Inertial Migration Research Articles

Dynamics of Small Particle Inertial Migration in Curved Square Ducts

Microchannels are well known in microfluidic applications for the control and separation of microdroplets and cells. Often the objects in the flow experience inertial effects, resulting in dynamics that is a departure from the underlying channel flow dynamics. This paper considers small neutrally buoyant spherical particles suspended in flow through a curved duct having a square cross-section. The particle experiences a combination of inertial lift force induced by the disturbance from the primary flow along the duct, and drag from the secondary vortices in the cross-section, which drive migration of the particle within the cross-section. We construct a simplified model that preserves the core topology of the force field yet depends on a single parameter $\kappa$, quantifying the relative strength of the two forces. We show that $\kappa$ is a bifurcation parameter for the dynamical system that describes motion of the particle in the cross-section of the duct. At large values of $\kappa$ there exists an attracting limit cycle in each of the upper and lower halves of the duct. At small $\kappa$ we find that particles migrate to one of four stable foci. Between these extremes, there is an intermediate range of $\kappa$ for which all particles migrate to a single stable focus. Noting that the positions of the limit cycles and foci vary with the value of $\kappa$, this behavior indicates that, for a suitable particle mixture, duct bend radius might be chosen to segregate particles by size. We evaluate the time and axial distance required to focus particles near the unique stable node, which determines the duct length required for particle segregation.

Numerical study of dielectrophoresis-modified inertial migration for overlapping sized cell separation.

Circulating tumor cells (CTCs) have been proven to have significant prognostic, diagnostic, and clinical values in early-stage cancer detection and treatment. The efficient separation of CTCs from peripheral blood can ensure intact and viable CTCs and can, thus, give proper genetic characterization and drug innovation. In this study, continuous and high-throughput separation of MDA-231 CTCs from overlapping sized white blood cells (WBCs) is achieved by modifying inertial cell focusing with dielectrophoresis (DEP) in a single-stage microfluidic platform by numeric simulation. The DEP is enabled by embedding interdigitated electrodes with alternating field control on a serpentine microchannel to avoid creating two-stage separation. Rather than using the electrokinetic migration of cells which slows down the throughput, the system leverages the inertial microfluidic flow to achieve high-speed continuous separation. The cell migration and cell positioning characteristics are quantified through coupled physics analyses to evaluate the effects of the applied voltages and Reynolds numbers (Re) on the separation performance. The results indicate that the introduction of DEP successfully migrates WBCs away from CTCs and that separation of MDA-231 CTCs from similar sized WBCs at a high Re of 100 can be achieved with a low voltage of magnitude 4 ×106 V/m. Additionally, the viability of MDA-231 CTCs is expected to be sustained after separation due to the short-term DEP exposure. The developed technique could be exploited to design active microchips for high-throughput separation of mixed cell beads despite their significant size overlap, using DEP-modified inertial focusing controlled simply by adjusting the applied external field.

Finite element analysis of inertial migration of polymer vesicles in microtubule flow

The finite element method based on fluid-structure interaction is used to systematically study the inertial migration of polymer vesicles in microtubule flow with a two-dimensional model, and the mechanism of the vesicles deformed by the fluid and the inertial migration phenomena are analyzed. The studies show that with the increase Reynolds number, the equilibrium position of vesicle inertial migration is farther and farther from its initial position; with the increase of blocking ratio, the equilibrium position of vesicle inertial migration is closer to the wall surface. For the modulus and viscosity of the vesicle membrane and for the membrane thickness, the results show that the modulus and viscosity determine the degree of deformation of the vesicle, and the modulus has little effect on the equilibrium position of the vesicle, but increases the viscosity, and the membrane thickness will promote the equilibrium position of the vesicle to be biased toward the center of the tube. This study helps to further clarify the deformation and equilibrium position of vesicles during inertial migration, and provides a reliable computational basis for the application of vesicles in drug transport, chemical reactions and physiological processes.

