We study the shear induced migration of microswimmers (primarily, active Brownian particles or ABP's) in a plane Poiseuille flow. For wide channels characterized by $U_b/HD_r \ll 1$, the separation between time scales characterizing the swimmer orientation dynamics (of O($D^{-1}_r$)) and those that characterize migration across the channel (of O($H^{2}D_r/U^{2}_b$)), allows for use of the method of multiple scales to derive a drift-diffusion equation for the swimmer concentration profile; here, $U_b$ is the swimming speed, $H$ is the channel half-width, and $D_r$ is the swimmer rotary diffusivity. The steady state concentration profile is a function of the Péclet number, $Pe = U_{f}/(D_r H)$ ($U_f$ being the channel centerline velocity), and the swimmer aspect ratio $κ$. Swimmers with $ κ\gg 1$ (with $ κ\sim$ O(1)), in the regime $1 \ll \textit{Pe} \ll κ^3$ ($Pe\sim$ O(1)), migrate towards the channel walls, corresponding to a high-shear trapping behavior. For $Pe \gg κ^3 $ ($Pe \gg $ 1 for $κ\sim$ O(1)), however, swimmers migrate towards the centerline, corresponding to a low-shear trapping behavior. Interestingly, within the low-shear trapping regime, swimmers with $κ< 2$ asymptote to a $Pe$-independent concentration profile for large $Pe$, while those with $κ\geq 2$ exhibit a `centerline-collapse' for $Pe \to \infty$. The prediction of low-shear-trapping, validated by Langevin simulations, is the first explanation of recent experimental observations [Barry $\textit{et al}$. (2015)]. We organize the high-shear and low-shear trapping regimes on a $Pe-κ$ plane, thereby highlighting the singular behavior of infinite-aspect-ratio swimmers.
With the continuing rapid development of artificial microrobots and active particles, questions of microswimmer guidance and control are becoming ever more relevant and prevalent. In both the applications and theoretical study of such microscale swimmers, control is often mediated by an engineered property of the swimmer, such as in the case of magnetically propelled microrobots. In this work, we will consider a modality of control that is applicable in more generality, effecting guidance via modulation of a background fluid flow. Here, considering a model swimmer in a commonplace flow and sim…
The emerging field of self-driven active particles in fluid environments has recently created significant interest in the biophysics and bioengineering communities owing to their promising future biomedical and technological applications. These microswimmers move autonomously through aqueous media where under realistic situations they encounter a plethora of external stimuli and confining surfaces with peculiar elastic properties. Based on a far-field hydrodynamic model, we present an analytical theory to describe the physical interaction and hydrodynamic couplings between a self-propelled act…
Many plankton species undergo daily vertical migration to large depths in the turbulent ocean. To do this efficiently, the plankton can use a gyrotactic mechanism, aligning them with gravity to swim downwards, or against gravity to swim upwards. Many species show passive mechanisms for gyrotactic stability. For example, bottom-heavy plankton tend to align upwards. This is efficient for upward migration in quiescent flows, but it is often sensitive to turbulence which upsets the alignment. Here we suggest a simple, robust active mechanism for gyrotactic stability, which is only lightly affected…
Biological microswimmers are known to navigate upstream of an external flow (positive rheotaxis) in trajectories ranging from linear, spiral to oscillatory. Such rheotaxis stems from the interplay between the motion and complex shapes of the microswimmers, e.g. the chirality of the rotating flagella, the shear flow characteristics, and the hydrodynamic interaction with a confining surface. Here, we show that an isotropic, active droplet microswimmer exhibits a unique oscillatory rheotaxis in a microchannel despite its simple spherical geometry. The swimming velocity, orientation, and the chemi…
For natural microswimmers, the interplay of swimming activity and external flow can promote robust motion, e.g. propulsion against ("upstream rheotaxis") or perpendicular to the direction of flow. These effects are generally attributed to their complex body shapes and flagellar beat patterns. Here, using catalytic Janus particles as a model experimental system, we report on a strong directional response that occurs for spherical active particles in a channel flow. The particles align their propulsion axes to be nearly perpendicular to both the direction of flow and the normal vector of a nearb…
Microswimmers (planktonic microorganisms or artificial active particles) immersed in a fluid interact with the ambient flow, altering their trajectories. In surface gravity waves, a common goal for microswimmers is vertical migration (e.g., to reach the free surface or to dive to deeper depths). By modelling microswimmers as spheroidal bodies with an intrinsic swimming velocity that supplements advection and reorientation by the flow, we investigate how shape and swimming affect vertical transport of microswimmers in waves. We find that it is possible for microswimmers to be initially swimming…
