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Complex flows in microfluidic geometries
Anke Lindner, PMMH-ESPCI, Paris, [email protected]
Peyresq, May 29th – June 2nd 2017
10 mm
10mm
Casanellas, AL et al, Soft Matter, 2016
Motivation
Often difficult to characterize: fluid and flow are complex
Often small Reynolds numbers (small size, high viscosity)
10mm
Blood flow
Paper fabrication
1m
1mm
Lava flow
Food processing
Microfluidic model
systems
Some examples….. elastic flow instabilities
incre
asin
g flo
w ra
te
Laminar flow
Secondary flow
Unstable time dependent flow
solution of PEO 4Mio
100 microns
𝑳 = 𝟒. 𝟗𝝁𝒎 𝜸 = 𝟐. 𝟕𝟕𝒔−𝟏
𝑳 = 𝟔. 𝟖𝝁𝒎 𝜸 = 𝟐. 𝟔𝟏𝒔−𝟏
𝑳 = 𝟐𝟔. 𝟔𝝁𝒎 𝜸 = 𝟏. 𝟕𝟔𝒔−𝟏
𝑳 = 𝟑𝟑. 𝟓𝝁𝒎 𝜸 = 𝟏. 𝟒𝟔𝒔−𝟏
Flow
Stage
Jeffery orbit – Rigid fiber
Fiber buckling –flexible fiber
U bending – very flexible fiber
S bending – very flexible fiber
Some examples….. deformation of semiflexible polymers
Yanan Liu, PMMH-ESPCI, 2017
Some examples….. viscosity of active suspensions
Outline
1.Rheology and complex fluids
2.Transport dynamics of complex particles
3.Suspension rheology
Some comments on microfluidics
Small scale
Low Reynolds number
Very good flow control by the geometry
High shear rates (high viscoelasticity)
Transparent: easy flow visualization or particle tracking
Small volumes required
Square channels
Typical dimensions: 100mmx100mm
Microchannel fabrication
Outline
1.Rheology and complex fluids
2.Transport dynamics of complex particles
3.Suspension rheology
Some examples of Newtonian fluids
Newtonian fluids
water h=1mPa.s
alcohol h~1mPa.s
acetone h~0.3 mPa.s
oil h=1 Pas
honey h=10 Pa.s
Newtonian fluids are scarce ……
….. but very wide spread!
Examples of non-Newtonian fluids
Biology blood (red blood cells aggregate and the orient with flow, shear thinning) saliva (polymers), long and stable filaments
Foodmayonnaise (emulsion, oil in lemon juice or vinegar), yield stress fluid chocolate mousse, yield stress fluid beer foam, dry or humid yoghurt (xanthane, polymers) shear thinning
Cosmetics tooth paste (polymers and particles), yield stress fluid hair gel(polymers), yield stress fluid creams (emulsions), yield stress fluid shampoo (polymers), normal stresses
Geology lava mud (non-Brownian suspensions), particles in water clay (Brownian suspensions, particles in water), very dense suspensions
Examples of non-Newtonian fluids
Building materials cement
Examples of non-Newtonian fluids
Examples of non-Newtonian fluid flow
Resistance to elongation: tubeless siphon and droplet detachment
Examples of non-Newtonian fluid flow
« Die Swell » – normal stresses« Rod climbing » - normal stresses
Examples of non-Newtonian fluid flow
« shear thinning »
Classical rheometers – shear measurements
Geometries
Couette
Cone - Plate
Plate - Plate
Shear thinning fluid – Xanthan (rigid polymer)
Viscosity plateau
Power law fluid
1/l
How to measure normal stress differences?
Cone and plate rheometer
Shear viscosity
Fz
Now: measure normal force on plate Fz
N1=2 Fz
a2 p
Keep in mind: shear rate is a constant!
Example: solution of flexible polymer
Oldroyd-B type model
N1 is quadratic in g, N2 negligibleViscosity is constant
Shear viscosity Normal stress difference N1
c
PEO solutions, c=125-1000ppm
p
Comment on simple shear flow
Combination between rotation and elongation!
