# Publications

## Book Chapters

- Theory of active suspensions D. Saintillan, M. J. Shelley, in
- Kinetic models for biologically active suspensions D. Saintillan, in

*Complex Fluids in Biological Systems*, S. Spagnolie (ed.), Springer (2015). [pdf]

Active suspensions, of which a bath of swimming microorganisms is a paradigmatic example, denote large collections of individual particles or macro- molecules capable of converting fuel into mechanical work and microstructural stresses. Such systems, which have excited much research in the last decade, exhibit complex dynamical behaviors such as large-scale correlated motions and pattern formation due to hydrodynamic interactions. In this chapter, we summarize efforts to model these systems using particle simulations and continuum kinetic theories. After reviewing results from experiments and simulations, we present a general kinetic model for a suspension of self-propelled rodlike particles and discuss its stability and nonlinear dynamics. We then address extensions of this model that capture the effect of steric interactions in concentrated systems, the impact of confinement and interactions with boundaries, and the effect of the suspending medium rheology. Finally, we discuss new active systems such as those that involve the interactions of biopolymers with immersed motor proteins and surface-bound suspensions of chemically powered particles.

*Natural Locomotion in Fluids and on Surfaces: Swimming, Flying, and Sliding*, S. Childress, A. Hosoi, W. W. Schultz, Z. J. Wang (eds.), Springer (2012). [pdf]

Biologically active suspensions, such as suspensions of swimming microorganisms, exhibit fascinating dynamics including large-scale collective motions and pattern formation, complex chaotic flows with good mixing properties, enhanced passsive tracer diffusion, among others. There has been much recent interest in modeling and understanding these effects, which often result from long-ranged fluid-mediated interactions between swimming particles. This paper provides a general introduction to a number of recent investigations on these systems based on a continuum mean-field description of hydrodynamic interactions. A basic kinetic model is presented in detail, and an overview of its applications to the analysis of coherent motions and pattern formation, chemotactic interactions, and the effective rheology in active suspensions, is given.

## Journal Publications

#### Under Review

- Flexible filaments buckle into helicoidal shapes in strong compressional flows B. Chakrabarti, Y. Liu, J. LaGrone, R. Cortez, L. Fauci, O. du Roure, D. Saintillan, A. Lindner, submitted (2019).
- Hydrodynamic synchronization of spontaneously beating filaments B. Chakrabarti, D. Saintillan, submitted (2019). [preprint]

#### 2019

- Computational mean-field modeling of confined active fluids M. Theillard, D. Saintillan,
- Interfacial instabilities in active viscous films R. Alonso-Matilla, D. Saintillan,
- Sharp numerical simulation of incompressible two-phase flows M. Theillard, F. Gibou, D. Saintillan,
- Spontaneous oscillations, beating patterns and hydrodynamics of active microfilaments B. Chakrabarti, D. Saintillan,
- Transport and dispersion of active particles in periodic porous media R. Alonso-Matilla, B. Chakrabarti, D. Saintillan,

*Journal of Computational Physics*,

**397**108841 (2019). [pdf]

We present a new framework for the efficient simulation of the dynamics of active fluids in complex two- and three-dimensional microfluidic geometries. Focusing on the case of a suspension of microswimmers such as motile bacteria, we adopt a continuum mean-field model based on partial differential equations for the evolution of the concentration, polarization and nematic tensor fields, which are nonlinearly coupled to the Navier-Stokes equations for the fluid flow driven by internal active stresses. A level set method combined with an adaptive mesh refinement scheme on Quad-/Octree grids is used to capture complex domain shapes while refining the solution near boundaries or in the neighborhood of sharp gradients. A hybrid finite volumes/finite differences method is implemented in which the concentration field is treated using finite volumes to ensure mass conservation, while the polarization and nematic alignment fields are treated using a combination of finite differences and finite volumes for enhanced accuracy. The governing equations for these fields are solved along with the Navier-Stokes equations, which are evolved using an unconditionally stable projection solver. We illustrate the versatility and robustness of our method by analyzing spontaneous active flows in various two- and three-dimensional systems. Our results show excellent agreement with previous models and experiments and pave the way for further developments in active microfluidics.

*Journal of Non-Newtonian Fluid Mechanics*,

**259**57 (2019). [pdf]

➠ Invited article, special issue on "Complex Fluids in Biological Systems"

The interfacial stability of an active viscous film is analyzed theoretically. The film, which rests on a flat substrate and is bounded from above by an air-liquid interface, contains a suspension of active particles such as swimming microorganisms that self-propel, diffuse, and exert active stresses on the suspending Newtonian medium. Using a continuum model for the configuration of the suspension coupled to the forced Stokes equations for the fluid motion, we analyze the growth of linearized normal mode fluctuations with respect to the quiescent base state. In the absence of gravity, puller suspensions are found to be always stable, whereas films containing pushers can become unstable above a critical activity level where active stresses overcome the damping effects of viscosity and surface tension and drive interfacial deformations.\ Confinement, diffusion and capillary forces all act to stabilize the system, and we characterize the transition to instability in terms of the dimensionless parameters of the problem. We also address the case of inverted films subject to the Rayleigh-Taylor instability, where we demonstrate that active stresses generated by pullers have the ability to stabilize gravitationally unstable films by counteracting the effect of the gravitational body force.

*Journal of Computational Physics*,

**391**91 (2019). [pdf]

We present a numerical method for simulating incompressible immiscible fluids, in two and three spatial dimensions. It is constructed as a modified pressure correction projection method on adaptive non-graded Oc/Quadtree Cartesian grids, using the level-set framework to capture the moving interface between the two fluids. The sharp treatment of the interface position, of fluid parameter discontinuities, and of the interfacial jump conditions ensures convergence in the *L*∞-norm. Using a novel construction for the pressure guess, we are able to alleviate the standard time step restriction incurred by capillary forces. The solver is validated numerically and employed to simulate the dynamics of physically relevant problems such as rising bubbles and viscous droplets in electric fields.

*Physical Review Fluids*,

**4**043102 (2019). [pdf]

Cilia and flagella are ubiquitous in nature and are known to help in transport and swimming at the cellular scale by performing oscillations. Fundamental to these periodic waveforms is the core internal structure of the filaments known as the axoneme, consisting of an array of microtubule doublets, protein linkers and dynein motors. In the presence of ATP, the collective action of the molecular motors drives internal sliding motions that are converted to spontaneous oscillations by a mechanism that still remains elusive. A sliding controlled axonemal feedback mechanism has recently been proposed and explored in the limit of small deformations, where it was shown to result in nonlinear amplitude selection through a mechanical regulation of dynein kinetics. Here, we build on that model to derive a more complete set of planar nonlinear governing equations that retains all the geometric nonlinearities, incorporates intrinsic biochemical noise and accounts for long-range, nonlocal hydrodynamic interactions. For a clamped filament, motor activity drives a Hopf bifurcation leading to traveling wave solutions that propagate from tip to base, in agreement with previous weakly nonlinear studies. Quite remarkably, our results demonstrate the existence of a second transition far from equilibrium, where nonlinearities cause a reversal in the direction of wave propagation and produce a variety of waveforms that resemble the beating patterns of swimming spermatozoa. We further extend the model to account for asymmetric ciliary beats and also allow for generalized dynein regulation mechanisms that can qualitatively reproduce *Chlamydomonas reinhardtii* flagellar dynamics. In the spirit of dimensional reduction, limit cycle representations are obtained for various waveforms and highlight the role of biochemical noise. We also analyze the velocity fields generated by the filaments and apply principal component analysis to derive low-order flow representations in terms of fundamental Stokes singularities that could be of use for constructing minimal models of swimming microorganisms.

*Physical Review Fluids*,

**4**043101 (2019). [pdf]

The transport of self-propelled particles such as bacteria and phoretic swimmers through crowded heterogeneous environments is relevant to many natural and engineering processes, from biofilm formation and contamination processes to transport in soils and biomedical devices. While there has been experimental progress, a theoretical understanding of mean transport properties in these systems has been lacking. In this work, we apply generalized Taylor dispersion theory to analyze the long-time statistics of an active self-propelled Brownian particle transported under an applied flow through the interstices of a periodic lattice that serves as an idealization of a porous medium. Our theoretical model, which we validate against Brownian dynamics simulations, is applied to unravel the roles of motility, fluid flow, and lattice geometry on asymptotic mean velocity and dispersivity. In weak flows, transport is dominated by active dispersion, which results from self-propulsion in the presence of noise and is hindered by the obstacles that act as entropic barriers. In strong flows, shear-induced Taylor dispersion becomes the dominant mechanism for spreading, with pillars now acting as regions of shear production that enhance dispersion. The interplay of these two effects leads to complex and unexpected trends, such as a non-monotonic dependence of axial dispersivity on flow strength and a reduction in dispersion due to swimming activity in strong flows. Brownian dynamics are used to cast light on the pre-asymptotic regime, where tailed distributions are observed in agreement with recent experiments on motile micro-organisms. Our results also highlight the subtle effects of pillar shape, which can be used to control the magnitude of dispersion and to drive a net particle migration in quiescent systems.

