Cross-streamline Migration Research Articles

Cross-streamline migration and near-wall depletion of elastic fibers in micro-channel flows.

The complex dynamics of elastic fibers in viscous fluids are central to many biological and industrial systems. Fluid-structure interactions underlying these dynamics govern the shape and transport of flexible fibers, and understanding these interactions can help tune flow properties in applications such as microfluidic separation, printing and clogging. In this work, we use slender-body theory to study micromechanical dynamics that arise from the coupling between the elastic backbone of a fiber and the local straining flow that contributes to filament flipping and cross-streamline migration. The resulting transverse drift is unbiased in either direction in simple shear flow. However, a non-uniform shear rate results in bias towards regions of high shear, which we connect to the shape transitions during flips. We discover a depletion layer that forms near the boundaries of pressure-driven channel flow due to the competition between such a cross-streamline drift and steric exclusion from the walls. Finally, we develop scaling laws for the curvature of filaments during flip events, demonstrating the origin of the drift bias in non-uniform flows, and confirm this behavior from our simulations. Put together, these results shed light on the role of a local and dominant coupling between elasticity and viscous resistance in dictating long-term dynamics and transport of elastic fibers in confined flows.

Cross-stream migration of a vesicle in vortical flows.

We use numerical simulations to systematically investigate the vesicle dynamics in two-dimensional (2D) Taylor-Green vortex flow in the absence of inertial forces. Vesicles are highly deformable membranes encapsulating an incompressible fluid and they serve as numerical and experimental proxies for biological cells such as red blood cells. Vesicle dynamics has been studied in free-space or bounded shear, Poiseuille, and Taylor-Couette flows in 2D and 3D. Taylor-Green vortex are characterized with even more complicated properties than those flows such as nonuniform flow line curvature, shear gradient. We study the effects of two parameters on the vesicle dynamics: the ratio of the interior fluid viscosity to that of the exterior one and the ratio of the shear forces on the vesicle to the membrane stiffness (characterized by the capillary number). Vesicle deformability nonlinearly depends on these parameters. Although the study is in 2D, our findings contribute to the wide spectrum of intriguing vesicle dynamics: Vesicles migrate inwards and eventually rotate at the vortex center if they are sufficiently deformable. If not, then they migrate away from the vortex center and travel across the periodic arrays of vortices. The outward migration of a vesicle is a new phenomenon in Taylor-Green vortex flow and has not been observed in any other flows so far. Such cross-streamline migration of deformable particles can be utilized in several applications such as microfluidics for cell separation.

Compressed vessels bias red blood cell partitioning at bifurcations in a hematocrit-dependent manner: Implications in tumor blood flow

The tumor microenvironment is abnormal and associated with tumor tissue hypoxia, immunosuppression, and poor response to treatment. One important abnormality present in tumors is vessel compression. Vessel decompression has been shown to increase survival rates in animal models via enhanced and more homogeneous oxygenation. However, our knowledge of the biophysical mechanisms linking tumor decompression to improved tumor oxygenation is limited. In this study, we propose a computational model to investigate the impact of vessel compression on red blood cell (RBC) dynamics in tumor vascular networks. Our results demonstrate that vessel compression can alter RBC partitioning at bifurcations in a hematocrit-dependent and flow rate-independent manner. We identify RBC focusing due to cross-streamline migration as the mechanism responsible and characterize the spatiotemporal recovery dynamics controlling downstream partitioning. Based on this knowledge, we formulate a reduced-order model that will help future research to elucidate how these effects propagate at a whole vascular network level. These findings contribute to the mechanistic understanding of hemodilution in tumor vascular networks and oxygen homogenization following pharmacological solid tumor decompression.