Effects of rheological properties of the fluids on migration behavior of double emulsions

Particle migration in microfluidic devices has received considerable attention because of its applications in medicine, chemistry, and engineering, including diagnostics, particle separation, and cell sorting. In this study, we observe inertial migration of soft particles in a straight tube at Re ranging from 100 to 1000. Uniform double emulsions as soft particles are produced by a flow focusing technique in a micro fluidic device. Mechanical properties of the double emulsions were measured prior to migration experiments. Equilibrium radial positions of the double emulsions were measured at several Re. Re affected the distance from the tube center and numbers of the equilibrium radial position. In Re < 250 range, a single equilibrium radial position was observed, which gradually shifted toward the channel wall. Two equilibrium positions were observed at Re ranging from 300 to 500. In Re > 500 range, a single equilibrium radial position closer to the center was observed. These different flow regimes and transitions were influenced by particle deformability.

Numerical investigation of proppant transport at hydraulic-natural fracture intersection

The transport mechanism of particles at fracture intersection is numerically studied by the coupled lattice Boltzmann-discrete element methods. First, the numerical method is validated via a benchmark test of the relative suspension viscosity. Second, a comprehensive parametric study on proppant transport through a T-junction is performed. The impacts of various parameters, including particle concentration, particle size, the Reynolds number and the fracture intersection angle are investigated. The results show that the proppant leak-off ratio decreases with particle concentration due to retardation and the Reynolds number due to inertial migration, and increases with fracture aperture. Particularly, the results also reveal a critical intersection angle of 60° at which the particle leak-off ratio reaches a maximum. Finally, an empirical expression is proposed to evaluate the particle leak-off ratio. The outcomes provide new insights into proppant transport in fracture networks and assist in an improved fracturing fluid design for naturally fractured reservoirs.

A numerical lift force analysis on the inertial migration of a deformable droplet in steady and oscillatory microchannel flows.

Inertial migration of deformable particles has become appealing in recent years due to its numerous applications in microfluidics and biomedicine. The physics underlying the motion of these particles is contingent upon the presence of lift forces in microchannels. Therefore, in this work, we present a lift force analysis for such migration of a deformable droplet in steady and oscillatory flow regimes and identify the effects of varying Capillary number and oscillation frequency on its dynamics. We then propose an expression that mimics the lift force behavior in oscillatory flows accurately. Finally, we introduce a procedure to derive and predict a simple expression for the steady and averaged oscillatory lift for any given combination of Capillary number and oscillation frequency within a continuous range.

Fluid-Structure Interaction Approach to Single Particle in a Square Microchannel

Inertial microfluidic technique has been widely applied on particle/cell manipulation and detection. To understand the physical principle of this technique more detailed, the interaction of fluid and particle was studied through the Fluid-Structure Interaction (FSI) method. The equilibrium positions of finite-size particles with different diameters were simulated at moderate Reynolds numbers. The flow structure around two typical particles was analysed. The vortex in the front of the particle retards particle’s translation leading to the lag velocity increasing. Finally, the rotation velocity and the rotational-induced force analysed quantitatively to demonstrate that particle’s self-rotation significantly promotes its inertial migration.

Research on the Inertial Migration Characteristics of Bi-Disperse Particles in Channel Flow

The inertial focusing effect of particles in microchannels shows application potential in engineering practice. In order to study the mechanism of inertial migration of particles with different scales, the motion and distribution of two particles in Poiseuille flow are studied by the lattice Boltzmann method. The effects of particle size ratio, Reynolds number, and blocking rate on particle inertial migration are analyzed. The results show that, at a high blocking rate, after the same scale particles are released at the same height of the channel, the spacing between the two particles increases monotonically, and the change in the initial spacing has little effect on the final spacing of inertial migration. For two different size particles, when the smaller particle is downstream, the particle spacing will always increase and cannot remain stable. When the larger particle is downstream, the particle spacing increases firstly and then decreases, and finally tends to be stable.

Instability of particle inertial migration in shear flow

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 interpreted before in terms of the unstable equilibrium.