#### 2018

- Extensile motor activity drives coherent motions in a model of interphase chromatin D. Saintillan, M. J. Shelley, A. Zidovska,
- Morphological transitions of flexible filaments in shear flow Y. Liu, B. Chakrabarti, D. Saintillan, A. Lindner, O. du Roure,
- Microfluidic flow actuation using magnetoactive suspensions R. Alonso-Matilla, D. Saintillan,
- Rheology of active fluids D. Saintillan,

*Proceedings of the National Academy of Sciences*,

**115**11442 (2018). [pdf]

➠ In the news: Phys.org, ScienceDaily

The 3D spatiotemporal organization of the human genome inside the cell nucleus remains a major open question in cellular biology. In the time between two cell divisions, chromatin - the functional form of DNA in cells - fills the nucleus in its uncondensed polymeric form. Recent in vivo imaging experiments reveal that the chromatin moves coherently, having displacements with long-ranged correlations on the scale of micrometers and lasting for seconds. To elucidate the mechanism(s) behind these motions, we develop a coarse-grained active polymer model where chromatin is represented as a confined flexible chain acted upon by molecular motors that drive fluid flows by exerting dipolar forces on the system. Numerical simulations of this model account for steric and hydrodynamic interactions as well as internal chain mechanics. These demonstrate that coherent motions emerge in systems involving extensile dipoles and are accompanied by large-scale chain reconfigurations and nematic ordering. Comparisons with experiments show good qualitative agreement and support the hypothesis that self-organizing long-ranged hydrodynamic couplings between chromatin-associated active motor proteins are responsible for the observed coherent dynamics.

*Proceedings of the National Academy of Sciences*,

**115**9438 (2018). [pdf]

The morphological dynamics, instabilities, and transitions of elastic filaments in viscous flows underlie a wealth of biophysical processes from flagellar propulsion to intracellular streaming and are also key to deciphering the rheological behavior of many complex fluids and soft materials. Here, we combine experiments and computational modeling to elucidate the dynamical regimes and morphological transitions of elastic Brownian filaments in a simple shear flow. Actin filaments are used as an experimental model system and their conformations are investigated through fluorescence microscopy in microfluidic channels. Simulations matching the experimental conditions are also performed using inextensible Euler–Bernoulli beam theory and nonlocal slender-body hydrodynamics in the presence of thermal fluctuations and agree quantitatively with observations. We demonstrate that filament dynamics in this system are primarily governed by a dimensionless elasto-viscous number comparing viscous drag forces to elastic bending forces, with thermal fluctuations playing only a secondary role. While short and rigid filaments perform quasi-periodic tumbling motions, a buckling instability arises above a critical flow strength. A second transition to strongly deformed shapes occurs at a yet larger value of the elasto-viscous number and is characterized by the appearance of localized high-curvature bends that propagate along the filaments in apparent "snaking" motions. A theoretical model for the as yet unexplored onset of snaking accurately predicts the transition and explains the observed dynamics. We present a complete characterization of filament morphologies and transitions as a function of elasto-viscous number and scaled persistence length and demonstrate excellent agreement between theory, experiments, and simulations.

*Europhysics Letters*,

**121**24002 (2018). [pdf]

The rheological behavior of magnetotactic bacterial suspensions is analyzed using a continuum kinetic theory. In both unbounded and confined geometries, the response of these suspensions under simple external flows can be controlled by applying a magnetic field and hinges in a subtle way on the interplay of magnetic alignment, rotation under shear, and wall-induced accumulation under confinement. By tuning magnetic field strength and direction, the apparent viscosity can either be enhanced or reduced, and the mechanisms for these trends are elucidated. In the absence of any applied flow, we further demonstrate the ability of magnetoactive suspensions to internally drive steady unidirectional flows upon application of a magnetic field, thus suggesting novel avenues for the design of microfluidic pumps and flow actuation devices.

*Annual Review of Fluid Mechanics*,

**50**563-592 (2018). [pdf]

An active fluid denotes a viscous suspension of particles, cells, or macro-molecules able to convert chemical energy into mechanical work by generating stresses on the microscale. By virtue of this internal energy conversion, these systems display unusual macroscopic rheological signatures, including a curious transition to an apparent superfluid-like state where internal activity exactly compensates viscous dissipation. These behaviors are unlike those of classical complex fluids and result from the coupling of particle configurations with both externally applied flows and internally generated fluid disturbances. Focusing on the well-studied example of a suspension of microswimmers, this review summarizes recent experiments, models, and simulations in this area and highlights the critical role played by the rheological response of these active materials in a multitude of phenomena, from the enhanced transport of passive suspended objects to the emergence of spontaneous flows and collective motion.

#### 2017

- Electrohydrodynamics of viscous drops in strong electric fields: Numerical simulations D. Das, D. Saintillan,
- Geometric control of active collective motion M. Theillard, R. Alonso-Matilla, D. Saintillan,
- A nonlinear small-deformation theory for transient droplet electrohydrodynamics D. Das, D. Saintillan,

*Journal of Fluid Mechanics*,

**829**127-152 (2017). [pdf]

Weakly conducting dielectric liquid drops suspended in another dielectric liquid and subject to an applied uniform electric field exhibit a wide range of dynamical behaviours contingent on field strength and material properties. These phenomena are best described by the Melcher-Taylor leaky dielectric model, which hypothesizes charge accumulation on the drop-fluid interface and prescribes a balance between charge relaxation, the jump in ohmic currents from the bulk and charge convection by the interfacial fluid flow. Most previous numerical simulations based on this model have either neglected interfacial charge convection or restricted themselves to axisymmetric drops. In this work, we develop a three-dimensional boundary element method for the complete leaky dielectric model to systematically study the deformation and dynamics of liquid drops in electric fields. The inclusion of charge convection in our simulations permits us to investigate drops in the Quincke regime, in which experiments have demonstrated a symmetry-breaking bifurcation leading to steady electrorotation. Our simulation results show excellent agreement with existing experimental data and small-deformation theories.

*Soft Matter*,

**13**363-375 (2017). [pdf]

Recent experimental studies have shown that confinement can profoundly affect self-organization in semi-dilute active suspensions, leading to striking features such as the formation of steady and spontaneous vortices in circular domains and the emergence of unidirectional pumping motions in periodic racetrack geometries. Motivated by these findings, we analyze the two-dimensional dynamics in confined suspensions of active self-propelled swimmers using a mean-field kinetic theory where conservation equations for the particle configurations are coupled to the forced Navier-Stokes equations for the self-generated fluid flow. In circular domains, a systematic exploration of the parameter space casts light on three distinct states: equilibrium with no flow, stable vortex, and chaotic motion, and the transitions between these are explained and predicted quantitatively using a linearized theory. In periodic racetracks, similar transitions from equilibrium to net pumping to traveling waves to chaos are observed in agreement with experimental observations and are also explained theoretically. Our results underscore the subtle effects of geometry on the morphology and dynamics of emerging patterns in active suspensions and pave the way for the control of active collective motion in microfluidic devices.

*Journal of Fluid Mechanics*,

**810**225-253 (2017). [pdf]

The deformation of a viscous liquid droplet suspended in another liquid and subject to an applied electric field is a classic multiphase flow problem best described by the Melcher-Taylor leaky dielectric model. The main assumption of the model is that any net charge in the system is concentrated on the interface between the two liquids as a result of the jump in Ohmic currents from the bulk. Upon application of the field, the drop can either attain a steady prolate or oblate shape with toroidal circulating flows both inside and outside arising from tangential stresses on the interface due to action of the field on the surface charge distribution. Since the pioneering work of Taylor (1966), there have been numerous computational and theoretical studies to predict the deformations measured in experiments. Most existing theoretical models, however, have either neglected transient charge relaxation or nonlinear charge convection by the interfacial flow. In this work, we develop a novel small-deformation theory accurate to second order in electric capillary number for the complete Melcher-Taylor model that includes transient charge relaxation, charge convection by the flow, as well as transient shape deformation. The main result of the paper is the derivation of coupled evolution equations for the induced electric multipoles and for the shape functions describing the deformations on the basis of spherical harmonics. Our results, which are consistent with previous models in the appropriate limits, show excellent agreement with fully nonlinear numerical simulations based on an axisymmetric boundary element formulation and with existing experimental data in the small-deformation regime.

