Complex flows in microfluidic geometries · 2017-06-21 · Complex flows in microfluidic geometries...

Complex flows in microfluidic geometries

Anke Lindner, PMMH-ESPCI, Paris, anke.lindner@espci.fr

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