Microscale instability and mixingin driven and active suspensions
Michael Shelley, Applied Math Lab, Courant Institute
Collaborators:Lisa Fauci TulaneChristel Hohenegger CIMSEric Keaveny CIMSDavid Saintillan UIUCJoseph Teran UCLABecca Thomases UC-Davis
Dynamics and interactions of micro-structure in complex fluids
Dynamics of non-Newtonian fluids Reinforced composite materialsBiological locomotionElastic “turbulence” & low Re mixingGroisman & Sternberg ’00, ’01, …
Microfluidic rectifiersGroisman, Enzelberger, & Quake ‘03
B. subtilis – one and manyC. Dombrowski et al ’05, 07
microscale mixing – Groisman & Steinbergmicrofluidic rectifier – Groisman & Quake
I-SA phase trans -- PPM
Experiments: V. Steinberg & A. GroismanViscoelastic fluid – Elastic “turbulence” - Efficient mixing
(Low Re, “High” Wi)Rotating plates
Mixing in micro channels
Arratia et al, PRL 2006Elastic fluid instabilities near hyperbolic points
Stokes-Oldroyd-B ( Re<<1 )• model of a “Boger” elastic fluid (normal stresses, no shear thinning)• derives from a microscopic, dilute theory of polymer coils• one of the standard viscoelastic flow models; Little known
about large data solutions.
0
( )
andp
Wi
β∇
−∇ + Δ = − ∇⋅ − ∇ ⋅ =
= − −p
p p
u σ f u
σ σ I
Upper convected time derivative
transport and dissipationof polymer stress
momentum and mass balance
T
Weissenberg number
coupling strength
polymer viscosity so
( )
;
lvent visco t
;
si y
p
f
f
p
DDt
Wi
G
GWi
ττ
τβ
μτ
βμ
∇•
•
= − ∇ ⋅ + ⋅∇
=
=
⋅ =
•
• =
pp p p
σσ u σ σ u
ratio of polymer relax. timeto flow time-scale
f LFμτ
ρ=
Material constant; fix to ½ as in expts of Arratia et al
solvent viscosity; = external force scale = polymer relaxation time; = background poly. stressp
FG
μτ=
( )21 1
1= tr2
Has decaying "strain" energy:
2
E
E Wi E β− − ⎡ ⎤+ = − ∇ + ⋅⎣ ⎦
∫
∫ ∫
pσ - I
u u f
ˆ ˆˆ ˆˆ( ) ( )
ˆ with
ppL P
t∂
= +∂
=
σk σ k f
k k/ | k |
Use the Fourier transformto solve the linearizedproblem
But lacks of scale dependent dissipation:
Properties:(1)
(2)
(4)
(3)
Existence of large-data solutions is unknown, even in 2d
Polymer stress tensor:
is s.p.d.
Cν= =pσ fr rrAssume linear
Hooke’s law for bead forces
Simulations: De-aliased Fourier based spectral method; second order time stepping.