#### 2016

- Microfluidic rheology of active particle suspensions: Kinetic theory R. Alonso-Matilla, B. Ezhilan, D. Saintillan,
- Effect of flexibility on the growth of concentration fluctuations in a suspension of sedimenting fibers: Particle simulations H. Manikantan, D. Saintillan,

*Biomicrofluidics*,

**10**043505 (2016). [pdf]

➠ Invited article, special issue on "Microfluidic Rheology"

We analyze the effective rheology of a dilute suspension of self-propelled slender particles confined between two infinite parallel plates and subject to a pressure-driven flow. We use a continuum kinetic model to describe the configuration of the particles in the system, in which the disturbance flows induced by the swimmers are taken into account, and use it to calculate estimates of the suspension viscosity for a range of channel widths and flow strengths typical of microfluidic experiments. Our results are in agreement with previous bulk models, and in particular, demonstrate that the effect of activity is strongest at low flow rates, where pushers tend to decrease the suspension viscosity whereas pullers enhance it. In stronger flows, dissipative stresses overcome the effects of activity leading to increased viscosities followed by shear-thinning. The effects of confinement and number density are also analyzed, and our results confirm the apparent transition to superfluidity reported in recent experiments on pusher suspensions at intermediate densities. We also derive an approximate analytical expression for the effective viscosity in the limit of weak flows and wide channels, and demonstrate good agreement between theory and numerical calculations.

*Physics of Fluids*,

**28**013303 (2016). [pdf]

Three-dimensional numerical simulations are performed to study the stability of a sedimenting suspension of weakly flexible fibers. It is well known that a suspension of rigid rods sedimenting under gravity at low Reynolds number is unstable to concentration fluctuations owing to hydrodynamic interactions. Flexible fibers, however, reorient while settling and even weak flexibility can alter their collective dynamics. In our recent work (Manikantan 2014), we developed a mean-field theory to predict the linear stability of such a system. Here, we verify these predictions using accurate and efficient particle simulations based on a slender-body model. We also demonstrate the mechanisms by which flexibility-induced reorientation alters suspension microstructure, and through it, its stability. Specifically, we first show that the anisotropy of the base state in the case of a suspension of flexible fibers has a destabilizing effect compared to a suspension of rigid rods. Second, a conflicting effect of flexibility is also shown to suppress particle clustering and slow down the growth of the instability. The relative magnitude of filament flexibility and rotational Brownian motion dictates which effect dominates, and our simulations qualitatively follow theoretically predicted trends. The mechanism for either effect is tied to the flexibility-induced reorientation of particles, which we illustrate using velocity and orientation statistics from our simulations. Finally, we also show that, in the case of an initially homogeneous and isotropic suspension, flexibility always acts to suppress the growth of the instability.

#### 2015

- Buckling transition of a semiflexible filament in extensional flow H. Manikantan, D. Saintillan,
- On the distribution and swim pressure of run-and-tumble particles in confinement B. Ezhilan, R. Alonso-Matilla, D. Saintillan,
- Spontaneous flows in suspensions of active cyclic swimmers T. Brotto, D. Bartolo, D. Saintillan,
- Vapor-driven propulsion of catalytic micromotors R. Dong, J. Li, I. Rozen, B. Ezhilan, T. Xu, C. Christianson, W. Gao, D. Saintillan, B. Ren, J. Wang,
- Transport of a dilute active suspension in pressure-driven channel flow B. Ezhilan, D. Saintillan,
- Emergent vortices in populations of confined colloidal rollers A. Bricard, J.-B. Caussin, D. Das, C. Savoie, V. Chikkadi, K. Shitara, O. Chepizhko, F. Peruani, D. Saintillan, D. Bartolo,
- Motion-based threat detection using microrods: Experiments and numerical simulations B. Ezhilan, W. Gao, A. Pei, I Rozen, R. Dong, B. Jurado-Sanchez, J. Wang, D. Saintillan,

*Physical Review E: Rapid Communications*,

**92**041002 (2015). [pdf]

An analytical expression for the fluctuation-rounded stretch-coil transition of semiflexible polymers in extensional flows is derived. The competition between elasticity and tension is known to cause a buckling instability in filaments placed near hyperbolic stagnation points and the effect of thermal fluctuations on this transition has yet to receive full quantitative treatment. Motivated by the findings of recent experiments as well as our simulations, we solve for the amplitude of the first buckled mode near the onset of the instability. This reveals a stochastic supercritical bifurcation, which is in excellent agreement with full numerical simulations.

*Journal of Fluid Mechanics: Rapids*,

**781**R4 (2015). [pdf]

The spatial and orientational distribution in a dilute active suspension of non-Brownian run-and-tumble spherical swimmers confined between two planar hard walls is calculated theoretically. Using a kinetic model based on coupled bulk/surface probability density functions, we demonstrate the existence of a concentration wall boundary layer with thickness scaling with the run length, the absence of polarization throughout the bulk of the channel, and the presence of sharp discontinuities in the bulk orientation distribution in the neighborhood of orientations parallel to the wall in the near-wall region. Our model is also applied to calculate the swim pressure in the system, which approaches the previously proposed ideal-gas behavior in wide channels but is found to decrease in narrow channels as a result of confinement. Monte-Carlo simulations are also performed for validation and show excellent quantitative agreement with our theoretical predictions.

*Journal of Nonlinear Science*,

**25**1125-1139 (2015). [pdf]

➠ Invited article, special issue on "Emergent Collective Behavior"

➠ Also see the editorial note by E. Kanso and D. Saintillan

Many swimming cells rely on periodic deformations to achieve locomotion. We introduce in this work a theoretical model and numerical simulations in order to elucidate the impact of these cyclic strokes on the emergence of mesoscale structures and collective motion in swimmer suspensions. The model extends previous kinetic theories for populations of identical swimmers to the case of self-propelled particles undergoing transitions between pusher and puller states, and is applied to quantify how the unsteadiness of the hydrodynamic velocity field, to which each swimmer population contributes, affects the onset and characteristics of spontaneous flows. A linear stability analysis reveals that the sign of the population-averaged dipole determines the stability of the uniform isotropic state, with suspensions dominated by pushers being subject to growing nematic bend fluctuations. Stochastic transitions, however, are also seen to provide an additional damping mechanism. To investigate the population dynamics above the instability threshold, we also perform direct particle simulations based on a slender-body model, where the growth or decay of the active power generated by the swimmers is found to be a robust measure of the structural and dynamical instability.

*Scientific Reports*,

**5**13226 (2015). [pdf]

Chemically-powered micromotors offer exciting opportunities in diverse fields, including therapeutic delivery, environmental remediation, and nanoscale manufacturing. However, these nanovehicles require direct addition of high concentration of chemical fuel to the motor solution for their propulsion. We report the efficient vapor-powered propulsion of catalytic micromotors without direct addition of fuel to the micromotor solution. Diffusion of hydrazine vapor from the surrounding atmosphere into the sample solution is instead used to trigger rapid movement of iridium-gold Janus microsphere motors. Such operation creates a new type of remotely-triggered and powered catalytic micro/nanomotors that are responsive to their surrounding environment. This new propulsion mechanism is accompanied by unique phenomena, such as the distinct off-on response to the presence of fuel in the surrounding atmosphere, and spatio-temporal dependence of the motor speed borne out of the concentration gradient evolution within the motor solution. The relationship between the motor speed and the variables affecting the fuel concentration distribution is examined using a theoretical model for hydrazine transport, which is in turn used to explain the observed phenomena. The vapor-powered catalytic micro/nanomotors offer new opportunities in gas sensing, threat detection, and environmental monitoring, and open the door for a new class of environmentally-triggered micromotors.