Comment on “Migration of an electrophoretic particle in a weakly inertial or viscoelastic shear flow”

A recent article [Khair and Kabarowski Phys. Rev. Fluids 5, 033702 (2020)] 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, 856 (2019); J. Fluid Mech. 898, A20 (2020)] 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 their calculation of first-order viscoelastic lift. When this integral is included, the resulting migration velocity matches exactly with that reported in our work [J. Fluid Mech. 898, A20 (2020)]. This qualitatively changes the migration direction that is reported by Khair and Kabarowski for viscoelastic flows. For inertial flows, we clarify that Khair and Kabarowski find the coefficient of lift to be $1.75\ensuremath{\pi}$ compared to $2.35\ensuremath{\pi}$ in our previous work [J. Fluid Mech. 874, 856 (2019)]. We show that this difference occurs because Khair and Kabarowski accurately include the effect of a rapidly decaying $\ensuremath{\sim}O(1/{r}^{4})$ velocity field (a correction to the stresslet field $\ensuremath{\sim}1/{r}^{2}$), which was neglected in our work.

Capsule Migration and Deformation in a Converging Micro-Capillary

The lateral migration of elastic capsules towards a microchannel centerline plays a major role in industrial and physiological processes. Via our computational investigation, we show that a constriction connecting two straight microchannels facilitates the lateral capsule migration considerably, which is relatively slow in straight channels. Our work reveals that the significant cross-streamline migration inside the constriction is dominated by the strong hydrodynamic forces due to the capsule size. However, in the downstream straight channel, the increased interfacial deformation at higher capillary numbers or a lower viscosity ratio and lower membrane hardness results in increased lateral cross-streamline migration. Thus, our work highlights the different migration mechanisms occurring over curved and straight streamlines.

Motion of a capsule in a curved tube

Abstract

Optimal Control of Colloidal Trajectories in Inertial Microfluidics Using the Saffman Effect.

In inertial microfluidics colloidal particles in a Poiseuille flow experience the Segré-Silberberg lift force, which drives them to specific positions in the channel cross section. An external force applied along the microchannel induces a cross-streamline migration to a new equilibrium position because of the Saffman effect. We apply optimal control theory to design the time protocol of the axial control force in order to steer a single particle as precisely as possible from a channel inlet to an outlet at a chosen target position. We discuss the influence of particle radius and channel length and show that optimal steering is cheaper than using a constant control force. Using a single optimized control-force protocol, we demonstrate that even a pulse of particles spread along the channel axis can be steered to a target and that particles of different radii can be separarted most efficiently.

Irreversible hydrodynamic trapping by surface rollers.

A colloidal particle driven by externally actuated rotation can self-propel parallel to a rigid boundary by exploiting the hydrodynamic coupling that surfaces induce between translation and rotation. As such a roller moves along the boundary it generates local vortical flows, which can be used to trap and transport passive cargo particles. However, the details and conditions for this trapping mechanism have not yet been fully understood. Here, we show that the trapping of cargo is accomplished through time-irreversible interactions between the cargo and the boundary, leading to its migration across streamlines into a steady flow vortex next to the roller. The trapping mechanism is explained analytically with a two dimensional model, investigated numerically in three dimensions for a wide range of parameters and is shown to be analogous to the deterministic lateral displacement (DLD) technique used in microfluidics for the separation of differently sized particles. The several geometrical parameters of the problem are analysed and we predict that thin, disc-like rollers offer the most favourable trapping conditions.

Focusing and splitting streams of soft particles in microflows via viscosity gradients.

Microflows are intensively used for investigating and controlling the dynamics of particles, including soft particles such as biological cells and capsules. A classic result is the tank-treading motion of elliptically deformed soft particles in linear shear flows, which do not migrate across straight streamlines in the bulk. However, soft particles migrate across straight streamlines in Poiseuille flows. In this work we describe a new mechanism of cross-streamline migration by using soft capsules with a spherical equilibrium shape. If the viscosity varies perpendicular to the streamlines then the soft particles migrate across streamlines towards regions of a lower viscosity, even in linear shear flows. An interplay with the repulsive particle-boundary interaction causes then focusing of particles in linear shear flows with the attractor streamline closer to the wall in the low viscosity region. Viscosity variations perpendicular to the streamlines in Poiseuille flows leads either to a shift of the particle attractor or even to a splitting of particle attractors, which may give rise to interesting applications for particle separation. The location of attracting streamlines depend on the particle properties, like their size and elasticity. The cross-stream migration induced by viscosity variations is explained by analytical considerations, Stokesian dynamics simulations with a generalized Oseen tensor and lattice-Boltzmann simulations.

