Emergence of coherent structures and large-scale flows in

Emergence of coherent structuresand large-scale flows in biologically

active suspensionsDavid Saintillan

Department of Mechanical Science and EngineeringUniversity of Illinois at Urbana-Champaign

IMA workshop: Natural locomotion in

fluids and on surfaces,

Minneapolis, MN, June 3 2010

Collaborator: Michael J. Shelley(Courant Institute, NYU)

Graduate student: Amir AlizadehPahlavan (MechSE, UIUC)

Interactions in active suspensions

E. Coli swarming on a surface

Active suspension: suspensions of self-propelled particles (e.g. bacteria, microalgae,artificial microswimmers).

Experiments exhibit:! correlated motions over length scales thatgreatly exceed the particle dimensions.! diffusive behavior! in some cases, large-scale densityfluctuations suggesting self-organization withinthe suspensions.Dombrowski et al., PRL (2004)Mendelson et al., J. Bacteriol. (1999)Wu and Libchaber, PRL (2000)Soni et al., Biophys. J. (2003)Sokolov et al., PRL (2007)and many others…

! What mechanisms lead to these large-scaleflows and pattern formation?! Under which conditions (concentration,system size, swimming mechanism) do theseoccur?! What are the characteristics of the resultingflows and patterns?

P. Mirabilis swarming on a surface

(Weibel Lab, University of Wisconsin)

Swimming at low Reynolds numberIn the absence of inertia, swimming must rely of non-reciprocal shape deformations. Innature, several mechanisms for swimming are observed:! flagellar propulsion (sperm, E Coli, B Subtilis…)! ciliary propulsion (Paramecium…)! body deformation (C Elegans…)

sperm Chlamydomonas Paramecium C Elegans

“Life at low Reynolds number”E.M. Purcell 1977

pusher particle

In all cases, the swimming particle exerts a propulsive force on the fluid, which isexactly balanced by the viscous drag: to leading order, it creates a force dipole.

puller particle

Kinetic theory: governing equations

! Continuity equation for the distribution function of particle center-of-mass and director :

! Flux velocities:

! Fluid disturbance velocity:

Particle extra stress:

Stress magnitude and swimming velocity arerelated on the particle scale by:

! Non-dimensionalization:

Saintillan & Shelley PRL 100 178103 (2008), Phys. Fluids 20 123304 (2008)Hohenegger & Shelley, PRE 81 046311 (2010)

Configurational entropy

! We define the system entropy as:

for an isotropic homogeneous suspension

! The evolution of the entropy is governed by the following equation:

diffusive damping

Pullers : the entropy is driven down by both diffusive processes and viscous dissipation

Pushers : there is a feedback loop wherein fluctuations create velocity gradients which increase fluctuations

Reduced equations for aligned case

! Consider a locally aligned suspension:


! If diffusion can be neglected, the full kinetic equations reduce to the followingequations for the concentration field and director field:

! The disturbance velocity field then satisfies:

Stability analysis: aligned case

! Consider a nearly aligneduniform suspension:

! The two growth rates have oppositesigns: the suspension is alwaysunstable.

! In the long wave limit, the solution ofSimha and Ramaswamy (2002) isrecovered:

! In real systems, diffusion will damphigh-wavenumber fluctuations.

Stability analysis: uniform isotropic case

! Nearly isotropic uniform suspension:

Eigenvalue problem for particle stress tensor:

Eigenvectors are of the form:

Dispersion relation:

Suspensions ofpushers areunstable for

For more details on this problem see Christel Hohenegger’s talk on Saturday.

Nonlinear simulations

Integrate the kineticequations using

! a spectral solution of theStokes equations

! 2nd-order finitedifferences forconservation equation

! 2nd order Adams-Bashforth time marching.

Parallel code, typicallyrun on 64 processors.

Concentration fluctuations

Concentration autocorrelation functionConcentration isosurfaces (c = 1.5)

! Concentration fluctuations are in the form of transient sheets that constantly formand break up in time in a quasi-periodic fashion.