*Journal of Fluid Mechanics*,

**777**482-522 (2015). [pdf]

Confined suspensions of active particles show peculiar dynamics characterized by wall accumulation, as well as upstream swimming, centerline depletion and shear-trapping when a pressure-driven flow is imposed. We use theory and numerical simulations to investigate the effects of confinement and non-uniform shear on the dynamics of a dilute suspension of Brownian active swimmers by incorporating a detailed treatment of boundary conditions within a simple kinetic model where the configuration of the suspension is described using a conservation equation for the probability distribution function of particle positions and orientations, and where particle-particle and particle-wall hydrodynamic interactions are neglected. Based on this model, we first investigate the effects of confinement in the absence of flow, in which case the dynamics is governed by a swimming Peclet number, or ratio of the persistence length of particle trajectories over the channel width, and a second swimmer-specific parameter whose inverse measures the strength of propulsion. In the limit of weak and strong propulsion, asymptotic expressions for the full distribution function are derived. For finite propulsion, analytical expressions for the concentration and polarization profiles are also obtained using a truncated moment expansion of the distribution function. In agreement with experimental observations, the existence of a concentration/polarization boundary layer in wide channels is reported and characterized, suggesting that wall accumulation in active suspensions is primarily a kinematic effect which does not require hydrodynamic interactions. Next, we show that application of a pressure-driven Poiseuille flow leads to net upstream swimming of the particles relative to the flow, and an analytical expression for the mean upstream velocity is derived in the weak flow limit. In stronger imposed flows, we also predict the formation of a depletion layer near the channel centerline, due to cross-streamline migration of the swimming particles towards high-shear regions where they become trapped, and an asymptotic analysis in the strong flow limit is used to obtain a scale for the depletion layer thickness and to rationalize the non-monotonic dependence of the intensity of depletion upon flow rate. Our theoretical predictions are all shown to be in excellent agreement with finite-volume numerical simulations of the kinetic model, and are also supported by recent experiments on bacterial suspensions in microfluidic devices.

*Nature Communications*,

**6**7470 (2015). [pdf]

Coherent vortical motion has been reported in a wide variety of populations including living organisms (bacteria, fishes, human crowds) and synthetic active matter (shaken grains, mixtures of biopolymers), yet a unified description of the formation and structure of this pattern remains lacking. Here we report the self-organization of motile colloids into a macroscopic steadily rotating vortex. Combining physical experiments and numerical simulations, we elucidate this collective behaviour. We demonstrate that the emergent-vortex structure lives on the verge of a phase separation, and single out the very constituents responsible for this state of polar active matter. Building on this observation, we establish a continuum theory and lay out a strong foundation for the description of vortical collective motion in a broad class of motile populations constrained by geometrical boundaries.

*Nanoscale*,

**7**7833-7840 (2015). [pdf]

Motion-based chemical sensing using microscale particles has attracted considerable recent attention. In this paper, we report on new experiments and Brownian dynamics simulations that cast light on the dynamics of both passive and active microrods (gold wires and gold-platinum micromotors) in a silver ion gradient. We demonstrate that such microrods can be used for threat detection in the form of a silver ion source, allowing for the determination of both the location of the source and concentration of silver. This threat detection strategy relies on the diffusiophoretic motion of both passive and active microrods in the ionic gradient and on the speed acceleration of the Au-Pt micromotors in the presence of silver ions. A Langevin model describing the microrod dynamics and accounting for all of these effects is pre- sented, and key model parameters are extracted from the experimental data, thereby providing a reliable estimate for the full spatiotemporal distribution of the silver ions in the vicinity of the source.

#### 2014

- The instability of a sedimenting suspension of weakly flexible fibres H. Manikantan, L. Li, S. Spagnolie, D. Saintillan,
- Bubble-propelled micromotors for enhanced transport of passive tracers J. Orozco, B. Jurado-Sanchez, G. Wagner, W. Gao, R. Vazquez- Duhalt, S. Sattayasamitsathit, M. Galarnyk, A. Cortes, D. Saintillan, J. Wang,
- Swimming in shear D. Saintillan,
- Globally aligned states and hydrodynamic traffic jams in confined suspensions of active asymmetric particles A. Lefauve, D. Saintillan,

*Journal of Fluid Mechanics*,

**756**935-964 (2014). [pdf]

Suspensions of sedimenting slender fibres in a viscous fluid are known to be unstable to fluctuations of concentration. In this paper we develop a theory for the role of fibre flexibility in sedimenting suspensions in the asymptotic regime of weakly-flexible bodies (large elasto-gravitation number). Unlike the behaviour of straight fibres, individual flexible filaments rotate as they sediment, leading to an anisotropic base state of fibre orientations in an otherwise homogeneous suspension. A mean-field theory is derived to describe the evolution of fibre concentration and orientation fields, and we explore the stability of the base state to perturbations of fibre concentration. We show that fibre flexibility affects suspension stability in two distinct and competing ways: the anisotropy of the base state renders the suspension more unstable to perturbations, while individual particle self-rotation acts to prevent clustering and stabilizes the suspension. In the presence of thermal noise, the dominant effect depends critically upon the relative scales of flexible fibre self-rotation compared to rotational Brownian motion.

*Langmuir*,

**30**5082-5087 (2014). [pdf]

Fluid convection and mixing induced by bubble-propelled tubular microengines are characterized using passive microsphere tracers. Enhanced transport of the passive tracers by bubble-propelled micromotors, indicated by their mean squared displacement (MSD), is dramatically larger than that observed in the presence of catalytic nanowires and Janus particle motors. Bubble generation is shown to play a dominant role in the effective fluid transport observed in the presence of tubular microengines. These findings further support the potential of using bubble-propelled microengines for mixing reagents and accelerating reaction rates. The study offers useful insights toward understanding the role of the motion of multiple micromotors, bubble generation, and additional factors (e.g., motor density and fuel concentration) upon the observed motor-induced fluid transport.

*Journal of Fluid Mechanics*,

**744**1 (2014). [pdf]

➠ Invited "Focus on Fluids" article

The complex patterns observed in experiments on suspensions of swimming cells undergoing bioconvection have fascinated biologists, physicists and mathematicians alike for over a century. Theoretical models developed over the last few decades have shown a strong similarity with Rayleigh-Benard thermal convection, albeit with a richer dynamical behavior owing to the orientational degrees of freedom of the cells. In a recent paper, Hwang and Pedley (2013) revisit previous models for bioconvection to investigate the effects of an external shear flow on pattern formation. In addition to casting light on new mechanisms for instability, their study demonstrates a subtle interplay between shear, swimming motions and bioconvection patterns.

*Physical Review E: Rapid Communications*,

**89**021002 (2014). [pdf]

Strongly confined active liquids are subject to unique hydrodynamic interactions due to momentum screening and lubricated friction by the confining walls. Using numerical simulations, we demonstrate that two-dimensional dilute suspensions of fore-aft asymmetric polar swimmers in a Hele-Shaw geometry can exhibit a rich variety of novel phase behaviors depending on particle shape, including coherent polarized density waves with global alignment, persistent counterrotating vortices, density shocks and rarefaction waves. We also explain these phenomena using a linear stability analysis and a nonlinear traffic flow model, both derived from a mean-field kinetic theory.

#### 2013

- The sedimentation of flexible filaments L. Li, H. Manikantan, D. Saintillan, S. E. Spagnolie,
- Active suspensions and their nonlinear models D. Saintillan, M. J. Shelley,
- Instabilities and nonlinear dynamics of concentrated active suspensions B. Ezhilan, M. J. Shelley, D. Saintillan,
- Report on the IUTAM Symposium on Mobile Particle Systems: Kinematics, Rheology and Complex Phenomena P. R. Nott, R. H. Davis, M. Reeks, D. Saintillan, S. Sundaresan,
- Subdiffusive transport of fluctuating elastic filaments in cellular flows H. Manikantan, D. Saintillan,
- Electrohydrodynamic interaction of spherical particles under Quincke rotation D. Das, D. Saintillan,

*Journal of Fluid Mechanics*,

**735**705 (2013). [pdf]

The dynamics of a flexible filament sedimenting in a viscous fluid are explored analytically and numerically. Compared with the well-studied case of sedimenting rigid rods, the introduction of filament compliance is shown to cause a significant alteration in the long-time sedimentation orientation and filament geometry. A model is developed by balancing viscous, elastic and gravitational forces in a slender-body theory for zero-Reynolds-number flows, and the filament dynamics are characterized by a dimensionless elasto-gravitation number. Filaments of both non-uniform and uniform cross-sectional thickness are considered. In the weakly flexible regime, a multiple-scale asymptotic expansion is used to obtain expressions for filament translations, rotations and shapes. These are shown to match excellently with full numerical simulations. Furthermore, we show that trajectories of sedimenting flexible filaments, unlike their rigid counterparts, are restricted to a cloud whose envelope is determined by the elasto-gravitation number. In the highly flexible regime we show that a filament sedimenting along its long axis is susceptible to a buckling instability. A linear stability analysis provides a dispersion relation, illustrating clearly the competing effects of the compressive stress and the restoring elastic force in the buckling process. The instability travels as a wave along the filament opposite the direction of gravity as it grows and the predicted growth rates are shown to compare favourably with numerical simulations. The linear eigenmodes of the governing equation are also studied, which agree well with the finite-amplitude buckled shapes arising in simulations.