The lift force on a charged sphere that translates and rotates in an electrolyte.

The distortion of the charge cloud around a uniformly charged, dielectric, rigid sphere that translates and rotates in an unbounded binary, symmetric electrolyte at zero Reynolds number is examined. The zeta potential of the particle ζ is assumed small relative to the thermal voltage scale. It is assumed that the equilibrium structure of the cloud is slightly distorted, which requires that the Péclet numbers characterizing distortion due to particle translation, , and rotation, , are small compared to unity. Here, a is radius of the particle; D is the ionic diffusion coefficient; and , where U and Ω are the rectilinear and angular velocities of the particle, respectively. Perturbation expansions for small and are employed to calculate the nonequilibrium structure of the cloud, whence the force and torque on the particle are determined. In particular, we predict that the sphere experiences a force orthogonal to its directions of translation and rotation. This "lift" force arises from the nonlinear distortion of the cloud under the combined actions of particle translation and rotation. The lift force is given by . Here, ε is the permittivity of the electrolyte; is the Debye length; and is a negative function that decreases in magnitude with increasing . The lift force implies that an unconstrained particle would follow a curved path; an electrokinetic analog of the inertial Magnus effect. Finally, the implication of the lift force on cross-streamline migration of an electrophoretic particle in shear flow isdiscussed.

Conditional stability of particle alignment in finite-Reynolds-number channel flow

Finite-size neutrally buoyant particles in a channel flow are known to accumulate at specific equilibrium positions or spots in the channel cross-section, if the flow inertia is finite at the particle scale. Experiments in different conduit geometries have shown that while reaching equilibrium locations, particles tend also to align regularly in the streamwise direction. In this paper, the force coupling method was used to numerically investigate the inertia-induced particle alignment, using square-channel geometry. The method was first shown to be suitable to capture the quasisteady lift force that leads to particle cross-streamline migration in channel flow. Then the particle alignment in the flow direction was investigated by calculating the particle relative trajectories as a function of flow inertia and of the ratio between the particle size and channel hydraulic diameter. The flow streamlines were examined around the freely rotating particles at equilibrium, revealing stable small-scale vortices between aligned particles. The streamwise interparticle spacing between aligned particles at equilibrium was calculated and compared to available experimental data in square-channel flow [Gao et al., Microfluid. Nanofluid. 21, 154 (2017)]. The new result highlighted by our numerical simulations is that the interparticle spacing is unconditionally stable only for a limited number of aligned particles in a single train, the threshold number being dependent on the confinement (particle-to-channel size ratio) and on the Reynolds number. For instance, when the particle Reynolds number is ≈1 and the particle-to-channel height size ratio is ≈0.1, the maximum number of stable aligned particles per train is equal to 3. This agrees with statistics realized on the experiments of [Gao et al., Microfluid. Nanofluid. 21, 154 (2017)]. It is argued that when several particles are hydrodynamically connected moving as a unique structure (the train) with a steady streamwise velocity, large-scale hydrodynamic perturbations induced at the train scale prohibit small-scale vortex connection between the leading and second particles, forcing the leading particle to leave the train.

Dynamics of cylindrical particles in the contraction flow of a second-order fluid

The finite volume method is used to numerically simulate the dynamics of cylindrical particles in the contraction flow of a second-order fluid. The effects of Stokes number, Deborah number, particle aspect ratio and contraction ratio on the spatial and orientation distributions of cylindrical particles are analyzed. Some numerical results are compared with the available experimental results in the contraction flow of Newtonian fluids. The results show that the spatial and orientation distributions of the particles are dependent on the competition among the inertia, viscoelasticity and effect of confined wall. The result of competition is responsible for the force and torque inducing a cross-streamline migration and rotation of particles. Along the flow direction, the particle spatial distribution in the cross section changes non-monotonically. The particle orientation distribution changes monotonically and particles gradually align themselves with the flow direction, and this phenomenon is more obvious than the case of the Newtonian fluid under the same conditions. High shear rate makes the particles align with their major axis near to the flow direction. With the increase of the Stokes number, the particle spatial distribution becomes more uniform, and orientation distribution function of particles becomes flatter and wider. With the increase of Deborah number and the contraction ratio, the particle spatial distribution becomes more non-uniform, in the meantime, more particles tend to align themselves with the flow direction. With the increases of the particle aspect ratio, the particle spatial distribution in the cross section becomes more uniform, and more particles align with their major axis near to the flow direction. In the parameter region studied here, the particle aspect ratio has a weaker effect on the spatial and orientation distributions of particles than Stokes number, Deborah number and contraction ratio.