Experimental observations: DNA molecules
Teixeira, Macromulecules, 2005
Normal stress differences for flexible polymers
under shearwithout shear
N1(g) = Sxx- Syy >0.
N2(g) = Syy- Szz=0.
stretched and slightly rotated
into direction of streamlines
Only non zero diagonal element: Sxx>0
“tension in direction of stream lines”
Other elements are zero: Syy =Szz=0
Normal stress differences
Droplet detachment
water solution of flexible polymer
• Flexible polymers strongly stabilize the filament
• Competition between surface tension and elongational viscosity determines the thinning dynamics
• Can be used to determine elongational viscosity
films slowed down
water
PEO solution
Droplet detachment
tp time of pinch off
CaBER rheometer
CaBER rheometer
• Filament created by pulling two plates apart
• Minimal diameter measured as a function of time using a laser
Molecular origine
Coil-stretch transition
Schroeder et al, Science, 2003
Predicted by de Gennes to take place at
Experimental observations
Turbulent drag reduction
… used by New York fireman …..
Molecular models: « bead and dumbbell »
Molecular model
Two bead connected by a spring
Transported by the flow
Evaluate polymer contribution to stress tensor
Obtain constitutive equation
Molecular models: Oldroyd B
Oldroyd-B or « 2nd order fluid » model
hookian springs single relaxation time t
Viscosity First normal stress difference
With n concentration (number of molecules/volume)
Elongational viscosity
• small departure from Newtonian behavior
• small extension rates• not realistic for elongational viscosities
Microfluidic rheometers
• Perfect control of flow geometry
• Small Reynolds number (due to
small size)
• Small volumes required
• Transparent, particles (polymers)
can be visualized directly in flow
• Channel flows
Characteristics of microfluidic rheometers
Types of rheometers
• Relying on measurement of flow
rate and pressure drop
• Indirect determination of non-
Newtonian property
Measuring shear viscosities
Most rheometers rely on a simultaneous
measure of the flow rate and the pressure
drop
Darcy’s law
x
pdhhQ
h12
2
Can be corrected for square channel geometry
Measuring flow velocities
Berthet et al. Lab on a Chip, 2010Koser et al. Lab on a Chip, 2013
Flow profiles Average ‘local’ flow velocities
(thermal flow rate sensors)
Pressure measurements
Local pressure measurements- example
Orth et al, Lab on Chip, 2011
Principe Calibration
Measuring shear viscosities
Shear thinning fluids
Correct Darcy’s law for shear
thinning
Direct measure of local flow
profile…Klessinger, Microfluidic Nanofluidic, 2013
« Transient viscosities »
Haward et al, PRL, 2012
Elongational viscosity
OSCER (Optimized Shape Cross Slot Extensional rheometer)
Flow field Birefringence measurements
Haward et al, PRL, 2012
Elongational viscosity
Measure of pressure difference (or birefringence) for a given flow rate.
Cross-slot – elongational viscosity
Filament thinning
Elongational viscosity from the thinning dynamics
Here thinning imposed by the flow
of the Newtonian fluid.
Arratia, New J. Physics, 2009
Comparative rheometer for shear viscosities
Principle
For a given pressure gradient the
flow rate is proportional to the
viscosity
P. Guillot et al., Langmuir (2006).
2
1
2
1
d
d
h
h
x
pdhQ
i
i
h
3
12
1
The more viscous fluid occupies
more space
Q
Q η1
η2
d1
d2
Simple approximation valid in the limit of Hele-
Shaw flow and small viscosity difference.
Y-channel
gh
1
stressshear
stresses normalWi
N
h
WU
viscosity
inertiaRe
Laminar flow || nontrivial coherent flow || turbulent flow
Newtonian fluids Re
Laminar flow || nontrivial coherent flow || turbulent flow
Visco-elastic fluids Wi
Normal stress differences - Elastic flow
instabilities (low Re)
A. Morozov, et al., Physics Reports, 2007
Elastic flow instabilities experimental observations in
solutions of flexible polymers
Elastic instability observed for:
• Curved streamlines
• Normal stress differences
Groisman and Steinberg, 2001.