*Comptes Rendus Physique*,

**14**497 (2013). [pdf]

➠ Invited review, special thematic issue on "Living Fluids"

Active suspensions, such as suspensions of self-propelled microorganisms and related synthetic microswimmers, are known to undergo complex dynamics and pattern formation as a result of hydrodynamic interactions. In this review, we summarize recent efforts to model these systems using continuum kinetic theories. We first derive a basic kinetic model for a suspension of self-propelled rodlike particles and discuss its stability and nonlinear dynamics. We then present extensions of this model to analyze the effective rheology of active suspensions in external flows, the effect of steric interactions in concentrated systems, and the dynamics of chemotactically responsive suspensions in chemical fields.

*Physics of Fluids*,

**25**070607 (2013). [pdf]

➠ Featured on the cover of

*Physics of Fluids*

➠ Invited paper, special section on "Mobile Particulate Systems"

Suspensions of active particles, such as motile microorganisms and artificial microswimmers, are known to undergo a transition to complex large-scale dynamics at high enough concentrations. While a number of models have demonstrated that hydrodynamic interactions can in some cases explain these dynamics, collective motion in experiments is typically observed at such high volume fractions that steric interactions between nearby swimmers are significant and cannot be neglected. This raises the question of the respective roles of steric vs hydrodynamic interactions in these dense systems, which we address in this paper using a continuum theory and numerical simulations. The model we propose is based on our previous kinetic theory for dilute suspensions, in which a conservation equation for the distribution function of particle configurations is coupled to the Stokes equations for the fluid motion (Saintillan and Shelley 2008). At high volume fractions, steric interactions are captured by extending classic models for concentrated suspensions of rodlike polymers, in which contacts between nearby particles cause them to align locally. In the absence of hydrodynamic interactions, this local alignment results in a transition from an isotropic base state to a nematic base state when volume fraction is increased. Using a linear stability analysis, we first investigate the hydrodynamic stability of both states. Our analysis shows that suspensions of pushers, or rear-actuated swimmers, typically become unstable in the isotropic state before the transition occurs; suspensions of pullers, or head-actuated swimmers, can also become unstable, though the emergence of unsteady flows in this case occurs at a higher concentration, above the nematic transition. These results are also confirmed using fully nonlinear numerical simulations in a periodic cubic domain, where pusher and puller suspensions are indeed both found to exhibit instabilities at sufficiently high volume fractions; these instabilities lead to unsteady chaotic states characterized by large-scale correlated motions and strong density fluctuations. While the dynamics in suspensions of pushers are similar to those previously reported in the dilute regime, the instability of pullers is novel and typically characterized by slower dynamics and weaker hydrodynamic velocities and active input power than in pusher suspensions at the same volume fraction.

*Physics of Fluids*,

**25**070501 (2013). [pdf]

This report summarizes the presentations and discussions conducted during the sym- posium, which was held under the aegis of the International Union of Theoretical and Applied Mechanics during 23-27 January 2012 in Bangalore, India.

*Physics of Fluids*,

**25**073603 (2013). [pdf]

The dynamics and transport properties of Brownian semiflexible filaments suspended in a two-dimensional array of counter-rotating Taylor-Green vortices are investigated using numerical simulations based on slender-body theory for low-Reynolds-number hydrodynamics. Such a flow setup has been previously proposed to capture some of the dynamics of biological polymers in motility assays. A buckling instability permits elastic filaments to migrate across such a cellular lattice in a Brownian-like manner even in the athermal limit. However, thermal fluctuations alter these dynamics qualitatively by driving polymers across streamlines, leading to their frequent trapping within vortical cells. As a result, thermal fluctuations, characterized here by the persistence length, are shown to lead to subdiffusive transport at long times, and this qualitative shift in behavior is substantiated by the slow decay of waiting-time distributions as a consequence of trapping events during which the filaments remain in a particular cell for extended periods of time. Velocity and mass distributions of polymers reveal statistically preferred positions within a unit cell that further corroborate this systematic shift from transport to trapping with increasing fluctuations. Comparisons to results from a continuum model for the complemen- tary case of rigid Brownian rods in such a flow also highlight the role of elastic flexibility in dictating the nature of polymer transport.

*Physical Review E*,

**87**043014 (2013). [pdf]

Weakly conducting dielectric particles suspended in a dielectric liquid of higher conductivity can undergo a transition to spontaneous sustained rotation when placed in a sufficiently strong dc electric field. This phenomenon of Quincke rotation has interesting implications for the rheology of these suspensions, whose effective viscosity can be controlled and reduced by application of an external field. While previous models based on the rotation of isolated particles have provided accurate estimates for this viscosity reduction in dilute suspensions, discrepancies have been reported in more concentrated systems where particle-particle interactions are likely significant. Motivated by this observation, we extend the classic description of Quincke rotation based on the Taylor-Melcher leaky dielectric model to account for pair electrohydrodynamic interactions between two identical spheres using the method of reflections. A coupled system of evolution equations for the dipole moments and angular velocities of the spheres is derived that accounts for electric dipole-dipole interactions and hydrodynamic rotlet interactions up to order O(R^{5}), where R is the separation distance between the spheres. A linear stability analysis of this system shows that interactions modify the value of the critical electric field for the onset of Quincke rotation: both electric and hydrodynamic interactions can either stabilize or destabilize the system depending on the orientation of the spheres, but the leading effect of interactions on the onset of rotation is hydrodynamic. We also analyze the dynamics in the nonlinear regime by performing numerical simulations of the governing equations. In the case of a pair of spheres that are fixed in space, we find that particle rotations always synchronize in magnitude at long times, though the directions of rotation of the spheres need not be the same. The steady-state angular velocity magnitude depends on the configuration of the spheres and electric field strength and agrees very well with an asymptotic estimate derived for corotating spheres. In the case of freely-suspended spheres, dipolar interactions are observed to lead to a number of distinct behaviors depending on the initial relative configuration of the spheres and on any infinitesimal initial perturbation introduced in the system: in some cases the spheres slowly separate in space while steadily rotating, while in other cases they pair up and either corotate or counterrotate depending on their orientation relative to the field.

#### 2012

- Chaotic dynamics and oxygen transport in thin films of aerotactic bacteria B. Ezhilan, A. Alizadeh Pahlavan, D. Saintillan,
- Concentration instability of sedimenting spheres in a second-order fluid R. Vishnampet, D. Saintillan,
- Emergence of coherent structures and large-scale flows in motile suspensions D. Saintillan, M. J. Shelley,

*Physics of Fluids*,

**24**091701 (2012). [pdf]

A kinetic model and three-dimensional numerical simulations are applied to study the dynamics in suspensions of run-and-tumble aerotactic bacteria confined in free-standing liquid films surrounded by air. In thin films, oxygen and bacterial concentration profiles approach steady states. In thicker films, a transition to chaotic dynamics is shown to occur and is characterized by unsteady correlated motions, the formation of bacterial plumes, and enhanced oxygen transport and consumption. This transition, also observed in previous experiments, arises as a result of the coupling between the aerotactic response of the bacteria and the flow fields they generate via hydrodynamic interactions.

*Physics of Fluids*,

**24**073302 (2012). [pdf]

The slow sedimentation of a dilute suspension of spherical particles in a second-order fluid is investigated using theory and numerical simulations. We first analyze the motion of a single isolated spherical particle sedimenting under gravity when placed in a linear flow field. In the limit of weak viscoelasticity (low Deborah number), the velocity of the particle is calculated, and the nonlinear coupling of the settling motion with the local flow field is shown to result in a lateral drift in a direction perpendicular to gravity. By the same effect, the mean flow driven by weak horizontal density fluctuations in a large-scale suspension of hydrodynamically interacting particles will also result in a horizontal drift, which has the effect of reinforcing the fluctuations as we demonstrate using a linear stability analysis. Based on this mechanism, an initially homogeneous suspension is expected to develop concentration fluctuations, a prediction supported by previous experiments on sedimentation in polymeric liquids. We further confirm this prediction using large-scale weakly nonlinear numerical simulations based on a point-particle model. Concentration fluctuations are indeed found to grow in the simulations, and are shown to result in an enhancement of the mean settling speed and velocity fluctuations compared to the Newtonian case.

