Microscale instability and mixing
in driven and active suspensions
Michael Shelley, Applied Math Lab, Courant Institute
Collaborators:
Lisa Fauci
Christel Hohenegger
Eric Keaveny
David Saintillan
Joseph Teran
Becca Thomases
Tulane
CIMS
CIMS
UIUC
UCLA
UC-Davis
Dynamics and interactions of
micro-structure in complex fluids
ƒ Dynamics of non-Newtonian fluids
ƒ Reinforced composite
materials
ƒ Biological locomotion
ƒ Elastic “turbulence”
& low Re mixing
B. subtilis – one and many
C. Dombrowski et al ’05, 07
Groisman & Sternberg ’00, ’01, …
ƒ Microfluidic rectifiers
Groisman, Enzelberger, & Quake ‘03
I-SA phase trans -- PPM
microfluidic rectifier – Groisman & Quake
microscale mixing – Groisman & Steinberg
Experiments: V. Steinberg & A. Groisman
Viscoelastic fluid – Elastic “turbulence” - Efficient mixing
Rotating plates (Low Re, “High” Wi)
Mixing in micro channels
Arratia et al, PRL 2006
Elastic 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.
−∇p + Δu = − β ∇ ⋅ σ p − f and ∇ ⋅ u = 0
Wi σ p ∇ = −(σ p − I )
∇
• σp =
Dσ p
Dt
momentum and mass balance
transport and dissipation
of polymer stress
− (∇u ⋅ σ p + σ p ⋅∇u T ) Upper convected time derivative
τp
• Wi = ; Weissenberg number ratio of polymer relax. time
to flow time-scale
τf
•β =
Gτ f
τf =
μ
ρ LF
μ = solvent viscosity; F = external force scale
; coupling strength τ = polymer relaxation time; G = background poly. stress
p
μ
Gτ p
polymer viscosity
• β ⋅ Wi =
=
solvent viscosity
μ
Material constant; fix to ½
as in expts of Arratia et al
Properties:
1
(1) Has decaying "strain" energy: E = ∫ tr ( σ p - I )
2
2
−1
−1 ⎡
E + Wi E = 2 β − ∫ ∇u + ∫ u ⋅ f ⎤
⎣
⎦
(2) But lacks of scale dependent dissipation:
∂σˆ p
= L(kˆ ) σˆ p + P (kˆ ) fˆ
∂t
with kˆ = k/ | k |
(3) Polymer stress tensor:
Use the Fourier transform
to solve the linearized
problem
σ p = ν fr = C rr
Assume linear
Hooke’s law for
bead forces
is s.p.d.
(4) Existence of large-data solutions is unknown, even in 2d
Simulations: De-aliased Fourier based spectral method;
second order time stepping.
Vorticity field for Newtonian fluid
⎛ −2sin x cos y ⎞
f =⎜
⎟
⎝ 2 cos x sin y ⎠
Four – roll mill geometry
6
With Newtonian fluid
yields
5
⎛ sin x cos y ⎞
u=⎜
⎟
⎝ − cos x sin y ⎠
4
3
Creates hyperbolic points
in background flow
ala Arratia et al., PRL 2006
2
1
0
Background force
0
1
2
3
4
5
Thomases & Shelley PF 2007
6
Also Berti et al ’08, Xi & Graham ‘09
Becherer, Morozov, van Saarloos ’08, 09
Evolution for
Wi=0.3
Wi=0.6
Wi=5.0
initial stress:
σ p (t = 0) = I
ω→
tr(σ p ) →
divergence
slices of
→
11
σp
smooth
cusp
Local Model – fix strain-rate α – determined by flow -- and advect
stress field by local straining velocity
u = (α x, −α y ); ϕ =σ ; t → Wi ⋅ t ; ε = α ⋅ Wi
11
p
ϕt + ε xϕ x − ε yϕ y + (1 − 2ε )ϕ − 1 = 0
General solution :
1
ϕ=
+ e(2ε −1)t H11 ( xe −ε t , yeε t )
1-2ε
Relevant solution :
H11 (a, b) = h(b) with h(b) ~| b |q as | b |→ ∞
Why? Choose q to eliminate long time t – dependence
⇒q =
1 − 2ε
ε
1− 2ε
steady states also studied
1
by Rallison & Hinch ‘88
⇒ ϕ |t →∞ =
+C | y| ε
1 − 2ε
and M. Renardy ‘06
ε = α (Wi ) Wi ; q =
1 − 2ε
ε
ε1 = 1/ 3 Wi1 ≈ 0.5
ε 2 = 1/ 2 Wi2 ≈ 0.9
q = -1
measured at central
hyperbolic point
Note ε < 1 implies q > -1 so
the stress is integrable.