Splitting and separation of colloidal streams in sinusoidal microchannels.

The control of the distribution of colloidal particles in microfluidic flows plays an important role in biomedical and industrial applications. A particular challenge is to induce cross-streamline migration in laminar flows, enabling the separation of colloidal particles according to their size, shape or elasticity. Here we show that viscoelastic fluids can mediate cross-streamline migration of deformable spherical and cylindrical colloidal particles in sinusoidal microchannels at low Reynolds numbers. For colloidal streams focused into the center of the channel entrance this leads to a symmetric stream-splitting and separation into four substreams. The degree of stream splitting and separation can be controlled via the flow rates, viscoelasticity of the focusing fluid, and the spatial microchannel modulation with an upper limit when reaching the microchannel walls. We demonstrate that this effect can be used to separate flexible particles of different size and shape. This methodology of cross-stream migration has thus great potential for the passive separation of colloids and cells in microfluidic channels.

Migration reversal of soft particles in vertical flows

Non-neutrally buoyant soft particles in vertical microflows are investigated. We find for light soft particles in downward Poiseuille flow cross-streamline migration (CSM) to off-center streamlines and for heavy particles CSM to the center. In both cases a reversal of the vertical flow direction and the related shear gradient causes a reversal of the migration direction. This gravitational driven CSM of soft particles occurs also in linear shear flows: heavy (light) particles migrate antiparallel (parallel) to the shear gradient. The surprising, flow-induced migration (reversal) is characterized by simulations and analytical approximations, confirming our plausible explanation of the effect. This might be applied for separating particles.

Focusing and alignment of erythrocytes in a viscoelastic medium

Viscoelastic fluid flow-induced cross-streamline migration has recently received considerable attention because this process provides simple focusing and alignment over a wide range of flow rates. The lateral migration of particles depends on the channel geometry and physicochemical properties of particles. In this study, digital in-line holographic microscopy (DIHM) is employed to investigate the lateral migration of human erythrocytes induced by viscoelastic fluid flow in a rectangular microchannel. DIHM provides 3D spatial distributions of particles and information on particle orientation in the microchannel. The elastic forces generated in the pressure-driven flows of a viscoelastic fluid push suspended particles away from the walls and enforce erythrocytes to have a fixed orientation. Blood cell deformability influences the lateral focusing and fixed orientation in the microchannel. Different from rigid spheres and hardened erythrocytes, deformable normal erythrocytes disperse from the channel center plane, as the flow rate increases. Furthermore, normal erythrocytes have a higher angle of inclination than hardened erythrocytes in the region near the side-walls of the channel. These results may guide the label-free diagnosis of hematological diseases caused by abnormal erythrocyte deformability.

Numerical simulations of deformable particle lateral migration in tube flow of Newtonian and viscoelastic media

The deformation and cross-streamline migration of an elastic particle in pressure-driven flows of Newtonian and viscoelastic (Oldroyd-B, Giesekus) fluids in a cylindrical tube are studied through 3D finite element method numerical simulations. The dependences of particle deformation and migration on geometric confinement, flow strength, and fluid rheology are investigated. If the particle is initially not at the channel axis, it attains an asymmetric shape and migrates. In a Newtonian liquid, the migration is always directed towards the tube axis. A project equation is proposed for the design of a microfluidic cylindrical device aimed at focusing elastic particles on the cylinder centerline. In a viscoelastic liquid, the migration direction and velocity depend on the competition among particle deformability, fluid elasticity, and fluid viscosity shear thinning (if any). In a certain range of parameters, an unstable radial position appears, which separates the region where the migration is directed towards the axis from the region where it is directed towards the wall.