*Journal of the Royal Society Interface*,

**9**571 (2012). [pdf]

The emergence of coherent structures, large-scale flows and correlated dynamics in suspensions of motile particles such as swimming micro-organisms or artificial microswimmers is studied using direct particle simulations. A detailed model is proposed for a slender rod-like particle that propels itself in a viscous fluid by exerting a prescribed tangential stress on its surface, and a method is devised for the efficient calculation of hydrodynamic interactions in large-scale suspensions of such particles using slender-body theory and a smooth particle-mesh Ewald algorithm. Simulations are performed with periodic boundary conditions for various system sizes and suspension volume fractions, and demonstrate a transition to large-scale correlated motions in suspensions of rear-actuated swimmers, or Pushers, above a critical volume fraction or system size. This transition, which is not observed in suspensions of head-actuated swimmers, or Pullers, is seen most clearly in particle velocity and passive tracer statistics. These observations are consistent with predictions from our previous mean-field kinetic theory, one of which states that instabilities will arise in uniform isotropic suspensions of Pushers when the product of the linear system size with the suspension volume fraction exceeds a given threshold. We also find that the collective dynamics of Pushers result in giant number fluctuations, local alignment of swimmers and strongly mixing flows. Suspensions of Pullers, which evince no large-scale dynamics, nonetheless display interesting deviations from the random isotropic state.

#### 2011

- From diffusive motion to local aggregation: Effect of surface contamination in dipolophoresis J. S. Park, D. Saintillan,
- Geometrically designing the kinematic behavior of catalytic nanomotors J. G. GIbbs, S. Kothari, D. Saintillan, Y.-P. Zhao,
- Electric-field-induced ordering and pattern formation in colloidal suspensions J. S. Park, D. Saintillan,
- Instability regimes in flowing suspensions of swimming micro-organisms A. Alizadeh Pahlavan, D. Saintillan,

*Soft Matter*,

**7**10720 (2011). [pdf]

We investigate the effects of surface contamination, modeled as a thin dielectric coating, on the dynamics in suspensions of ideally polarizable spheres in an applied electric field using large-scale direct particle simulations. In the case of clean particles (no contamination), the suspensions are known to undergo dipolophoresis, or a combination of dielectrophoresis, which tends to cause particle chaining and aggregation, and induced-charge electrophoresis, which dominates the dynamics and drives transient pairings, chaotic motions, and hydrodynamic diffusion at long times. As surface contamination becomes significant, induced-charge electrophoresis is gradually suppressed, which results in the simulations in a transition from diffusive dynamics to local aggregation and chaining as a result of dielectrophoresis. This effect has a strong impact on the suspension microstructure, as well as on particle velocities, which are strongly reduced for contaminated particles. This transition is also visible in the particle mean-square displacements, which become sub-diffusive in the case of strong contamination. We explain this sub-diffusive regime as a consequence of the slow dynamics of the particles trapped inside clusters and chains, which result in non-integrable local waiting time distributions.

*Nano Letters*,

**11**2543 (2011). [pdf]

Catalytic nanomotors with silica microbead heads and TiO2 arms are systematically designed by dynamic shadowing growth. The swimming trajectories are fine tuned by altering the arm length and orientation exploiting geometry-dependent hydrodynamic interactions at low Reynolds number. The curvature, angular frequency, and radius of curvature of the trajectories change as a function of arm length. Simulations based on the method of regularized Stokeslets are also described and correctly capture the trends observed in the experiments.

*Physical Review E*,

**83**041409 (2011). [pdf]

The long-time dynamics and pattern formation in semidilute suspensions of colloidal spheres in a viscous electrolyte under a uniform electric field are investigated using numerical simulations. The rapid chain formation that occurs in the field direction as a result of dipolar interactions is found to be followed by a slow coarsening process by which chains coalesce into hexagonal sheets and eventually rearrange to form mesoscale cellular structures, in qualitative agreement with recent experiments. The morphology and characteristic wavelength of the patterns that emerge at steady state are shown to depend on the suspension's volume fraction, electrode spacing, and field strength, suggesting additional ways of controlling effective suspension properties in practical applications.

*Physics of Fluids*,

**23**011901 (2011). [pdf]

➠ Featured as a "Research Highlight" on the

*Physics of Fluids*website

The effects of an external shear flow on the dynamics and pattern formation in a dilute suspension of swimming micro-organisms are investigated using a linear stability analysis and three-dimensional numerical simulations, based on the kinetic model previously developed by Saintillan and Shelley (2008). The external shear flow is found to damp the instabilities that occur in these suspensions by controlling the orientation of the particles. We demonstrate in our simulations that the rate of damping is direction-dependent: it is fastest in the flow direction, but slowest in the direction perpendicular to the shear plane. As a result, transitions from three- to two- to one-dimensional instabilities are observed to occur as shear rate increases, and above a certain shear rate the instabilities altogether disappear. The density patterns and complex flows that arise at long time in the suspensions are also analyzed from the numerical simulations using standard techniques from the literature on turbulent flows. The imposed shear flow is found to have an effect on both density patterns and flow structures, which typically align with the extensional axis of the external flow. The disturbance flows in the simulations are shown to exhibit similarities with turbulent flows, and in particular two of the seemingly universal characteristics of turbulent flows also occur, namely: (i) the bias of Q-R plots toward the second and fourth quadrants, corresponding to stable focus/stretching and unstable node/saddle/saddle flow topologies, respectively, and (ii) the alignment of the vorticity vector with the intermediate strain-rate eigenvector. However, the flows described herein also significantly differ from turbulent flows owing to the strong predominance of large scales, as exemplified by the very rapid decay of the kinetic energy spectrum, an effect further enhanced after the transitions to two- and one-dimensional instabilities.

#### 2010

- The dilute rheology of swimming suspensions: A simple kinetic model D. Saintillan,
- A quantitative look into microorganism hydrodynamics D. Saintillan,
- Dipolophoresis in large-scale suspensions of ideally polarizable spheres J. S. Park, D. Saintillan,
- Extensional rheology of active suspensions D. Saintillan,

*Experimental Mechanics*,

**50**1275 (2010). [pdf]

➠ Special issue on "Mechanical Aspects of Biological Locomotion"

A simple kinetic model is presented for the shear rheology of a dilute suspension of particles swimming at low Reynolds number. If interparticle hydrodynamic interactions are neglected, the configuration of the suspension is characterized by the particle orientation distribution, which satisfies a Fokker-Planck equa- tion including the effects of the external shear flow, rotary diffusion, and particle tumbling. The orientation distribution then determines the leading-order term in the particle extra stress in the suspension, which can be evaluated based on the classic theory of Hinch and Leal (1972), and involves an additional contribution arising from the permanent force dipole exerted by the particles as they propel themselves through the fluid. Numerical solutions of the steady-state Fokker-Planck equation were obtained using a spectral method, and results are reported for the shear viscosity and normal stress difference coefficients in terms of flow strength, rotary diffusivity, and correlation time for tumbling. It is found that the rheology is characterized by much stronger normal stress differences than for passive suspensions, and that tail-actuated swimmers result in a strong decrease in the effective shear viscosity of the fluid.

*Physics*,

**3**84 (2010). [pdf]

➠ Invited "Viewpoint" article

Direct measurements show that the fluid flow around swimming microorganisms is more complex than previ- ously thought, with important implications for how they interact and behave.

*Journal of Fluid Mechanics*,

**662**66 (2010). [pdf]

The nonlinear dynamics of uncharged ideally polarizable spheres freely suspended in a viscous electrolyte in a uniform electric field are analysed using theory and numerical simulations. When a sphere polarizes under the action of the field, it acquires a non-uniform surface charge, which results in an electro-osmotic flow near its surface that scales quadratically with the applied field magnitude. While this so-called induced-charge electrophoresis yields no net motion in the case of a single sphere, it can drive relative motions by symmetry breaking when several particles are present. In addition, Maxwell stresses in the fluid also result in non-zero dielectrophoretic forces, which also cause particle motions. The combination of these two nonlinear electrokinetic effects, termed dipolophoresis, is analysed in detail by using numerical simulations. An efficient simulation method based on our previous analysis of pair interactions is presented and accounts for both far-field and near-field electric and hydrodynamic interactions in the thin-Debye-layer limit, as well as steric interactions using a novel contact algorithm. Simulation results in large-scale suspensions with periodic boundary conditions are presented. While the dynamics under dielectrophoresis alone are shown to be characterized by particle chaining along the field direction, in agreement with previous investigations, chaining is not found to occur under dipolophoresis, which instead causes transient particle pairings and results in a non-uniform microstructure with large number of density fluctuations, as we demonstrate by calculating pair distribution functions and particle occupancy statistics. Dipolophoresis is also found to result in significant hydrodynamic dispersion and velocity fluctuations, and the dependence of these two effects on suspension volume fraction is investigated.