Taylor Couette flow
Groisman & Steinberg, 2000
Microfluidic ChannelPlate – plate set-up
Larson, Shaqfeh & Muller, 1990
Elastic flow instabilities – Pakdel-McKinley criterium
lgh
ULM
NLcrit ;
5,0
1
U typical velocity
l polymer relaxation time
typical radius of curvature of streamlines
Unified instability criterium
Definitions
P. Pakdel, G.H. McKinley, Phys. Rev. Lett. (1996)
Hoop
stress
Example: instability onset in a serpentine microchannel
Use of microfluidic systems:
– Easy to change geometry
– High shear rates (and Wi)
at low Re (small size)
Well known solution of flexible
polymer:
– PEO, MW= 2x106 , 2x106
+ varying percentage of Glycerol
– dilute regime
W=H=100µm
R=50µm - 1950µm
Experiments
Numerical simulations
– Same geometry
– 3D simulations
p
pN
hh
glh
2
1 2
UCM model to describe rheology
N
Wi 1 glgh
Experimental observations
incre
asin
g flo
w ra
te
Laminar flow
Unstable time dependent flow
solution of PEO 4Mio
Instability onset
• Solution with c=125ppm 2x106 MW
PEO and varying percentages of
Glycerol
• Zimm relaxation time l=0,36ms in
water, varies with solvent viscosity
1000
500
0
Sh
ea
r ra
te (
1/s
)
2000150010005000
Radius (microns)
25% Glycerol
40% Glycerol
50% Glycerol
60% Glycerol1.0
0.5
0.0
We
isse
nb
erg
nu
mb
er
20151050
R/W
25% Glycerol
50% Glycerol
40% Glycerol
60% Glycerol
0% Glycerol (W=60microns)
Rodd et al, JNNFM, 143 (2007) 170-191
Critical shear rate Critical Weissenberg number
W
Ug
Using Zimm relaxation time and
average shear rate:
Dependence of instability onset on radius of curvature
critMNU
5,0
1
gh
l
Pakdel-McKinley criterium
W/Wicrit
Critical Weissenberg number
Flow profile is parabolic
Shear rate is not constant
Radius of curvature varies
• R>>W, (y)Ri
• R/W0
Use local values, maximize, combine the two limits
For channel flow
a
Critical Wi as a function of the radius of curvature
Good agreement between experiments, simple theory and numerical simulations.
Zilz, AL et al. “Geometric scaling of purely-elastic flow instabilities”, J. Fluid. Mech, 2012
Calibrate the serpentine rheometer
PEO, Mw=2x106, 400ppmNote: one has to correct for the solvent viscosity!
.
1 / ghl pN
Relaxation time from viscosity
and first normal stress difference
Classical rheology
2
W
RCc 1/
.
lg
Critical shear rate
Serpentine channel
Calibration
Calibration factor C=0.05
Calibrate with classical rheology measurements – PEO 2 Mio
Serpentine rheometer can now be used to access relaxation times
Classical rheology 400ppm
Serpentine 125ppm
Serpentine 400ppm
Zilz, AL et al, Serpentine channels: micro–rheometers for fluid relaxation times, Lab on Chip, 2013
2.5
2.0
1.5
1.0
0.5
0.0
l(
ms)
Rhe
om
ete
r
121086420h
s (mPas)
60
50
40
30
20
10
0
a (m
s) S
erp
entin
e c
han
ne
l
fits proportional to hs
l/C
Relaxation time measurements
2.0
1.5
1.0
0.5
0.0
lam
bd
a (
ms)
6005004003002001000
concentration (ppm)
PEO, Mw=2Mio, solvent viscosity 4,9mPas
Very good resolution even at small concentration.