! The concentration field is correlated on large-length scales, with anautocorrelation length of the order of 1/4 of the box size.

Disturbance velocity field

! Velocity fields are chaotic, correlated onscales of the order of the system size, andstrongly dominated by large-scales (rapiddecay of velocity power spectrum withwavenumber)

-4.5

Velocity power spectrum

Velocity autocorrelation function

Evolution of Fourier modes

! Decay of high-wavenumber fluctuations with oscillations

! Growth of low-wavenumber fluctuations. At long-times, quasi-periodic oscillationsare observed over long time scales.

! Shear stresses and director divergence grow first, followed by concentration.

Mechanism for concentration fluctuations

Evolution equation for concentration field:

source term

Mechanism for the formation of concentration fluctuations:

! The stress instability causes the particles to align locally (particles stress = orderparameter)

! Particles swim towards regions of negative

Entropy and power input

Note:

power input rate of viscousdissipation

pushers pushers

pullers pullers

Slender-body theory for swimming rod

tangentialtractions

normaltractions

! Surface tractions

! Integrated traction (force per unit length):

! Force and torque balances

Saintillan & Shelley, Phys. Rev. Lett. 99 058102 (2007)

Slender-body theory for swimming rod

prescribed unknownSlip velocity(unknown)

Disturbancefluid velocity

Green’s function for Stokes flow

unknown

Local slender-body equation

! with prescribed stress:

! with no-slip boundary condition:

Disturbance fluid velocity (hydrodynamic interactions):

! Coupled linear system of integral equations, solved by spectral methods:

- Legendre polynomial expansion of the force distribution - Smooth particle mesh algorithm (Saintillan, Darve & Shaqfeh, Phys. Fluids 2005)

Batchelor JFM (1971)

Single particle flow fields

Particle dynamicsPushers

800 particles

Pushers

4000 particles

Pushers

8000 particles

Pullers

8000 particles

Disturbance velocity field

Onset of collectiveswimming

Pushers Pushers

Pushers Pullers

Large-scale correlated flows appearin suspensions of pushers witheffective volume fraction above 0.5.

Velocity power spectrum

Number density fluctuations! Particle occupancy statistics: statistics of the number of particles falling inside acubic interrogation cell placed at an arbitrary location inside the simulation box.

! Poisson law for a random distribution of particles:

Number density fluctuations

Onset ofcollectiveswimming

! Standard deviation of number density fluctuations exhibitspower law dependence on the mean number of particles:

! For a Poisson law,

Orientation correlations

Orientationally correlated regions

Pair correlation function for orientations

Mean particle velocities


Velocities and local structure

High particle velocities correlatewith:

! high local density (clusters)

! local polar alignment

Orientational diffusion

Define rotary diffusion coefficientas the inverse time scale fororientation decorrelation:

Rotary diffusion coefficient

Linear increase of rotary diffusion coefficient at low concentration as a result ofpair interactions.

Translational diffusionIsotropic translational diffusion coefficientTwo definitions for translational

diffusion coefficient:

Decrease of translational diffusion coefficient with concentration: diffusion arisesthrough randomization of orientation by angular diffusion (generalized Taylordispersion process, cf. Brenner 1980).

Fluid mixingPushers

800 particles

Pushers

4000 particles

Pullers

8000 particles

Pushers

8000 particles

Fluid mixing


Multiscale mixing norm:

(Mathew, Mezic & Petzold, Physica D 211 23-46 2005)

mixing efficiency b

Tracer diffusion


Rheology of swimming suspensions

Recent experiments show a decrease in viscosity in suspensions of swimmingB. Subtilis (bacterium, pusher)

Bacterial concentration Bacterial mean swimming speed

Sokolov & Aranson, “Reduction of viscosity in a suspension ofswimming bacteria”, PRL 103, 148101 (2009)

Rheology of swimming suspensions

In suspensions of Chlamydomonas (alga, puller), however, a viscosityenhancement is observed.