The current work studies the dynamics of a microswimmer in pressure-driven flow of a weakly viscoelastic fluid. Employing the second-order fluid model, we show that the self-propelling swimmer experiences a viscoelastic swimming lift in addition to the well-known passive lift that arises from its resistance to shear flow. Using the reciprocal theorem, we evaluate analytical expressions for the swimming lift experienced by neutral and pusher/puller-type swimmers and show that they depend on the hydrodynamic signature associated with the swimming mechanism. We find that for neutral swimmers focu…
The motion of microswimmers in complex flows is ruled by the interplay between swimmer propulsion and the dynamics induced by the fluid velocity field. Here we study the motion of a chiral microswimmer whose propulsion is provided by the spinning of a helical tail with respect to its body in a simple shear flow. Thanks to an efficient computational strategy that allowed us to simulate thousands of different trajectories, we show that the tail shape dramatically affects the swimmer's motion. In the shear dominated regime, the swimmers carrying an elliptical helical tail show several different J…
In a shear flow particles migrate to their equilibrium positions in the microchannel. Here we demonstrate theoretically that if particles are inertial, this equilibrium can become unstable due to the Saffman lift force. We derive an expression for the critical Stokes number that determines the onset of instable equilibrium. We also present results of lattice Boltzmann simulations for spherical particles and prolate spheroids to validate the analysis. Our work provides a simple explanation of several unusual phenomena observed in earlier experiments and computer simulations, but never interpret…
Collinear Swimming of a Squirmer Pair in Newtonian and Shear-Thinning Fluids
Santa Clara University（圣克拉拉大学）；University of Illinois Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Towson University（陶森大学）；New Jersey Institute of Technology（新泽理工学院）；Texas A&M University–Corpus Christi（德克萨斯农工大学科珀斯克里斯蒂分校）
Mixing induced by microswimmers as probed by mutual information
Max Planck Institute for Dynamics and Self-Organization（马克斯·普朗克动力学与自组织研究所）；Institute of Science and Technology Austria（奥地利科学技术研究所）；Department of Mathematics, Kyoto University（京都大学数学系）；Jožef Stefan Institute（约瑟夫·斯特凡研究所）；Rudolf Peierls Centre for Theoretical Physics, University of Oxford（牛津大学鲁道夫·皮尔斯理论物理中心）
We experimentally investigate particle migration in a non-Brownian suspension sheared in a Taylor-Couette configuration and in the limit of vanishing Reynolds number. Highly resolved index-matching techniques are used to measure the local particulate volume fraction. In this wide-gap Taylor-Couette configuration, we find that for a large range of bulk volume fraction, $ϕ_b \in [20\% - 50\%]$, the fully developed concentration profiles are well predicted by the suspension balance model of Nott \& Brady (J. Fluid Mech., vol. 275, 1994, pp. 157-199). Moreover, we provide systematic measurements o…
We present experimental results on dense corn-starch suspensions as examples of non-Brownian, nearly-hard particles that undergo continuous and discontinuous shear thickening (CST and DST) at intermediate and high densities respectively. Our results offer strong support for recent theories involving a stress-dependent effective contact friction among particles. We show however that in the DST regime, where theory might lead one to expect steady-state shear bands oriented layerwise along the vorticity axis, the real flow is unsteady. To explain this, we argue that steady-state banding is generi…
It has recently been argued that steady-state vorticity bands cannot arise in shear thickening suspensions, because the normal stress imbalance across the interface between the bands will set up particle migrations. In this Letter, we develop a simple continuum model that couples shear thickening to particle migration. We show by linear stability analysis that homogeneous flow is unstable towards vorticity banding, as expected, in the regime of negative constitutive slope. In full nonlinear computations, we show however that the resulting vorticity bands are unsteady, with spatiotemporal patte…
We report direct measurements of spatially resolved stress at the boundary of a shear thickening cornstarch suspension revealing persistent regions of high local stress propagating in the flow direction at the speed of the top boundary. The persistence of these propagating fronts enables precise measurements of their structure, including the profile of boundary stress measured by Boundary Stress Microscopy (BSM) and the non-affine velocity of particles at the bottom boundary of the suspension measured by particle image velocimetry (PIV). In addition, we directly measure the relative flow betwe…