5
4
3
2
1
0
lam
bd
a (
ms)
se
rpe
ntin
e543210
lambda (ms) classical rheometer
PEO 2Mio
PEO 4Mio-1
PEO 4Mio-2
As a function of concentration For different molecular weights
Outline
1.Rheology and complex fluids
2.Transport dynamics of complex particles
3.Suspension rheology
Motivation
Locomotion at small
Reynolds-numbers
Biofluids
Red blood cell under flow,
Stefano Guido, Naples
Bacteria: E. ColiArtificial swimmers
Dreyfus et al., Nature, 2005
Biofilm in microchannel
Rusconi et al, J R Soc Interface, 2011
Separation and clogging
Clogging of a microfilter
PhD, Gbedo, 2011, ToulouseLost circulation
problems in oil wells
Schlumberger
Properties of complex suspensions
Normal stresses in fiber suspensions
Becker & Shelley, PRL, (2001)
Transport dynamics of complex particles
Rigid particles with complex shape
Spheres
Fibers
Helices
Microswimmers
Microfabrication
Flexible particles
Single translating sphere
Guazzelli, Morris, An introduction to suspension dynamics, Cambridge University Press
All illustrations from:
Flow field – point force
Guazzelli, Morris, An introduction to suspension dynamics, Cambridge University Press
Single sphere freely transported in shear flow
Guazzelli, Morris, An introduction to suspension dynamics, Cambridge University Press
Rotation and straining
Flow induced by a point stresslet - dipole
Guazzelli, Morris, An introduction to suspension dynamics, Cambridge University Press
Flow around a sphere in a shear flow
Guazzelli, Morris, An introduction to suspension dynamics, Cambridge University Press
Force and Stresslet
Sedimenting sphere
Stokes drag
Freely transported sphere in shear
Stresslet
Sedimenting fiber
Horizontal fiber Vertical fiber Inclined fiber
Falls two times quicker!
v1
v2?
Sedimenting fiber
Fiber drifts due to
anisotropic friction
coefficient!
Horizontal fiber Vertical fiber Inclined fiber
Falls two times quicker!
v1
v2
Elongated objects in shear flows
Fiber dynamics in simple shear?
Jeffery orbits!
Center of mass is transported with the fluid velocity along the stream lines
Fiber rotates with given dynamics and period around its axis.
Jeffery, 1922
Jeffery orbits
Solutions for an ellipsoid in simple shear in 2D (only motion in x-y-plane)
f=acrtan{r tan (t/T)}
a
b
r=a/b (=L/(2R))
Dynamics of angle f:
with period T=2 p(r+1/r)/g
3 2 1 1 2 3
1.5
1.0
0.5
0.5
1.0
1.5
r=2
3 2 1 1 2 3
1.5
1.0
0.5
0.5
1.0
1.5
r=10
3 2 1 1 2 3
1.5
1.0
0.5
0.5
1.0
1.5
r=100
f vs time/T for more and more elongated particles
For more elongated particles the particle spends more time aligned with the flow direction!
Period increases with increasing elongation.
More complex orbits in 3D!
aligned with z-axis
in x-y plane
Guazzelli & Morris, A Physical Introduction to Suspension
Dynamics, Cambridge 2012
Normal stress differences dilute rigid fiber suspensions
Line tension of a rigid fiber in shear flow
Fiber perturbs the flow by its presence, but in average (over one Jeffery
orbit) the contribution to the normal stresses is zero.
No experiments….
Shear induced migration ….
• Spherical particles do not drift in simple shear or Poiseuille flows
(reversibility of Stokes flows).
• Axis-symmetric particles follow the stream lines, but perform complex
Jeffery orbits.
• What happens for non axis-symmetric particles?
• Curved fibers (non-chiral objects)
• Spirals (chiral objects)
• Deformable objects
Can isolated particles migrate across streamlines?
Importance for particle separation devices?
Fiber drift together with wall interaction can lead to stable
equilibrium positions function of particle properties.
Spirals drift in vorticity direction!
Jeffery orbit aligns helix with stream lines
In the reference frame of the helix
upper part and lower part see flows of
opposite directions
Due to the anisotropy in drag, both
segments lead to a drift velocity in the –z
direction
Direction of drift
Spirals drift in vorticity direction, as a function of chirality!