Contribution of Particle–Wall Distance and Rotational Motion of a Single Confined Elliptical Particle to the Effective Viscosity in Pressure-Driven Plane Poiseuille Flows

Rheological properties of the suspension flow, especially effective viscosity, partly depend on spatial arrangement and motion of suspended particles. It is important to consider effective viscosity from the microscopic point of view. For elliptical particles, the equilibrium position of inertial migration in confined state is unclear, and there are few studies on the relationship between dynamics of suspended particles and induced local effective viscosity distribution. Contribution of a single circular or elliptical particle flowing between parallel plates to the effective viscosity was studied, focusing on the particle–wall distance and particle rotational motion using the two-dimensional regularized lattice Boltzmann method and virtual flux method. As a result, confinement effects of the elliptical particle on the equilibrium position of inertial migration were summarized using three definitions of confinement. In addition, the effects of particle shape (aspect ratio and confinement) on the effective viscosity were assessed focusing on the particle–wall distance. The contribution of particle shape to the effective viscosity was found to be enhanced when the particle flowed near the wall. Focusing on the spatial and temporal variation of relative viscosity evaluated from wall shear stress, it was found that the spatial variation of the local relative viscosity was larger than temporal variation regardless of the aspect ratio and particle–wall distance.

Inertial migration of a non-neutrally buoyant particle in a linear shear flow with thermal convection

Inertial migration of a nonneutrally buoyant disk particle in a linear shear flow under thermal convection (TC) is studied numerically. Particle dynamics and equilibrium position depend on competition between the inertial lift force, the gravitational force (buoyancy), and the TC lift force. When the particle is heavier than the fluid, inertial lift-off is observed and a modified phase diagram is established in terms of the channel Reynolds number Re and density ratio \ensuremath{\sigma}. When the particle is lighter than the fluid, TC induced re-suspension and sedimentation are observed, where the normalized critical Grashof number, Gr${}_{\mathrm{crit}}$/\ensuremath{\sigma}, depends linearly on Re for all particle densities studied.

Inertial migration of a neutrally buoyant circular particle in a planar Poiseuille flow with thermal fluids

This work presents an extensive numerical study on the inertial migration of a neutrally buoyant circular particle in a planar Poiseuille flow with thermal convection using the lattice Boltzmann method. The numerical analysis confirms that, once the channel Grashof number Gr reaches a critical value Grc, the particle lateral equilibrium position transits from above the centerline to below. Moreover, the critical Grashof number Grc depends on the channel Reynolds number and the particle-to-channel size ratio. A force scaling analysis is performed, from which a unified Grashof number Gr* is introduced to consider the synergetic effects of thermal convection and particle size. In addition, it is also found that the occurrence of equilibrium position transition is not sensitive to the initial conditions, but the exact transition point depends on the initial conditions, such as the initial position of the particle and the fluid flow state.

Study on the binding focusing state of particles in inertial migration

In this paper, the inertial migration of particles in a periodic straight channel is studied numerically based on the immersed boundary-lattice Boltzmann method in two-dimensional and three-dimensional Poiseuille flow. In the model, on considering the inertial migration of particles, the interaction of a soft particle and a hard particle that are similar in size is studied. We observe an interesting particle interaction phenomenon named the binding focusing state (BFS). That is, when the focused soft particle gets close to the hard particle, both particles will switch their original runways, and then run together on a new equilibrium position. To investigate the characteristics of the BFS, the effects of the soft particle deformability, the Reynolds numbers and the initial particle positions on the BFS are studied in detail. The soft particle deformability and Reynolds numbers can affect the formation of the BFS. Moreover, the streamline and pressure around the particles are analyzed to show the formation laws of the BFS; this is more intuitive and aids in understanding the BFS.