We study the coupled hydrodynamics between a motile slender microswimmer and a deformable interface that separates two Newtonian fluid regions. From the disturbance field generated by the swimming motion, we quantitatively characterize the interface deformation and the manner in which the coupling modifies the microswimmer translation itself. We treat the role of the swimmer type (pushers and pullers), size and model an interface that can deform due to both surface tension and bending elasticity. Our analysis reveals a strong dependence of the hydrodynamics on the swimmer orientation and posit…
We examine the longitudinal dispersion of spheroidal microswimmers in pressure-driven channel flow. When time scales corresponding to swimmer orientation relaxation, and diffusion in the gradient and flow directions, are well separated, a multiple scales analysis leads to the shear-enhanced diffusivity governing the long-time spread of the swimmer population along the flow\,(longitudinal) direction. For large $Pe_r$, $Pe_r$ being the rotary Peclet number, this diffusivity scales as $O(Pe_r^4D_t)$ for $1 \leq κ\lesssim 2$, and as $O(Pe_r^{\frac{10}{3}}D_t)$ for $κ= \infty$, $D_t$ being the (bar…
We study the steady-state behavior of active, dipolar, Brownian spheroids in a planar channel subjected to an imposed Couette flow and an external transverse field, applied in the 'downward' normal-to-flow direction. The field-induced torque on active spheroids (swimmers) is taken to be of magnetic form by assuming that they have a permanent magnetic dipole moment, pointing along their self-propulsion (swim) direction. Using a continuum approach, we show that a host of behaviors emerge over the parameter space spanned by the particle aspect ratio, self-propulsion and shear/field strengths, and…
A recent article (Khair and Kabarowski; Phys. Rev. Fluids 5, 033702) has studied the cross-streamline migration of electrophoretic particles in unbounded shear flows with weak inertia or viscoelasticity. That work compares their results with those reported in two of our previous studies (Choudhary et al.: J. Fluid Mech. 874; J. Fluid Mech. 898) and reports a disagreement in the derived analytical expressions. In this comment, we resolve this discrepancy. For viscoelastic flows, we show that Khair and Kabarowski have not accounted for a leading order surface integral of polymeric stress in thei…
The use of tank-treading as a means of propulsion for microswimmers in viscous shear flows is taken into exam. We discuss the possibility that a vesicle be able to control the drift in an external shear flow, by varying locally the bending rigidity of its own membrane. By analytical calculation in the quasi-spherical limit, the stationary shape and the orientation of the tank-treading vesicle in the external flow, are determined, working to lowest order in the membrane inhomogeneity. The membrane inhomogeneity acts in the shape evolution equation as an additional force term, that can be used t…
Experiments of particle-laden inclined film flow [Zhou, Dupuy, Bertozzi, and Hosoi, Phys. Rev. Lett. 94 (2005)] have displayed different settling behaviors depending on the particle concentration $ϕ$ and angle of inclination $θ$, in which particles accumulate on the substrate or near the advancing contact line, or remain mixed. Zhou et al. presented a lubrication model that captures the qualitative behavior of the high-$ϕ$, high-$θ$ regime, characterized by a particle-rich ridge near the contact line, but cannot explain the other observed settling behaviors. This work presents a model in which…
Suspension flows are ubiquitous in nature (hemodynamics, subsurface fluid mechanics, etc.) and industrial applications (hydraulic fracturing, CO$_2$ storage, etc.). However, such flows are notoriously difficult to model due to the variety of fluid-particle and particle-particle interactions that can occur. In this work, we focus on non-Brownian shear-dominated suspensions, where kinetic collisions are negligible and frictional effects play a dominant role. Under these circumstances, irreversible phenomena such as particle diffusion and migration arise, requiring anisotropic stress models to de…
The motion of an initially spherical capsule in a wall-bounded oscillating shear flow is investigated via an accelerated boundary integral implementation. The neo-Hookean model is used as the constitutive law of the capsule membrane. The maximum wall-normal migration is observed when the oscillation period of the imposed shear is of the order of the relaxation time of the elastic membrane; hence, the optimal capillary number scales with the inverse of the oscillation frequency and the ratio agrees well with the theoretical prediction in the limit of high-frequency oscillation. The migration ve…
We develop a two-fluid model (TFM) for heat transfer in dense non-Brownian suspensions. Specifically, we propose closure relations for the inter-phase heat transfer coefficient and the thermal diffusivity of the particle phase based on calibration against experimental data. The model is then employed to simulate non-isothermal flow in an annular Couette cell. We find that, when the shear rate is controlled by the rotation of the inner cylinder, both the shear and thermal gradients are responsible for particle migration. Within the TFM framework, we identify the origin and functional form of a …
