collective behaviour of small swimmers:
J-P Hernandez-Ortiz, C G Stoltz & M D Graham
Transport and collective dynamics in suspensions of confined
s
swimming particles
(PRL 95, 204501, 2005)
*
D Saintillan & M Shelley
Orientational order and instabilities in suspensions of self-locomoting
r
rods
(PRL 99, 058102, 2007)
T Ishikawa, M P Simmonds & T J Pedley
Hydrodynamic interaction of two swimming model micro-organisms
(J Fluid Mech, 568:119-160, 2006)
T Ishikawa & T J Pedley
- The rheology of a semi-dilute suspension of swimming model
micro-organisms (J Fluid Mech, 2007)
- Diffusion of swimming model micro-organisms in a dilute suspension
(J Fluid Mech, 2007)
Copyright T. Pedley, 2007
Saintillan & Shelley
Model the organism as a long prolate spheroid, with a given
tangential shear stress over part of the surface, the rear for a
pusher, the front for a puller. An “elongated squirmer”.
shear stress
no slip
p
They use resistive force theory, which ignores near-field
hydrodynamic interactions, and simulate a 3D (fairly shallow)
suspension with periodic boundary condiditons. (They state
that putting in the near-field interactions makes little difference.)
Also, pushers tend to align with neighbours; pullers don’t.
Copyright T. Pedley, 2007
Copyright T. Pedley, 2007
29 June 2004
Fluid dynamical interaction of two
swimming model micro-organisms
T. Ishikawa*1 and T. J. Pedley*2
*1
Tohoku University, Sendai, Japan
*2 University of Cambridge, UK
Copyright T. Pedley, 2007
2. Modelling a micro-organism
Envelope model
Ciliate
The model micro-organism will be assumed to propel
itself by generating tangential velocities on its surface.
squirmer
by J.R.Blake (1971)
B1
us = V1 (cos θ ) + B2V2 (cos θ )
3
2sin θ ′
Vn =
Pn
n ( n + 1)
θ
a micro-organism is modeled
as a rigid sphere with
squirming surface velocity P : Legendre polynominal of order n
n
Bn : coefficient of mode n
Copyright T. Pedley, 2007
β = B2/B1
p
B1 ~ swimming speed
B2 ~ stresslet strength
(a) β = 1
β > 1 : puller
β < 1 : pusher
p
(b) β = 5
Figure 1. Velocity vectors relative to the translational velocity vector of a squirmer. Uniform flow of speed 1.0, in dimensionfree form, coming from far right. The scales of vectors in (a) and (b) are the same.
Copyright T. Pedley, 2007
A Closer View
Fluorescence
PIV for Volvox
050504EIGHT.AVIvv
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1. Interaction between two squirmers (Simulation)
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θ1= π
Case1: θ1 = π
Two squirmers are
facing each other.
e1
lx=10
θ2=0
e2
Sample movie with
dly = 1 and β =5.
e2
e2
dly|ini=1,2,3,5,10,
Copyright T. Pedley, 2007
Case2: θ1 = π/2
e1 θ1= π /2
10
Two squirmers are
crossing in a right angle.
10
θ2=0
Sample movie with
dly = -1 and β =5.
Sample movie with
dly = -5 and β =5.
e2
e2
e2
dly|ini=-1,-2,-3,-5,-10
Copyright T. Pedley, 2007
Numerical Results : Effect of 3D orientation
When θ1 = π/4 and β =5, there is a stable
condition if we restrict their motion in 2D.
Sample movie with dly = -1.
In case with a gap in
z direction, there is
no stable condition.
Sample movie
with dly = -1
and dlz|ini= -0.1
e1
θ1= π /4
24.14
10
θ2=0
e2
x
z
y
dly|ini= -1
Copyright T. Pedley, 2007
1
2
Compute the motion of many
interacting squirmers, using the
database of pairwise interactions
Use full Stokesian Dynamics
Copyright T. Pedley, 2007
Simulation shows aggregation in 2D, not in 3D
2D-c01
2D-c05
3D-c01
3D-c04
Bottom-heavy
2D-bh-c01
2D-bh-c05
3D-bh-c01
Copyright T. Pedley, 2007
2D – aggregation or alignment
3D - diffusive spreading?
- some aggregation as well
(see paper by J T Locsei)?
Copyright T. Pedley, 2007
Diffusion tensor, D
∞
D = ∫ V (t )V (t − t ' ) dt ' = lim
T
t →∞
0
1
≅
MN
[r (t + t0 ) − r (t0 )][r (t + t0 ) − r (t0 )]
M
N
∑∑
2t
[ri (t + tk ) − ri (tk )][ri (t + tk ) − ri (tk )]
2t
k =1 i =1
∞
D = ∫ Ω (t ) Ω (t − t ' ) dt ' = lim
R
[ω(t + t0 ) − ω(t0 )][ω(t + t0 ) − ω(t0 )]
t →∞
0
2t
ω = ∫ Ωdt
In case of non-BH squirmers, their orientation is isotropic.