Only works when spirals are preferentially aligned with flow!
Marcos, PRL, 2009
Can be used to separate particles of different chirality in microfluidic devices!
Spirals drift in vorticity direction!
Marcos, PRL, 2009
E-coli bacteria in shear flows
Combination between shape and activity leads to “rheotaxis”
Marcos et al., PNAS, 2011
E-coli bacteria swim towards a given direction in simple shear flows
(opposite to simple helices)…..
Transport dynamics of complex particles
Rigid particles with complex shape
Spheres
Fibers
Helices
Microswimmers
Microfabrication and 3D tracking
Flexible particles
• Projecting a fiber 2D shape into channel
• Photo sensitive fluid of PEGDA with
photo-initiator: crosslinks under UV
exposure
Projection photo-lithographie
Control of fiber confinement by the channel height:Control of size,
concentration, orientation:
Microfabrication of polymeric fibers
P. Doyle group, MIT
In situ beam bending experiment Deflection as a function of flow speed
Balancing viscous and elastic forces allows to determine the Youngs modulus
Mechanical properties
Deformation of the beam
The viscous flow exerts a force per length on the
fiber (due to pressure gradient and viscous
friction):
Euler Bernoulli equation for a slender
beam :
leads to
• The Young’s modulus E can be measured from the deflection!
• Strong dependence on channel and fiber geometry!
leads to
Young’s modulus vs exposure time
Young’s modulus varies strongly with exposure time
Mechanical properties
Duprat, AL et al, Lab on Chip, 2016
Micro-helix fabrication - I
Flexible ribbons when released ON TOP OF waterFlow coating – nano-ribbons
Pham et al, Advanced Materials, 2013
Lee et al, Advanced Materials, 2013Al Crosby, UMass, Amherst
Kim et al, Advanced Materials, 2010
10 μmCdSe Quantum Dots
500 µm
Long, flexible ribbons
Fluorescent PMMA
t
Spontaneaous helix formation when
relased IN water
Spontaneous helix formation
Ribbon dimensions determine the radius R of the helix
Ribbon cross section Helix
Mechanical characterization
Pham et al, PRE 2015
View from side
Glas
s
PDMS
50 μm
1
cm
Pham et al, 2015
Stretching of helices under flow
Helix extension (linear)
Stretching of helices under flow
Pham, AL, et al, PRE, 2015
Micro-helix fabrication - II
3D printed using Nanoscribe
Francesca Tesser, Justine Laurent PMMH-ESPCI
Lagrangian tracking of swimming E.coli
T.Darnige, AL, et al. Review of Scientific Instrument, 2017
3D automatic tracker
Lagrangian tracking of swimming E.coli
Obtain 3D trajectories
• in the bulk
• at surfaces
• with /without flow
• varying environmental conditionsT.Darnige, AL, et al. Review of Scientific Instrument, 2017
N.Figueroa-Morales (2017)
Flow geometry
H
W
Plug flow in the
channel width
Poiseuille flow in
the channel height
Hele-Shaw cell
lateral confinement
transverse confinement
Top view
Cross-section
Fiber geometry
Fiber transport in confined geometries
Fiber is faster in perpendicular than in parallel direction!
Experimental observations
Single fiber transport
Transport velocities
Berthet, AL, et al, PoF, 2014, Nagel, AL, et al, under revision, JFM, 2017
Anisotropic transport velocity leads to fiber drift of inclined fibers
Consequences of transport anisotropy
Transported and sedimenting fibers drift in opposite directions!
Sedimenting fiber drifts due to anisotropic friction coefficient
Anisotropic transport velocity leads to fiber drift of inclined fibers
g
Consequences of transport anisotropy
Wall effects: oscillations
Wall effects: oscillations
Rotation and drift are observed.
Stable orientations reached are function of fiber shape and confinement.
More complex shaped fibers
Transport dynamics of complex particles
Rigid particles with complex shape
Spheres
Fibers
Helices
Microswimmers
Microfabrication and 3D tracking
Flexible particles
Fluid-structure interactions
Elastic objects can
be deformed by
viscous flows.