0 1 2 3 4 5 60
1
2
3
4
5
6
Background force
With Newtonian fluid yields
2sin cos2cos sin
x yx y
−⎛ ⎞= ⎜ ⎟⎝ ⎠
f
Vorticity field for Newtonian fluid
Four – roll mill geometry
sin coscos sin
x yx y
⎛ ⎞= ⎜ ⎟−⎝ ⎠
u
Creates hyperbolic pointsin background flow ala Arratia et al., PRL 2006
Also Berti et al ’08, Xi & Graham ‘09Becherer, Morozov, van Saarloos ’08, 09
Thomases & Shelley PF 2007
tr( )p →σ
Wi=5.0Wi=0.6Wi=0.3
ω →
( )
Evolution for initial stress:
0p t = =σ I
smooth cusp
divergence
11
slices of
p
→σ
Local Model – fix strain-rate α – determined by flow -- and advectstress field by local straining velocity
11pu = ( , ); = ; ;
(1 2 ) 1 0 t x y
x y t Wi t Wi
x y
α α ϕ σ ε α
ϕ ε ϕ ε ϕ ε ϕ
− → ⋅ = ⋅
+ − + − − =
(2 1)11
1= ( , )1-2
t t te H xe yeε ε εϕε
− −+
General solution :
11 with ( , ) ( ) ( ) ~| | |as |qH a b h b h b b b= →∞Relevant solution :
Why? Choose q to eliminate long time t – dependence 1 21 2 1| | |
1 2tq C yε
εε ϕε ε
−
→∞
−⇒ = ⇒ = +
−
steady states also studiedby Rallison & Hinch ‘88
and M. Renardy ‘06
( ) 1 2;Wi Wi q εε αε−
= =
q > 1 0 < q < 1cusp in stress
q < 0Divergence in stress
1 1
2 2
1/ 3 0.51/ 2 0.9
WiWi
εε= ≈= ≈
Note ε < 1 implies q > -1 so the stress is integrable.
measured at centralhyperbolic point
q = -1
Full disclosure: Small amount of polymer stress diffusion added to control gradient growth
x
y
0 1 2 3 4 5 60
1
2
3
4
5
6
Mixing and Symmetry-Breaking: Thomases & Shelley ’09 The SOB system is also unstable to symmetry-breaking;see Poole et al ’07, Xi & Graham ‘08
0 2 4 60
1
2
3
4
5
6
−5
0
5
10
x 10−3
( ) ( )2p 0 ~ 10O −−σ I ( )0 Stokesω ω−
Long-time behavior with increasing Wi:Wi=0.5 Wi=5 Wi=6 Wi=10
Slow relaxationto asymmetric state
Persistentoscillations
Wi=6.0
relaxation tosymmetric state
Arratia et al ‘06
ptr(σ )ω
2 primary frequencies
Larger Wi:• multiple frequencies
of oscillation• robust GRS of viscoelastic flows• well-mixed fluid outside of GRS
Need new experiments,stability analyses.
Wi=6, t=2000
Smaller Wi:symmetry breaking, little mixing
Update:(1) 1 of 10 simulations using random amplitude/phase initial
perturbations for polymer stress.
(2) What if the number of vortex cells is increased?(3) Now investigating in a new expt’l rig in the AML
16 counter-rotatingrotors driving a PAAviscoelastic solutionw. Bin Liu, J. Zhang
Collective dynamics of active suspensions (bacterial baths)
R. Goldstein, J. Kessler, and coworkers150 μm
• A complex fluid driven by dynamics of its microstructure –many body interactions mediated by fluid.
• collective behavior leads to strong mixing.• Role of body geometry? Emergence or role of orientational ordering?• Competition of hydrodynamic coupling vs. attractive gradients?