*Physical Review E*,

**81**056307 (2010). [pdf]

➠ Featured in

*Physical Review*E's Kaleidoscope

A simple model is presented for the effective extensional rheology of a dilute suspension of active particles, such as self-propelled microswimmers, extending previous classical studies on suspensions of passive rodlike particles. Neglecting particle-particle hydrodynamic interactions, we characterize the configuration of the sus- pension by an orientation distribution, which satisfies a Fokker-Planck equation including the effects of an external flow field and of rotary diffusion. Knowledge of this orientation distribution then allows the determi- nation of the particle extra stress as a configurational average of the force dipoles exerted by the particles on the fluid, which involve contributions from the imposed flow, rotary diffusion, and the permanent dipoles resulting from activity. Analytical expressions are obtained for the stress tensor in uniaxial extensional and compressional flows, as well as in planar extensional flow. In all types of flows, the effective viscosity is found to increase as a result of activity in suspensions of head-actuated swimmers (pullers) and to decrease in suspensions of tail-actuated swimmers (pushers). In the latter case, a negative particle viscosity is found to occur in weak flows. In planar extensional flow, we also characterize normal stresses, which are enhanced by activity in suspensions of pullers but reduced in suspensions of pushers. Finally, an energetic interpretation of the seemingly unphysical decrease in viscosity predicted in suspensions of pushers is proposed, where the decrease is explained as a consequence of the active power input generated by the swimming particles and is shown not to be directly related to viscous dissipative processes.

#### 2009

- Falling jets of particles in viscous fluids F. Pignatel, M. Nicolas, E. Guazzelli, D. Saintillan,
- Hydrodynamic interactions in metal rod-like particle suspensions due to induced charge electroosmosis K. A. Rose, B. Hoffman, D. Saintillan, E. S. G. Shaqfeh, J. G. Santiago,

*Physics of Fluids*,

**21**123303 (2009). [pdf]

➠ Featured on the cover of

*Physics of Fluids*

We have investigated the time evolution of a jet of non-Brownian particles falling under the action of gravity in a viscous liquid at low Reynolds number. Different regimes have been observed depending on volume fraction and particle-to-jet diameter ratio. In particular, the jet has been found to be unstable and to develop varicose modulation of its diameter at low volume fractions. The dominant wavelength and saturated amplitude have been measured and are observed to decrease with increasing volume fraction. A simple numerical simulation using point particles is able to capture the major features of the instability.

*Physical Review E*,

**79**011402 (2009). [pdf]

We present a theoretical and experimental study of the role of hydrodynamic interactions on the motion and dispersion of metal rodlike particles in the presence of an externally applied electric field. In these systems, the electric field polarizes the particles and induces an electroosmotic flow relative to the surface of each particle. The simulations include the effect of the gravitational body force, buoyancy, far-field hydrodynamic interactions, and near-field lubrication forces. The particles in the simulations and experiments were observed to experience repeated pairing interactions in which they come together axially with their ends approaching each other, slide past one another until their centers approach, and then push apart. These interactions were confirmed in measurements of particle orientations and velocities, pair distribution functions, and net dispersion of the suspension. For large electric fields, the pair distribution functions show accumulation and depletion regions consistent with many pairing events. For particle concentrations of 108 particles/mL and higher, dispersion within the suspension dramatically increases with increased field strength.

#### 2008

- Instabilities, pattern formation, and mixing in active suspensions D. Saintillan, M. J. Shelley,
- Nonlinear interactions in electrophoresis of ideally polarizable particles D. Saintillan,
- Instabilities and pattern formation in active particle suspensions: Kinetic theory and continuum simulations D. Saintillan, M. J. Shelley,

*Physics of Fluids*,

**20**123304 (2008). [pdf]

➠ Also selected to appear in the

*Virtual Journal of Biological Physics Research*, January 1 2009

Suspensions of self-propelled particles, such as swimming micro-organisms, are known to undergo complex dynamics as a result of hydrodynamic interactions. To elucidate these dynamics, a kinetic theory is developed and applied to study the linear stability and the nonlinear pattern formation in these systems. The evolution of a suspension of self-propelled particles is modeled using a conservation equation for the particle configurations, coupled to a mean-field description of the flow arising from the stress exerted by the particles on the fluid. Based on this model, we first investigate the stability of both aligned and isotropic suspensions. In aligned suspensions, an instability is shown to always occur at finite wavelengths, a result that extends previous predictions by Simha and Ramaswamy (2002). In isotropic suspensions, we demonstrate the existence of an instability for the active particle stress, in which shear stresses are eigenmodes and grow exponentially at long scales. Nonlinear effects are also investigated using numerical simulations in two dimensions. These simulations confirm the results of the stability analysis, and the long-time nonlinear behavior is shown to be characterized by the formation of strong density fluctuations, which merge and breakup in time in a quasiperiodic fashion. These complex motions result in very efficient fluid mixing, which we quantify by means of a multiscale mixing norm.

*Physics of Fluids*,

**20**067104 (2008). [pdf]

In the classical analysis of electrophoresis, particle motion is a consequence of the interfacial fluid slip that arises inside the ionic charge cloud or Debye screening layer surrounding the particle surface when an external field is applied. Under the assumptions of thin Debye layers, weak applied fields, and zero polarizability, it can be shown that the electrophoretic velocity of a collection of particles with identical zeta potential is the same as that of an isolated particle, unchanged by interactions (Reed and Morrison 1976). When some of these assumptions are relaxed, nonlinear effects may also arise and result in relative motions. First, the perturbation of the external field around the particles creates field gradients, which may result in nonzero dielectrophoretic forces due to Maxwell stresses in the fluid. In addition, if the particles are able to polarize, they can acquire a nonuniform surface charge, and the action of the field on the dipolar charge clouds surrounding them drives disturbance flows in the fluid, causing relative motions by induced-charge electrophoresis. These two nonlinear effects are analyzed in detail in the prototypical case of two equal-sized ideally polarizable spheres carrying no net charge, using accurate boundary-element simulations, along with asymptotic calculations by the method of twin multipole expansions and the method of reflections. It is found that both types of interactions result in significant relative motions and can be either attractive or repulsive depending on the configuration of the spheres.

*Physical Review Letters*,

**100**178103 (2008). [pdf]

➠ Also selected to appear in the

*Virtual Journal of Biological Physics Research*, May 1 2008

We use kinetic theory and nonlinear continuum simulations to study the collective dynamics in suspensions of self-propelled particles. The stability of aligned suspensions is first analyzed, and we demonstrate that such suspensions are always unstable to fluctuations, a result that generalizes previous predictions by Simha and Ramaswamy. Isotropic suspensions are also considered, and it is shown that an instability for the particle stress occurs in that case. Using simulations, nonlinear effects are investigated, and the long-time behavior of the suspensions is observed to be characterized by the formation of strong density fluctuations, resulting in efficient fluid mixing.

#### 2007

- Orientational order and instabilities in suspensions of self-locomoting rods D. Saintillan, M. J. Shelley,

*Physical Review Letters*,

**99**058102 (2007). [pdf]

➠ Also selected to appear in the

*Virtual Journal of Biological Physics Research*, August 1 2007

The orientational order and dynamics in suspensions of self-locomoting slender rods are investigated numerically. In agreement with previous theoretical predictions, nematic suspensions of swimming particles are found to be unstable at long wavelengths as a result of hydrodynamic fluctuations. Nevertheless, a local nematic ordering is shown to persist over short length scales and to have a significant impact on the mean swimming speed. The consequences of the large-scale orientational disorder for particle dispersion are also discussed.