Deformation can
change transport
properties.
Here: study mainly deformation and transport of slender objects (fibers).
Buckling instability
Elastic elongated objects show a buckling instability under compression
For comparison: bulk objects are compressed without instability!!!
It is energetically more favorable to
bend than to compress the object
above a threshold in deformation
(force).
Buckling of elastic fibers in viscous flows
Competition between viscous forces….
Fv ~ h L2 g.
… and elastic forces
Fel ~ E/L2 with E the bending modulus
Control parameter elasto-visous number
Strong dependence on fiber length
(aspect ratio)!
for elastic filaments
E=Y*I
Y=Youngs modulus,
I=moment of inertia Ipr4/4
for semi-flexible polymers
E=kT lp
lp=persitance length
B
L
c
4.
~
~gh
h
Flexible fibers in shear flows
Actin filament - a semi flexible polymer
𝐿𝑐 = 17𝜇𝑚 𝐿𝑐 = 15𝜇𝑚 𝐿𝑐 = 14𝜇𝑚
𝜂 = 1𝑚𝑃𝑎 ⋅ 𝑠 𝜂 = 28𝑚𝑃𝑎 ⋅ 𝑠𝜂 = 7𝑚𝑃𝑎 ⋅ 𝑠
Typical length: 5 mm – 20 mm
Width: 6 nm
Persistence length lp: 17 mm
Flow
Motorized
Stage Objective
63X
y
z
W/2
H=500~800μm
W=200μm
Z=200μm
Top
view
𝑧~200𝜇𝑚
Characterization Flow geometry
kTlB psee also Harasim PRL 2013 and Kantsler, PRL, 2012
𝑳 = 𝟒. 𝟗𝝁𝒎 𝜸 = 𝟐. 𝟕𝟕𝒔−𝟏
𝑳 = 𝟔. 𝟖𝝁𝒎 𝜸 = 𝟐. 𝟔𝟏𝒔−𝟏
𝑳 = 𝟐𝟔. 𝟔𝝁𝒎 𝜸 = 𝟏. 𝟕𝟔𝒔−𝟏
𝑳 = 𝟑𝟑. 𝟓𝝁𝒎 𝜸 = 𝟏. 𝟒𝟔𝒔−𝟏
Flow
Stage
Experimental observations
Jeffery orbit – Rigid fiber
Fiber buckling –flexible fiber
U bending – very flexible fiber
S bending – very flexible fiber
Yanan Liu, PMMH
𝐿𝑒𝑒/𝐿End to end distance over
length
𝐸 Bending energy
𝜑/𝜋Angle between 𝐿𝑒𝑒 and 𝑢𝑥over 𝜋
Jeffery orbit C shape buckling
Characteristic of typical dynamics
U shape bending S shape bending
Characteristic of typical dynamics
Evolution of typical dynamics
ζ=8𝜋𝜂 𝛾𝐿4/𝑐
𝐵
Evolution of typical dynamics – comparison to simulations
Simulations: Chakrabarti B. & Saintillan D. (non-linear slender body + Brownian fluctuations)
Evolution of typical dynamics – comparison to simulations
Buckling
transition
U and S
Transition?
Role of
Brownian
fluctuations?
Becker et al,
PRL, (2001)
Fiber in shear flow - simulations
Becker & Shelley, PRL, (2001)
Jeffery orbit and buckling instability
Buckling threshold h*152,6
Stretch – coil transition leads to normal stress differences….
First Normal Stress difference over one period
First normal stress difference
Shear stress
Tornberg et al, J. Comp. Phys, 2004
Flexible fiber: confined geometry
Slight deformation of the fiber (inverse C shape).
Fiber transported in plug flow
Jean Cappello, PMMH
Flexible fiber: confined geometry
Slight deformation of the fiber (inverse C shape).
Fiber transported in plug flow
Why do they deform?
Spheres on the outside lack neighbors, so they feel more friction!