Observation: meandering jet and vortices of scale 50-100 μm, speeds 50-100 μm/sec in jetsScale of B. subtilis ~ 4 μm (plus tail); swimming speed 20-30 μm/sec
Some of the experiments:• Wu & Libchaber ’00:“brownian” motion of test particles in bacterial baths.• Dombrowski et al ’04: large-scale flow structures (many body lengths).• Kim & Breuer ’04, enhanced mixing using bacteria in micro-fluidic device.• Paxton et al, ’04, fabricated chemically-driven nano-rod-swimmers.• Dreyfus et al, ’05, bio-mimetic swimmers driven by magnetic fields• Short et al, 06, expts and model of Volvox swimming.• Sokolov et al, ’07, expts on concentration dependencies in thin films.•…
Some of the theory:• bioconvection: Childress & Spiegel, Pedley and many others• Simha & Ramaswamy ’02: predict instability of long-wave oriented states• Hernandez-Ortiz et al, ’05: simulations of force-dipole suspensions show
emergence of large-scale structures• Toner et al, ’05: models of flocking.• Sambelashvili, Lau, & Cai ’07, ordering of 2d rod locomotors by local
steric interactions• Pedley, Ishikawa et al, interactions of squirmers (specified surface velocity)• Saintillan & Shelley, 07, ‘08, particle simulations, kinetic theory of moving rod suspensions• Keaveny & Maxey, ’08, theory and simulations for bio-mimetic swimmers• Kanevsky et al, ’09, simulations of interacting stress-actuated swimmers•…
Surface tractions:
prescribed unknown
Integrated traction (force per unit length):
prescribed unknown
Force and torque balances:
Slender-body swimmer driven by surface stressSaintillan & Shelley PRL 2007 , motivated by Volvox model of Short, Goldstein, et al;(simulation of multi-V interactions by Kanevsky, Shelley, Tornberg, ’08)
Single particle flow fields
Saintillan & Shelley, PRL ‘072500 swimming “pushers” in periodic box of dimensions
10 x 10 x 3effective volume fractionn (L/2)3 = 1; n = # density(strongly interacting)
All initially aligned in the z direction – nematic order –with randomized positions
10
Spatially organized instability destroys long-range order. Predicted bySimha & Ramaswamy ‘02
( )21 2
Loss of global orientational order:order p
1 3 1
arameters:
& 2
S S= ⋅ = ⋅ −p z p z
Emergence of large-scale dynamical flowas in Dombrowski et al, Hernandez-Ortiz et al
( )
( ) ( )t
Pose Fokker-Planck equation for distribution function of particle
center of mass and (unit) swimming director (rod theory, Doi & Edwards,
, ,
'86) :
with
w. "
1 0 x p x p
t
dV dS nV
Ψ
Ψ +∇ Ψ +∇ Ψ = Ψ =∫ ∫
x p
x p
x pi i
( ) ( )( )( ) ( )
( ) ( )[ ]
0
0
Background fluid velocity:
driven by active swimming stress (Kirkwood theory; Batchelor '70):
:
particle" fluxes , ln
ln
0
, , , /
and
3
x
p
a
ap
Pushers
U t D
d
q
t dS t
γ
σ
= + −∇ Ψ
= − + −∇ Ψ
∇ −Δ = ∇ ∇ =
= Ψ −∫
x p u x
p I pp E W p
u Σ u
Σ x x p pp I
i i
( )
0 0
0 0
: ;
Important d'less parameter
0 0
U: 1 , /s 1, c
Pullers
O L L l
σ σ
σ α
< >
→ → = →
A kinetic theory for active suspensions S&S, PRL ‘08, PF ‘08
A useful special caseNeglecting diffusion, consider a locally aligned suspension:
Setting D=d=0 The full kinetic equations reduce exactly to:
with 2 , 0ax x x xq∇ −∇ = −∇ ∇ =u Σ ui i
( )( ) , / 3p c tα= −Σ x nn I
( )( )
( ) ( ) (pr
0
eserv ) s 1e
x
x x
c ct
t
∂+∇ =
∂∂
+ ∇ = ∇ =∂
n + u
n n + u n I - nn un n n
i
i i
and particleextra stress
( ) ( ) ( )( ), , ,t c t tδΨ = −x,p x p n x
Stability analysis II: uniform isotropic case
( ) ( ) ( )
( ) ( ) ( )
2
'
A nearly isotropic uniform suspension:
Derive relation:
where
App
1, , 14
ˆ3 (1) = 2
ˆ ˆ ˆ'
lyi g
'
n
'
p
i tt e
ii Dk
dS
λε
π
αγπ λ
⋅ +⎡ ⎤Ψ = + Ψ⎢ ⎥⎣ ⎦
⋅ ⎡ ⎤Ψ − ⋅ Ψ⎣ ⎦+ ⋅ +
⎡ ⎤Ψ = − ⋅ Ψ⎣ ⎦ ∫
k
k k
k k
k xx p p
k p p Fk p
F I kk p k p p
( ) ( )3 4 2 2
operator to , and evaluation of the integral, yields the eigen
3 4 12 log 1 - /2 3
value relation:
w. 1
(1)
i aa a a a a i Dk kk aαγ λ−⎡ ⎤− + − = = +⎢ ⎥+⎣ ⎦
F
( )Eigenfunctions: 0
p
a
c dS
⊥ ⊥
= Ψ =
Σ = +∫k k
k
p
kk k k
( )(pushers),1 =1 rod 0s , Dα γ= − =
Reλ Imλ
Suspensions of pushers are unstable at long wavelengths.pullers are stable
(eigen-solutions do not describe small-scale behavior – Hohenegger & Shelley ’09)
no concentration fluctuationsin linear theory.