Transport of a dilute active suspension in pressure-driven channel flow

Confined suspensions of active particles show peculiar dynamics characterized by wall accumulation, as well as upstream swimming, centreline 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 Péclet 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 that 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 centreline, 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.

Lubrication analysis of interacting rigid cylindrical particles in confined shear flow

Lubrication analysis is used to determine analytical expressions for the elements of the resistance matrix describing the interaction of two rigid cylindrical particles in two-dimensional shear flow in a symmetrically confined channel geometry. The developed model is valid for non-Brownian particles in a low-Reynolds-number flow between two sliding plates with thin gaps between the two particles and also between the particles and the walls. Using this analytical model, a comprehensive overview of the dynamics of interacting cylindrical particles in shear flow is presented. With only hydrodynamic interactions, rigid particles undergo a reversible interaction with no cross-streamline migration, irrespective of the confinement value. However, the interaction time of the particle pair substantially increases with confinement, and at the same time, the minimum distance between the particle surfaces during the interaction substantially decreases with confinement. By combining our purely hydrodynamic model with a simple on/off non-hydrodynamic attractive particle interaction force, the effects of confinement on particle aggregation are qualitatively mapped out in an aggregation diagram. The latter shows that the range of initial relative particle positions for which aggregation occurs is increased substantially due to geometrical confinement. The interacting particle pair exhibits tangential and normal lubrication forces on the sliding plates, which will contribute to the rheology of confined suspensions in shear flow. Due to the combined effects of the confining walls and the particle interaction, the particle velocities and resulting forces both tangential and perpendicular to the walls exhibit a non-monotonic evolution as a function of the orientation angle of the particle pair. However, by incorporating appropriate scalings of the forces, velocities, and doublet orientation angle with the minimum free fraction of the gap height and the plate speed, master curves for the forces versus orientation angle can be constructed.

Migration of finite sized particles in a laminar square channel flow from low to high Reynolds numbers

The migration of neutrally buoyant finite sized particles in a Newtonian square channel flow is investigated in the limit of very low solid volumetric concentration, within a wide range of channel Reynolds numbers Re = [0.07-120]. In situ microscope measurements of particle distributions, taken far from the channel inlet (at a distance several thousand times the channel height), revealed that particles are preferentially located near the channel walls at Re > 10 and near the channel center at Re < 1. Whereas the cross-streamline particle motion is governed by inertia-induced lift forces at high inertia, it seems to be controlled by shear-induced particle interactions at low (but finite) Reynolds numbers, despite the low solid volume fraction (<1%). The transition between both regimes is observed in the range Re = [1-10]. In order to exclude the effect of multi-body interactions, the trajectories of single freely moving particles are calculated thanks to numerical simulations based on the force coupling method. With the deployed numerical tool, the complete particle trajectories are accessible within a reasonable computational time only in the inertial regime (Re > 10). In this regime, we show that (i) the particle undergoes cross-streamline migration followed by a cross-lateral migration (parallel to the wall) in agreement with previous observations, and (ii) the stable equilibrium positions are located at the midline of the channel faces while the diagonal equilibrium positions are unstable. At low flow inertia, the first instants of the numerical simulations (carried at Re = O(1)) reveal that the cross-streamline migration of a single particle is oriented towards the channel wall, suggesting that the particle preferential positions around the channel center, observed in the experiments, are rather due to multi-body interactions.

Viscous Marangoni migration of a drop in a Poiseuille flow at low surface Péclet numbers

AbstractThe motion of a spherical drop with a bulk-insoluble surfactant immersed in a background flow in the limits of low surface Péclet number and low Reynolds number is investigated. We develop a reciprocal theorem that applies to any prescribed background flow and provide a specific example of an unbounded Poiseuille flow. Analytical formulas for the migration velocity of the drop are obtained perturbatively in powers of the surface Péclet number. We show that the redistribution of surfactant due to the background flow acts to retard the motion of the drop, with the magnitude of this slip velocity being independent of the drop’s position in the Poiseuille flow. Moreover, a surfactant-induced cross-streamline migration of the drop occurs towards the centre of the Poiseuille flow, with its magnitude depending linearly on the distance of the drop from the centre of the Poiseuille flow.