Viscous force not uniform (higher order terms in 1/ln(L/R)) - > deformation!
Total viscous drag balances gravity -> sedimentation speed
But…..
Why do they deform?
Li et al, JFM, 2013
Flexible fiber: C-shape
Force per length on the fiber Resulting fiber shape (using Euler elastica)
Predicted fiber shape in good agreement with experimental observations!
Fiber is a local pressure distribution sensor!
with F. Gallaire, EPFL, Lausanne
Fiber buckling in sedimentation
Numerical simulations (Saintillan, UCSD and Spanoglie, Wisconsin)
Buckling threshold can be determined
Filament is under compression in the
bottom part and under extension in the
upper part! Li et al, JFM, 2013
Why do they deform?
Spheres on the outside lack neighbors, so they feel more friction .
Viscous force not uniform (higher order terms in 1/ln(L/R))
Filament is under compression in the bottom part and under extension in the
upper part -> buckling instability can occur!
Total viscous drag balances gravity -> sedimentation speed, but….
Flexible fiber: buckling
Jean Cappello, PMMH-ESPCI, 2017
Confined fiber in plug flow
Flexible fiber: buckling
Jean Cappello, PMMH-ESPCI, 2017
Confined fiber in plug flow
Outline
1.Rheology and complex fluids
2.Transport dynamics of complex particles
3.Suspension rheology
Active suspensions – E-coli bacteria
Far field description – velocity field
Shun Pak and Lauga, Theoretical models in low Reynolds-number locomotion, 2014
Swimming at low Reynolds number
• No net force
• No net torque
Low Reynolds number
Sign of dipole depends on
swimming strategie
Pusher Puller
Clamydomonas reinhardtiiE.Coli
Drescher et al, PNAS, 2011 Drescher et al. PRL (2010)
Microswimmers
• elongated
• activePredictions of the
effective viscosity
• orientation under shear
• force dipole
Effective shear viscosity of a suspension of
microswimmers
Particles
Mean orientation of elongated particles in shear
flow
Mean orientation including noise under shear
D.Saintillan, Exp. Mech. (2010)
Jeffery orbit
• Elongated objects
rotate under shear
• Spend most of the
time aligned with
shear rate
Noise for bacteria:
• rotary diffusion
• tumbling
Increasing shear rate
Very little direct measurements for bacteria up to now.
Anisotropic orientation and force dipole
Viscosity decrease for pusher like bacteria
Theoretical models rely on description of distribution of orientation of individual bacteria.
Hatwalne et al, PRL, 2004
Pushers
n
Consequence of disturbance field
2
1
2
1
d
d
h
h
1
2
6
dh
Qm g
x
pdhQ
i
i
h
3
12
1
Bacterial suspension
Suspending fluid
Q
Qη
1
η2
d1
d2
w=600mm
h=100mmmeasurement region
Adapt a microfluidic rheometer
Advantages of the measurement technique:
• Very good resolution on the viscosity
• Reasonable to impose small shear rates
• Small volumes needed
• In-situ visualization P.Guillot et al., Langmuir (2006)
1 10 1000.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
1.4
1.5
f=0.8%
Non-Motile Bacteria
Motile BacteriaRe
lative v
isco
sity : h
r
Shear rate : g (Hz)
Viscosity measurements
Non-Newtonian viscosity of active suspensions revealed : non-monotonic behavior!
Maximum at shear rate of 20s-1 (comparable to V/L~ 10 s-1)
Compare motile to non-motile bacteria
Gachelin et al, Non-Newtonian viscosity of E-coli suspensions, Phys. Rev. Lett. 2013
f=0.8%
Viscosity measurements
Non-Newtonian viscosity of active suspensions revealed : non-monotonic behavior!
Maximum at shear rate of 20s-1 (comparable to V/L~ 10 s-1)
Compare motile to non-motile bacteria
Gachelin et al, Non-Newtonian viscosity of E-coli suspensions, Phys. Rev. Lett. 2013
f=0.8%
Saintillan and Shelley, CRAS, 2013
Theoretical predictions