The migration of a Brownian fluid droplet in a parallel-plate microchannel was investigated using dissipative particle dynamics computer simulations. In a Newtonian solvent, the droplet migrated toward the channel walls due to inertial effects at the studied flow conditions, in agreement with theoretical predictions and recent simulations. However, the droplet focused onto the channel centerline when polymer chains were added to the solvent. Focusing was typically enhanced for longer polymers and higher polymer concentrations with a nontrivial flow-rate dependence due to droplet and polymer de…
The distribution of Red Blood Cells in a confined channel flow is inhomogeneous and shows a marked depletion near the walls due to a competition between migration away from the walls and shear-induced diffusion resulting from interactions between particles. We investigated the lift of RBCs in a shear flow near a wall and measured a significant lift velocity despite the tumbling motion of cells. We also provide values for the collective and anisotropic shear-induced diffusion of a cloud of RBCs, both in the direction of shear and in the direction of vorticity. A generic down-gradient subdiffusi…
Interactions between microorganisms and their complex flowing environments are essential in many biological systems. We develop a model for microswimmer dynamics in non-Newtonian Poiseuille flows. We predict that swimmers in shear-thickening (-thinning) fluids migrate upstream more (less) quickly than in Newtonian fluids and demonstrate that viscoelastic normal stress differences reorient swimmers causing them to migrate upstream at the centreline, in contrast to well-known boundary accumulation in quiescent Newtonian fluids. Based on these observations, we suggest a sorting mechanism to selec…
Dynamics of non-Brownian flexible fibers in Poiseuille flow between two parallel planar solid walls is evaluated from the Stokes equations, solved numerically by an accurate multipole code HYDROMULTIPOLE. Fibers migrate towards a critical distance from the wall zc, which depends significantly on the fiber length N and bending stiffness A. Therefore, the calculated values of zc can be used to sort fibers. Three modes of the dynamics are found, depending on a shear-to-bending parameter Gamma. In the first mode, stiff fibers deform only a little and accumulate close to the wall, as the result of …
The optimal strategy for a microscopic swimmer to migrate across a linear shear flow is discussed. The two cases, in which the swimmer is located at large distance, and in the proximity of a solid wall, are taken into account. It is shown that migration can be achieved by means of a combination of sailing through the flow and swimming, where the swimming strokes are induced by the external flow without need of internal energy sources or external drives. The structural dynamics required for the swimmer to move in the desired direction is discussed and two simple models, based respectively on th…
We analyze the migration characteristics of a droplet in an oscillatory flow field in a parallel plate micro-confinement. Using phase filed formalism, we capture the dynamical evolution of the droplet over a wide range of the frequency of the imposed oscillation in the flow field, drop size relative to the channel gap, and the capillary number. The latter two factors imply the contribution of droplet deformability, commonly considered in the study of droplet migration under steady shear flow conditions. We show that the imposed oscillation brings in additional time complexity in the droplet mo…
We investigate a model for the dynamics of ellipsoidal microswimmers in an externally imposed, laminar Kolmogorov flow. Through a phase-space analysis of the dynamics without noise, we find that swimmers favor either cross-stream or rotational drift, depending on their swimming speed and aspect ratio. When including noise, i.e. rotational diffusion, we find that swimmers are driven into certain parts of phase space, leading to a nonuniform steady-state distribution. This distribution exhibits a transition from swimmer aggregation in low-shear regions of the flow to aggregation in high-shear re…
Recent findings of possible applications of bio-friendly synthetic self-phoretic swimmers, have motivated the researchers in investigating the various motion-generating mechanisms to optimize the operating characteristics of the same. In this paper we model the auto-electrophoretic motion of a bimetallic (Au-Pt) spherical swimmer in a non-Newtonian medium. In view of the fact that many bio-fluids closely follow the generalized Newtonian rheology, the rheology of the surrounding swimming medium is considered to follow the Carreau-Yasuda viscosity model. Further to capture the experimentally obs…