active stress eigen-modes areshear-stresses.
Non-linear simulations (2-d)
Initial condition:
Concentration field c Mean director field n
Long-time dynamics: velocity field
concentration bands
The concentration bands are located inside shear layers.These shear layers become unstable, leading to the formation of vortices and to the
break-up of the bands, which then reform in the transverse direction.
( )
0 0
0
22
0 0
2 2
0 0
only for
But ... from the momentum
l
equations:
n
0; =03 1: ln ln
: 2
1 ln
:
6
x p
ax x p x p
aa x x
x x p x p
S dV dS
dS dV dV dS D ddt
P t dV dV
dS dV dV dS D ddt
α
α
⎛ ⎞ ⎛ ⎞Ψ Ψ= ⎜ ⎟ ⎜ ⎟Ψ Ψ⎝ ⎠ ⎝ ⎠≥ Ψ ≡ Ψ
⎡ ⎤= − Ψ ∇ Ψ + ∇ Ψ⎢ ⎥⎣ ⎦Ψ Ψ
= − =
−= − Ψ ∇ Ψ + ∇
Ψ Ψ
⇒
⇒
∫ ∫
∫ ∫ ∫
∫ ∫
∫
E Σ
E Σ E E
E2
( ): fluctuations, as measured by , will dissipate.( ): the input power increases fluctuations,
until limited by diffusive proc
ln
>0
esse0
s.
SPullersPushers
αα
⎡ ⎤Ψ⎢ ⎥⎣ ⎦
<
∫ ∫
Configurational entropy:
( ) rate of viscous dissipation balances the active power input of the swi rst mmeaP
total entropypushers
pullers
Pa(t)
( ) ( )( ) ( ) ( ) ( )( )
Active swimmer power density:
For to be positive w. , expect to be aligned with extensional a
, , , ; ,
<0 xis of a x
a
p t dp t t P t dV p t
P t
α
α
= − Ψ =∫ ∫Tx p E x p x,p x
p E
entropy growth saturates;system in statistical equil.
fluctuations in pullersuspension quicklydissipate
Efficient convective fluid mixing is achieved by stretching and folding of fluid elements during the formation and break-up of the concentration bands. After approximately 4 cycles, good mixing is achieved in the suspension.
Mixing by active suspensions
1/ 2
From Mathew '07:
mixing norm": || ||H
et al
s −
ConclusionsAligned suspensions of swimming rods destabilize as a result of hydrodynamic
interactions. The chaotic flow fields arising in suspensions of swimming rods are dominated
locally by near uniaxial extensional (pushers) and compressional (pullers) flows.At steady state, particle orientations show a clear correlation at short length
scales owing to the disturbance flow and to hydrodynamic interactions. This correlation results in an enhancement (or decrease) of the mean particle swimming speed.
Dynamics in thin liquid films are characterized by a strong particle migration towards the gas/liquid interfaces.
Kinetic theory predicts instabilities for both aligned and isotropic suspensions. In the isotropic case, the instability is driven by the particle shear stress.
Non-linear simulations show that active suspensions evolve toward non-uniform distributions as a result of these instabilities. More precisely, the shear stress instability causes the local polar alignment of the particles, which in turn results in the formation of concentration inhomogeneities.