q>1 0<q<1
cusp in
stress
q<0
Divergence in
stress
Mixing and Symmetry-Breaking: Thomases & Shelley ’09
The SOB system is also unstable to symmetry-breaking;
see Poole et al ’07, Xi & Graham ‘08
σ p ( 0 ) − I ~ O (10−2 )
ω ( 0 ) − ωStokes
6
−3
x 10
6
5
10
5
y
4
5
4
3
3
0
2
2
1
−5
1
0
0
1
2
3
x
4
5
6
0
0
2
4
6
Full disclosure: Small amount of polymer stress diffusion added to control
gradient growth
Long-time behavior with increasing Wi:
Wi=0.5
Wi=5
Wi=6
Wi=6.0
Wi=10
relaxation to
symmetric state
Slow relaxation
to asymmetric state
Arratia et al ‘06
Persistent
oscillations
ω
tr(σ p )
2 primary frequencies
Smaller Wi:
symmetry breaking, little mixing
Wi=6, t=2000
Larger Wi:
• multiple frequencies
of oscillation
• robust GRS of viscoelastic flows
• well-mixed fluid outside of GRS
Need new experiments,
stability analyses.
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-rotating
rotors driving a PAA
viscoelastic solution
w. Bin Liu, J. Zhang
Collective dynamics of active suspensions (bacterial baths)
150 μm
R. Goldstein, J. Kessler, and coworkers
Observation: meandering jet and vortices of scale 50-100 μm, speeds 50-100 μm/sec in jets
Scale of B. subtilis ~ 4 μm (plus tail); swimming speed 20-30 μm/sec
• 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?
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
•…
Slender-body swimmer driven by surface stress
Saintillan & Shelley PRL 2007 , motivated by Volvox model of Short, Goldstein, et al;
(simulation of multi-V interactions by Kanevsky, Shelley, Tornberg, ’08)
Surface tractions:
prescribed
unknown
Integrated traction (force per unit length):
prescribed
Force and torque balances:
unknown
Single particle flow fields
Saintillan & Shelley, PRL ‘07
2500 swimming “pushers” in
periodic box of dimensions
10 x 10 x 3
effective volume fraction
n (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 by
Simha & Ramaswamy ‘02
Loss of global orientational order:
order parameters:
1
2
S1 = p ⋅ z & S 2 = 3 p ⋅ z − 1
2
(
Emergence of large-scale dynamical flow
as in Dombrowski et al, Hernandez-Ortiz et al
)
A kinetic theory for active suspensions
S&S, PRL ‘08, PF ‘08
Pose Fokker-Planck equation for distribution function Ψ ( x, p, t ) of particle
center of mass x and (unit) swimming director p (rod theory, Doi & Edwards, '86) :
1
Ψ t + ∇ x i( xΨ ) + ∇ p i( pΨ ) = 0 with
dVx ∫ dS p Ψ = n
∫
V
w. "particle" fluxes x = U 0 p + u ( x, t ) − ∇ x ( D ln Ψ )
p = ( I − pp )( γ E + W ) p − ∇ p ( d ln Ψ )
Background fluid velocity:
∇q − Δu = ∇i Σ a and ∇iu = 0
driven by active swimming stress (Kirkwood theory; Batchelor '70):
Σ a ( x, t ) = σ 0 ∫ dS p Ψ ( x, p, t ) [pp − I / 3]
Pushers: σ 0
< 0; Pullers: σ 0 > 0
Important d'less parameters: U 0 → 1, σ 0 → α = O (1) , L → L / lc
A useful special case
Neglecting diffusion, consider a locally aligned suspension:
Ψ ( x,p, t ) = c ( x, t ) δ ( p − n ( x, t ) )
Setting D=d=0 The full kinetic equations reduce exactly to:
∂c
+ ∇ x i( ( n + u ) c ) = 0
∂t
∂n
+ ( n + u )i∇ x n = ( I - nn ) ∇ x u n
∂t
with ∇ x q − ∇ 2x u = −∇ x i Σ a ,
and particle
extra stress
∇x i u = 0
Σ p = α c ( x, t )( nn − I / 3)
(preserves nin = 1)
Stability analysis II: uniform isotropic case
A nearly isotropic uniform suspension:
i (k ⋅x + λt ) ⎤
1 ⎡
1 + ε Ψk (p ) e
Ψ ( x, p , t ) =
⎢
⎥⎦
4π ⎣
Derive relation:
3iαγ
kˆ ⋅ p
Ψk = −
p ⋅ F ⎡⎣ Ψ k ⎤⎦
2
2π λ + ik ⋅ p + Dk
where
(
ˆˆ
F ⎡⎣ Ψ k ⎤⎦ = I − kk
) ∫ dS p ' ( kˆ ⋅ p ') Ψ
p'
k
(1)
( p ')
Applying F operator to (1), and evaluation of the integral,
yields the eigenvalue relation:
3iαγ
2k
a − 1⎤
⎡ 3 4
4
2
2
a
a
a
a
a
i
Dk
2
log
λ
1
w.