Transport of Non-Spherical Particles in Square Microchannel Flows: A Review

Understanding the behavior of a single particle flowing in a microchannel is a necessary step in designing and optimizing efficient microfluidic devices for the separation, concentration, counting, detecting, sorting, or mixing of particles in suspension. Although the inertial migration of spherical particles has been deeply investigated in the last two decades, most of the targeted applications involve shaped particles whose behavior in microflows is still far from being completely understood. While traveling in a channel, a particle both rotates and translates: it translates in the streamwise direction driven by the fluid flow but also in the cross-section perpendicular to the streamwise direction due to inertial effects. In addition, particles’ rotation and translation motions are coupled. Most of the existing works investigating the transport of particles in microchannels decouple their rotational and lateral migration behaviors: particle rotation is mainly studied in simple shear flows, whereas lateral migration is neglected, and studies on lateral migration mostly focus on spherical particles whose rotational behavior is simple. The aim of this review is to provide a summary of the different works existing in the literature on the inertial migration and the rotational behavior of non-spherical particles with a focus and discussion on the remaining scientific challenges in this field.

Inertial Migration of Neutrally Buoyant Spherical Particles in Square Channels at Moderate and High Reynolds Numbers

The inertial migration of particles in microchannel flows has been deeply investigated in the last two decades. In spite of numerous reports on the inertial focusing patterns in a square channel, the particle inertial focusing and longitudinal ordering processes remain unclear at high Reynolds numbers (>200) in square microchannels smaller than 100 µm in width. Thus, in this work, in situ visualization of particles flowing in square micro-channels at Reynolds numbers Re ranging from 5 to 280 has been conducted and their migration behaviors have been analyzed. The obtained results confirm that new equilibrium positions appear above a critical Re depending on the particle to channel size ratio and the particle volume fraction. It is also shown that, for a given channel length, an optimal Reynolds number can be identified, for which the ratio of particles located on equilibrium positions is maximal. Moreover, the longitudinal ordering process, i.e., the formation of trains of particles on equilibrium positions and the characterization of their length, has also been analyzed for the different flow conditions investigated in this study.

The influence of cell elastic modulus on inertial positions in Poiseuille microflows

Microchannels are used as a transportation highway for suspended cells both in vivo and ex vivo. Lymphatic and cardiovascular systems transfer suspended cells through microchannels within the body, and microfluidic techniques such as lab-on-a-chip devices, flow cytometry, and CAR T-cell therapy utilize microchannels of similar sizes to analyze or separate suspended cells ex vivo. Understanding the forces that cells are subject to while traveling through these channels are important because certain applications exploit these cell properties for cell separation. This study investigated the influence that cytoskeletal impairment has on the inertial positions of circulating cells in laminar pipe flow. Two representative cancer cell lines were treated using cytochalasin D, and their inertial positions were investigated using particle streak imaging and compared between benign and metastatic cell lines. This resulted in a shift in inertial positions between benign and metastatic as well as treated and untreated cells. To determine and quantify the physical changes in the cells that resulted in this migration, staining and nanoindentation techniques were then used to determine the cells’ size, circularity, and elastic modulus. It was found that the cells’ exposure to cytochalasin D resulted in decreased elastic moduli of cells, with benign and metastatic cells showing decreases of 135 ± 91 and 130 ± 60 Pa, respectively, with no change in either size or shape. This caused benign, stiffer cancer cells to be more evenly distributed across the channel width than metastatic, deformable cancer cells; additionally, a decrease in the elastic moduli of both cell lines resulted in increased migration toward the channel center. These results indicate that the elastic modulus may play more of a part in the inertial migration of such cells than previously thought.

Resolving dynamics of inertial migration in straight and curved microchannels by direct cross-sectional imaging.