Rafai, Jibuti & Peyla, “Effective viscosity of microswimmersuspensions”, Phys. Rev. Lett. 104 098102 (2010)

Dilute rheology: Kinetic theory (1/3)

! Interparticle hydrodynamic interactionsare neglected.

! We consider a single particle swimmingin an imposed shear flow at low Reynoldsnumber.

! The configuration of the suspension isentirely determined by the orientationdistribution .

! Fokker-Planck equation for the orientation distribution:

shear flow diffusion tumbling

! Rotational velocity (Jeffery 1922):

(Subramanian andKoch JFM 2009)

Saintillan, Exp. Mech. (to appear, 2010), Phys. Rev. E 81 056307 (2010)

Dilute rheology: Kinetic theory (2/3)! Stress calculation: ensemble average of the force dipoles on the particles

Newtonian stress particle extra stress

Total stress:

Batchelor (Annu. Rev. Fluid Mech. 1974), Hinch & Leal (JFM 1972, 1975)


Three contributions: External flow

“Brownian” rotations

Dipole due to swimming

For a slender-body:(Hinch & Leal, JFM 1976)

Pusher: Puller:

Dilute rheology: Kinetic theory (3/3)! Characteristic scales and non-dimensionalization

Particle viscosity

First and secondnormal stressdifferencecoefficients

! Dimensionless equations

Characteristic scales

Dimensionless parameters

Steady-state FP equation

Particle extra stress

Pusher:

Puller:

Orientation distributions

! The steady-state Fokker-Planckequation is solved spectrally byexpanding the orientationdistribution on the basis ofspherical harmonics.

! Numerical solutions wereobtained using harmonics ofdegree up to 40 (corresponding to861 modes), which ensured anexcellent accuracy for theparameters considered here.

Chen & Koch (Phys. Fluids 1996)Doi & Edwards (J. Chem. Soc. 1978)

Rheology of “smooth” swimmers

Smooth slender swimmers(no particle tumbling)

Rheology of “smooth” swimmers

Smooth slender swimmers(no particle tumbling)

Negative particle viscosity

! Zero-shear-rate particle viscosity ( ):

! Negative particle viscosities occur for: ( for slender swimmers)

! Critical shear ratebelow which negativeviscosities occur vs.rotary diffusivity

Effect of particle tumbling

Slender swimmerswith fixed rotary diffusivity

Particle tumbling has the samequalitative effect as rotary diffusion.

Extensional rheology

! The same model can be developed for uniaxial extensional and compressional flows,as well as planar extensional flows, and is amenable to analytical solutions.

Saintillan, Phys. Rev. E 81 056307 (2010)

Stability in shear flow

Substituting in the governing equations, the linearized equation becomes:

This is a generalized eigenvalue problem (integro-differential equation) which can beturned into a system of algebraic eigenvalue problems using spherical harmonics.

Saintillan and AlizadehPahlavan, in preparation 2010

Stability in shear flow

S = 0 S = 0.1

S = 0.142D instability

S = 0.21D instability

Summary! A kinetic theory for suspensions of self-propelled microswimmers predicts aninstability in isotropic suspensions of pushers, which occurs above a certainsystem size and/or suspension volume fraction. The instability is characterized bythe growth of density fluctuations, in the form of dense clusters that form andbreak up in time.

! Particle simulations confirm the results of the kinetic model, and show theemergence of coherent structures and large-scale flows in concentratedsuspensions of pushers, as demonstrated by an increase in density fluctuations,velocity correlation length, and mean particle velocity above a threshold volumefraction. These effects are not observed for pullers.

! The large-scale flows that result from the instability cause very efficient fluidmixing.

! The effective rheology of active suspensions is characterized by a decrease inviscosity in weak flows of pushers, and by an enhancement for pullers.

! An external shear flow has a stabilizing effect. Kinetic simulations show atransition from 3D to 2D to 1D instability as shear rate increases, followed bycomplete stabilization.

! Acknowledgments: NSF Grant DMS-0920931-ARRA, National Center forSupercomputing Applications

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Emergence of coherent structures and large-scale flows in