In this work, we analyze the inertial migration of an electrophoretic particle in a 2-D Poiseuille flow with an electric field applied parallel to the walls. For a thin electrical double layer, the particle exhibits a slip-driven electrokinetic motion along the direction of the applied electric field, which causes the particle to lead or lag the flow (depending on its surface charge). The fluid disturbance caused by this slip-driven motion is characterized by a rapidly decaying source-dipole field which alters the inertial lift on the particle. We determine this inertial lift using the recipro…
Shear-thinning is an important rheological property of many biological fluids, such as mucus, whereby the apparent viscosity of the fluid decreases with shear. Certain microscopic swimmers have been shown to progress more rapidly through shear-thinning fluids, but is this behavior generic to all microscopic swimmers, and what are the physics through which shear-thinning rheology affects a swimmer's propulsion? We examine swimmers employing prescribed stroke kinematics in two-dimensional, inertialess Carreau fluid: shear-thinning "Generalized Stokes" flow. Swimmers are modeled, using the method…
Particles migrate in the transverse direction of the flow due to the existence of normal stress anisotropy in weakly viscoelastic liquids. We test the ability of theoretical predictions to predict the transverse velocity migration of particles in a confined Poiseuille flow according to the viscoelastic constitutive parameters of dilute polymers solutions. Firstly, we carefully characterize the viscoelastic properties of two families of dilute polymer solutions at various concentrations using shear rheometry and capillary breakup experiments. Secondly, we develop a specific 3D particle tracking…
Cilia and flagella are actively bending slender organelles, performing functions such as motility, feeding and embryonic symmetry breaking. We review the mechanics of viscous-dominated microscale flow, including time-reversal symmetry, drag anisotropy of slender bodies, and wall effects. We focus on the fundamental force singularity, higher order multipoles, and the method of images, providing physical insight and forming a basis for computational approaches. Two biological problems are then considered in more detail: (1) left-right symmetry breaking flow in the node, a microscopic structure i…
Inertial focusing is the migration of particles in fluid toward equilibrium, where current theory predicts that shear-induced and wall-induced lift forces are balanced. First reported in 1961, this Segre-Silberberg effect is particularly useful for microfluidic isolation of cells and particles. Interestingly, recent work demonstrated particle focusing at high Reynolds numbers that cannot be explained by current theory. In this work, we show that non-monotonous velocity profiles, such as those developed in curved channels, create peripheral velocity maxima around which opposing shear-induced fo…
Motivated by the complex rheological behaviors observed in small/micro scale blood vessels, such as the Fahraeus effect, plasma-skimming, shear-thinning, etc., we develop a non-linear suspension model for blood. The viscosity is assumed to depend on the volume fraction (hematocrit) and the shear rate. The migration of the red blood cells (RBCs) is studied using a concentration flux equation. A parametric study with two representative problems, namely simple shear flow and a pressure driven flow demonstrate the ability of this reduced-order model to reproduce several key features of the two-flu…
Shear stress is an important physical factor that regulates proliferation, migration and morphogenesis. In particular, the homeostasis of blood vessels is dependent on shear stress. To mimic this process ex vivo, efforts have been made to seed scaffolds with vascular and other cell types in the presence of growth factors and under pulsatile flow conditions. However, the resulting bioreactors lack information on shear stress and flow distributions within the scaffold. Consequently, it is difficult to interpret the effects of shear stress on cell function. Such knowledge would enable researchers…
In this study, we report the use of digital holography microscopy (DHM) for 3D-resolved flow kinematics and shear rheometry of viscoelastic polymeric fluids. We computationally reconstruct the recorded holograms to visualize the tracer imbued flow volume in microchannels, followed by implementation of particle tracking velocimetry (PTV) to quantitate spatially-resolved velocity fields in 3D. In order to select optimal parameters for DHM-PTV characterization of complex fluids, we studied the effect of hologram recording distance, seeding density and particle size. Using the optimal parameters, …
The possibility of microscopic swimming by extraction of energy from an external flow is discussed, focusing on the migration of a simple trimer across a linear shear flow. The geometric properties of swimming, together with the possible generalization to the case of a vesicle, are analyzed.The mechanism of energy extraction from the flow appears to be the generalization to a discrete swimmer of the tank-treading regime of a vesicle. The swimmer takes advantage of the external flow by both extracting energy for swimming and "sailing" through it. The migration velocity is found to scale linearl…