The explosive development of inertial microfluidic systems for label-free sorting and isolation of cells demands improved understanding of the underlying physics that dictate the intriguing phenomenon of size-dependent migration in microchannels. Despite recent advances in the physics underlying inertial migration, migration dynamics in 3D is not fully understood. These investigations are hampered by the lack of easy access to the channel cross section. In this work, we report on a simple method of direct imaging of the channel cross section that is orthogonal to the flow direction using a common inverted microscope, providing vital information on the 3D cross-sectional migration dynamics. We use this approach to revisit particle migration in both straight and curved microchannels. In the rectangular channel, the high-resolution cross-sectional images unambiguously confirm the two-stage migration model proposed earlier. In the curved channel, we found two vertical equilibrium positions and elucidate the size-dependent vertical and horizontal migration dynamics. Based on these results, we propose a critical ratio of blockage ratio (β) to Dean number (De) where no net lateral migration occurs (β/De ∼ 0.01). This dimensionless number (β/De) predicts the direction of lateral migration (inward or outward) in curved and spiral channels, and thus serves as a guideline in design of such channels for particle and cell separation applications. Ultimately, the new approach to direct imaging of the channel cross section enables a wealth of previously unavailable information on the dynamics of inertial migration, which serves to improve our understanding of the underlying physics.

Inertial migration of a deformable capsule in an oscillatory flow in a microchannel.

Dynamics of a deformable capsule in an oscillatory flow of a Newtonian fluid in a microchannel has been studied numerically. The effects of oscillation frequency, capsule deformability, and channel flow rate have been explored by simulating the capsule within a microchannel. In addition, the simulation captures the effect of the type of imposed pressure oscillations on the migration pattern of the capsule. An oscillatory channel flow enables the focusing of extremely small biological particles by eliminating the need to design impractically long channels. The presented results show that the equilibrium position of the capsule changes not only by the addition of an oscillatory component to the pressure gradient but it also is influenced by the capsule deformability and channel flow rate. Furthermore, it has been shown that the amplitude of oscillation of capsules decreases as the channel flow rate and the rigidity of the capsule increases.

A simulation method for particle migration in microfluidic spirals with application to small and medium particle concentrations

This paper treats the separation of particles in microchannels relevant to biological and industrial process engineering. To elucidate the mechanisms creating uneven distribution of particles over the cross section, simulations are conducted with the particles being geometrically resolved and coupled to the fluid by an immersed-boundary method. In a first step, the method is validated for particle focusing in straight channels. Beyond validation, new information not previously available is reported for these cases. Next, an efficient approach is presented to simulate the motion of particles in spiral ducts of small curvature by means of a well-controlled set of approximate equations. It is applied here to situations with spherical particles and validated with reference data for inertial migration in curved channels achieving good agreement. The simulation data provide new rich information on the details of the separation process concerning migration time, particle positioning in the cross section, streamwise particle spacing, and velocity field of the continuous phase. For concentrations smaller than 1%, three different focusing modes are observed: single position, two symmetric positions, and periodic trajectories oscillating between two focusing points. Another set of results is obtained with particle concentrations up to 10% in a curved channel. Here, the spatial distribution of particles is determined in a statistical sense and related to the mean flow of the continuous phase. While focusing is reduced with increasing particle concentration, the distribution of particles is found to be still far from uniform up to the investigated concentration level.

Mapping inertial migration in the cross section of a microfluidic channel with high-speed imaging

The wide adoption of inertial microfluidics in biomedical research and clinical settings, such as rare cell isolation, has prompted the inquiry of its underlying mechanism. Although tremendous improvement has been made, the mechanism of inertial migration remains to be further elucidated. Contradicting observations are not fully reconciled by the existing theory, and details of the inertial migration within channel cross sections are missing in the literature. In this work, for the first time, we mapped the inertial migration pathways within channel cross section using high-speed imaging at the single-particle level. This is in contrast to the conventional method of particle streak velocimetry (PSV), which provides collective information. We also applied smoothed particle hydrodynamics (SPH) to simulate the transient motion of particles in 3D and obtained cross-sectional migration trajectories that are in agreement with the high-speed imaging results. We found two opposing pathways that explain the contradicting observations in rectangular microchannels, and the force analysis of these pathways revealed two metastable positions near the short walls that can transition into stable positions depending on the flow condition and particle size. These new findings significantly improve our understanding of the inertial migration physics, and enhance our ability to precisely control particle and cell behaviors within microchannels for a broad range of applications.