#### 2006

- Stabilization of a suspension of sedimenting rods by induced-charge electrophoresis D. Saintillan, E. S. G. Shaqfeh, E. Darve,
- The effect of stratification on the wave number selection in the instability of sedimenting spheroids D. Saintillan, E. S. G. Shaqfeh, E. Darve,
- Hydrodynamic interactions in the induced-charge electrophoresis of colloidal rod dispersions D. Saintillan, E. Darve, E. S. G. Shaqfeh,
- Effect of flexibility on the shear-induced migration of short-chain polymers in parabolic channel flow D. Saintillan, E. S. G. Shaqfeh, E. Darve,
- The growth of concentration fluctuations in dilute dispersions of orientable and deformable particles under sedimentation D. Saintillan, E. S. G. Shaqfeh, E. Darve,

*Physics of Fluids*,

**18**121701 (2006). [pdf]

We use numerical simulations to investigate the dynamics in suspensions of ideally polarizable rods sedimenting under gravity in a vertical electric field. While such suspensions are unstable to concentration fluctuations when no field is applied, we show that the induced-charge electrophoresis that results from the application of the field provides control over the concentration instability by causing particle alignment in the field direction. A phase diagram is obtained for the occurrence of the instability in terms of field strength and volume fraction. In stable suspensions velocity hindrance is shown to occur, and results for the hindered settling function are presented.

*Physics of Fluids*,

**18**121503 (2006). [pdf]

It is well known that a dilute suspension of spheroids sedimenting under gravity at low Reynolds number is unstable to density fluctuations as a result of hydrodynamic interactions (Koch and Shaqfeh 1989). Using a linear stability analysis, it is shown that a vertical density gradient in such a suspension can lead to a wave number selection by damping fluctuations at long wavelengths. A scaling for the most unstable wavelength, or characteristic size of the density fluctuations, is obtained in terms of the background stratification and volume fraction, and is compared to results from numerical simulations in stratified particulate suspensions using methods that we have developed previously. In initially homogeneous suspensions, simulations show a continuous decay of the size of the density fluctuations over time, which we demonstrate can be attributed to the development of stratification inside the suspension.

*Journal of Fluid Mechanics*,

**563**223 (2006). [pdf]

The behaviour of a dispersion of infinitely polarizable slender rods in an electric field is described using theory and numerical simulations. The polarization of the rods results in the formation of dipolar charge clouds around the particle surfaces, which in turn drive nonlinear electrophoretic flows under the action of the applied field. This induced-charge electrophoresis causes no net migration for uncharged particles with fore-aft symmetry, but can lead to rotations and to relative motions as a result of hydrodynamic interactions. A slender-body formulation is derived that accounts for induced-charge electrophoresis based on a thin double layer approximation, and shows that the effects of the electric field on a single rod can be modelled by a linear slip velocity along the rod axis, which causes particle alignment and drives a stresslet flow in the surrounding fluid. Based on this slender-body model, the hydrodynamic interactions between a pair of aligned rods are studied, and we identify domains of attraction and repulsion, which suggest that particle pairing may occur. An efficient method is implemented for the simulation of dispersions of many Brownian rods undergoing induced-charge electrophoresis, that accounts for far-field hydrodynamic interactions up to the stresslet term, as well as near-field lubrication and contact forces. Simulations with negligible Brownian motion show that particle pairing indeed occurs in the suspension, as demonstrated by sharp peaks in the pair distribution function. The superposition of all the electrophoretic flows driven on the rod surfaces is observed to result in a diffusive motion at long times, and hydrodynamic dispersion coefficients are calculated. Results are also presented for colloidal suspensions, in which Brownian fluctuations are found to hinder particle pairing and alignment. Orientation distributions are obtained for various electric field strengths, and are compared to an analytical solution of the Fokker–Planck equation for the orientation probabilities in the limit of infinite dilution.

*Journal of Fluid Mechanics*,

**557**297 (2006). [pdf]

We use Brownian dynamics to investigate the effect of chain flexibility on the cross- streamline migration of short polymers in a pressure-driven flow between two infinite flat plates. A simulation method is described that models a polymer molecule at the Kuhn step level as a chain of N freely jointed Brownian rods, and includes multibody hydrodynamic interactions between the chain segments and the surrounding channel walls. Our study confirms the existence of shear-induced migration away from the solid boundaries toward the channel centreline as a result of wall hydrodynamic interactions. At a fixed ratio H/Rg of the channel width to the bulk radius of gyration, and at a fixed value of the Weissenberg number Wi, simulations show that migration is not significantly influenced by flexibility for chains of length N greater than 2. Much weaker migration is observed however for fully rigid chains (N = 1), and a mechanism is discussed to explain migration in that case.

*Journal of Fluid Mechanics*,

**553**347 (2006). [pdf]

Theory and numerical simulations are used to investigate the concentration fluctuations and the microstructure in dilute sedimenting suspensions of orientable and deformable particles at zero Reynolds number. The case of orientable particles is studied using prolate and oblate spheroids, while viscous droplets in the small deformation regime illustrate the effects of deformability. An efficient method based on a point-particle approximation and on smooth localized force representations is implemented to simulate full-scale suspensions with both periodic and slip boundaries, where the latter are used to qualitatively reproduce the effects of horizontal walls. The concentration instability predicted theoretically for suspensions of spheroids is captured in the simulations, and we find that including horizontal walls provides a mechanism for wavenumber selection, in contrast to periodic systems in which the longest wavelength set by the size of the container dominates. A theoretical model for the case of slightly deformable particles is developed, and a linear stability analysis shows that such suspensions are also unstable to concentration fluctuations under sedimentation. In the absence of diffusion, the model predicts that density fluctuations are equally unstable at all wavelengths, but we show that diffusion, whether Brownian or hydrodynamic, should damp high-wavenumber fluctuations. Simulations are also performed for deformable particles, and again an instability is observed that shows a similar mechanism for the wavenumber selection in finite containers. Our results demonstrate that all sedimentation processes of orientable or deformable particles are subject to spontaneous concentration inhomogeneities, which control the sedimentation rates in these systems.

#### 2005

- A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: The sedimentation of fibers D. Saintillan, E. Darve, E. S. G. Shaqfeh,

*Physics of Fluids*,

**17**033301 (2005). [pdf]

Large-scale simulations of non-Brownian rigid fibers sedimenting under gravity at zero Reynolds number have been performed using a fast algorithm. The mathematical formulation follows the previous simulations by Butler and Shaqfeh (2002). The motion of the fibers is described using slender-body theory, and the line distribution of point forces along their lengths is approximated by a Legendre polynomial in which only the total force, torque, and particle stresslet are retained. Periodic boundary conditions are used to simulate an infinite suspension, and both far-field hydrodynamic interactions and short-range lubrication forces are considered in all simulations. The calculation of the hydrodynamic interactions, which is typically the bottleneck for large systems with periodic boundary conditions, is accelerated using a smooth particle-mesh Ewald (SPME) algorithm previously used in molecular dynamics simulations. In SPME the slowly decaying Green's function is split into two fast-converging sums: the first involves the distribution of point forces and accounts for the singular short-range part of the interactions, while the second is expressed in terms of the Fourier transform of the force distribution and accounts for the smooth and long-range part. Because of its smoothness, the second sum can be computed efficiently on an underlying grid using the fast Fourier transform algorithm, resulting in a significant speed-up of the calculations. Systems of up to 512 fibers were simulated on a single-processor workstation, providing a different insight into the formation, structure, and dynamics of the inhomogeneities that occur in sedimenting fiber suspensions.

## Conference Proceedings

- Direct numerical simulations of electrophoretic deposition of charged colloidal suspensions J. S. Park, D. Saintillan,
- Instabilite d'un jet de particules dans un fluide visqueux F. Pignatel, E. Guazzelli, M. Nicolas, D. Saintillan,
- Microstructure in the sedimentation of anisotropic and deformable particles D. Saintillan, E. Darve, E. S. G. Shaqfeh,
- Induced-charge electrophoresis in suspensions of rodlike particles: theory and simulations D. Saintillan, E. Darve, E. S. G. Shaqfeh,
- Dynamic simulations of the instability of sedimenting fibers D. Saintillan, E. Darve, E. S. G. Shaqfeh,

*Proceedings of the 4th International Conference on Electrophoretic Deposition*, in

*Key Engineering Materials*,

**507**47 (2012).

➠ Featured on the cover of

*Key Engineering Materials*

*19eme Congres Francais de Mecanique*, Marseille, France, August 24-28 2009.

*ASME International Mechanical Engineering Congress and Exposition 2005*(IMECE205-80119).

*ASME International Mechanical Engineering Congress and Exposition 2005*(IMECE205-80104).

*21st International Congress of Theoretical and Applied Mechanics*, Springer (2004).

## Ph.D. Thesis

##### Collective dynamics in dispersions of anisotropic and deformable particles

D. Saintillan, Stanford University, August 2006. [reprint]➠ Recipient of the 2009 Andreas Acrivos Dissertation Award