Therefore, the diffusion tensor is also isotropic. Hence we
discuss only the following quantities:
DT =
DxxT + D Tyy + DzzT
3
, DR =
DxxR + D yyR + DzzR
3
Copyright T. Pedley, 2007
Diffusion tensor, D
diffusion coefficient
101
100
translational
rotational
y=ax
10-1
10-2
10-3 -1
10
100
101
time interval
102
Copyright T. Pedley, 2007
SCALING
A squirmer’s velocity changes in direction, not (much) in
magnitude, due to interactions with others.
In a semi-dilute suspension such interactions can be considered
as pairwise collisions, interspersed with more-or-less straight runs.
Hence model the system as molecules in a gas (kinetic theory):
a random walk of runs separated by near collisions.
Copyright T. Pedley, 2007
Speed of squirmer ~ U (constant)
Distance travelled between collisions ~ L mfp
Time between collisions ~ t mfp
= Lmfp/U
Volume swept out in this time V
= π a 2 U t mfp
This will contain one other squirmer if
volume fraction
c = 4 π a /3V
3
i.e. t mfp = 4a/(3Uc) = 4/(3c) in dimensionless terms
Copyright T. Pedley, 2007
But the time required for a significant change of orientation
Will be the effective duration of a collision, independent of
the number of collisions.
Copyright T. Pedley, 2007
T
D /D
T
inf
10 0
10 -1
β=5
c= 0.1
c= 0.075
c= 0.05
c= 0.025
10 -2
10 -2
10 -1
Δ t / Δ t mf p
10 0
10 1
(a) translational diffusivity replotted against Δt /Δtmfp
R
D /D
R
imf
10 0
β= 5
c= 0.1
c= 0.075
c= 0.05
c= 0.025
10 -1
10 -2
10 -1
Δ t / Δ t mf p
10 0
10 1
(b) rotational diffusivity replotted against Δt / Δtmfp
Figure 9. The results of figure 6 are replotted by using different characteristic time Δtmfp and Δtmps. The vertical
axis is normalized by Dinf.
Copyright T. Pedley, 2007
R
D /D
R
inf
10 0
β= 5
10 -1
10 -1
c=0.1
c=0.075
c=0.05
c=0.025
10 0
Δt
10 1
(c) rotational diffusivity replotted against normal Δt
Figure 9. The results of figure 6 are replotted by using different characteristic time Δtmf. The vertical axis is
normalized by Dinf.
Copyright T. Pedley, 2007
Magnitudes?
Random walk model:
DTinf
= U L mfp /3 ~ t mfp /3 = 4/(9c)
DRinf =
2
<
Δ
ω
>
_______
6t
∼ t -1
mfp
~ c
mfp
Copyright T. Pedley, 2007
β =5
y=0.24/x
DTinf
101
100 -2
10
10-1
c
(a) translational diffusivity
10 -1
β =5
D
R
inf
y=0.85x
10 -2 -2
10
c
10 -1
(b) rotational diffusivity
Figure 10. Correlation between Dinf and c (β = 5). Dinf is the converged diffusivity after a sufficiently long time.
Copyright T. Pedley, 2007
Bottom-heavy squirmers ?
They all have a preferred swimming direction – upwards
Mean free path not significant – squirmer interaction
dominated by configuration of surrounding squirmers, which
changes on a length scale proportional to the particle
spacing:
L
=
mps
t mps ~
3
4πa
_____
3c
( )
1/3
(4π/3c)
1/3
Copyright T. Pedley, 2007
Gbh =10, β =5
c=0.1
c=0.075
c=0.05
c=0.025
10-1
D
T
xx
/D
T
xx,inf
100
10-2
10-2
10-1
100
101
Δ t / Δ t mps
(a) translational diffusivity DTxx
R
yy
/D
R
yy,inf
10 0
D
G bh=10, β =5
c=0.1
c=0.075
c=0.05
c=0.025
10 -1
10 -2
10 -1
Δ t / Δ tmps
10 0
10 1
(b) rotational diffusivity DRyy
Figure 18. The diffusivities are plotted in terms of Δt / Δtmps. The vertical axis is normalized by Dinf.
Copyright T. Pedley, 2007
β =5
0.1 Gbh=10,
xx component
DRinf
0.08
yy component
y=bx
0.06
0.04
0.02
0
0.05
c
0.1
(b) bottom-heavy squirmers (Gbh = 100)
Figure 21. Correlation between DRinf and c (Gbh = 100, β = 5). DRinf is the converged rotational diffusivity after a
sufficiently long time.
Copyright T. Pedley, 2007
2D – aggregation or alignment
3D - diffusive spreading?
- some aggregation as well
(see paper by J T Locsei)?
Copyright T. Pedley, 2007