/k
−
+
−
=
=
+
(
)
(
)
⎢⎣
⎥
a + 1⎦
3
α = −1 (pushers), γ =1
Re λ
( rods ) , D = 0
Im λ
Suspensions of pushers are unstable at long wavelengths.
pullers are stable
(eigen-solutions do not describe small-scale behavior – Hohenegger & Shelley ’09)
Eigenfunctions:
ck = ∫ dS p Ψ k ( p ) = 0
Σka = kk ⊥ + k ⊥k
no concentration fluctuations
in linear theory.
active stress eigen-modes are
shear-stresses.
Non-linear simulations (2-d)
ƒ Initial condition:
Concentration field c
Mean director field n
Long-time dynamics: velocity field
concentration bands
ƒ The concentration bands are located inside shear layers.
ƒ 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.
Configurational entropy:
⎛ Ψ ⎞ ⎛ Ψ ⎞
S = ∫ dVx ∫ dS p ⎜
⎟ ln ⎜
⎟
⎝ Ψ0 ⎠ ⎝ Ψ0 ⎠
≥ 0; =0 only for Ψ ≡ Ψ 0
2
dS
3
1
2
a
⎡
⎤
dV
dV
dS
D
d
:
ln
ln
=
−
Ψ
∇
Ψ
+
∇
Ψ
E
Σ
⇒
x
x
p
x
p
∫
⎢⎣
⎥⎦
Ψ0 ∫
dt α Ψ 0 ∫
But ... from the momentum equations:
Pa ( t ) = − ∫ dVx E : Σ = 2∫ dVx E : E
a
rate of viscous dissipation balances the
active power input Pa ( t ) of the swimmers
2
1
−6
dS
2
2
⎡
⎤
ln
ln
Ψ
E
=
−
Ψ
∇
Ψ
+
∇
⇒
dV
dV
dS
D
d
x
x
p
x
p
∫
⎢⎣
⎥⎦
Ψ0 ∫
dt α Ψ 0 ∫
Pullers (α >0): fluctuations, as measured by S , will dissipate.
Pushers (α < 0): the input power increases fluctuations,
until limited by diffusive processes.
Pa(t)
total entropy
pushers
entropy growth saturates;
system in statistical equil.
pullers
fluctuations in puller
suspension quickly
dissipate
Active swimmer power density:
p ( x, t ) = − ∫ dp (α p T E ( x, t ) p ) Ψ ( x,p, t ); Pa ( t ) = ∫ dVx p ( x, t )
For Pa ( t ) to be positive w. α <0, expect p to be aligned with extensional axis of E
Mixing by active suspensions
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.
From Mathew et al '07:
mixing norm":
|| s ||H −1/ 2
Conclusions
ƒ Aligned suspensions of swimming rods destabilize as a result of hydrodynamic
interactions.
ƒ The chaotic flow fields arising in suspensions of swimming rods are dominated
locally by near uniaxial extensional (pushers) and compressional (pullers) flows.
ƒ 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.
ƒ Dynamics in thin liquid films are characterized by a strong particle migration
towards the gas/liquid interfaces.
ƒ Kinetic theory predicts instabilities for both aligned and isotropic suspensions.
In the isotropic case, the instability is driven by the particle shear stress.
ƒ Non-linear simulations show that active suspensions evolve toward nonuniform 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.