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 $κ$, quantifying the relative strength of the two forces. We show that $κ$ 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 $κ$ there exists an attracting limit cycle, in each of the upper and lower halves of the duct. At small $κ$ we find that particles migrate to one of four stable foci. Between these extremes, there is an intermediate-range of $κ$ 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 $κ$, 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.
Inertial focusing dynamics of spherical particles in curved microfluidic ducts with a trapezoidal cross-section
School of Mathematics and Statistics, Victoria University of Wellington（维多利亚大学惠灵顿校区数学与统计学院）；School of Mathematical Sciences, University of Adelaide（阿德莱德大学数学科学学院）
We study the inertial migration of finite-size neutrally buoyant spherical particles in dilute and semi-dilute suspensions in laminar square duct flow. We perform several direct numerical simulations using an immersed boundary method to investigate the effects of the bulk Reynolds number $Re_b$, particle Reynolds number $Re_p$ and duct to particle size ratio $h/a$ at different solid volume fractions $ϕ$, from very dilute conditions to $20\%$. We show that the bulk Reynolds number $Re_b$ is the key parameter in inertial migration of particles in dilute suspensions. At low solid volume fraction …
We study numerically the inertial migration of a single rigid sphere and an oblate spheroid in straight square and rectangular ducts. A highly accurate interface-resolved numerical algorithm is employed to analyse the entire migration dynamics of the oblate particle and compare it with that of the sphere. Similarly to the inertial focusing of spheres, the oblate particle reaches one of the four face-centred equilibrium positions, however they are vertically aligned with the axis of symmetry in the spanwise direction. In addition, the lateral trajectories of spheres and oblates collapse into an…
Particles suspended in fluid flow through a closed duct can focus to specific stable locations in the duct cross-section due to hydrodynamic forces arising from the inertia of the disturbed fluid. Such particle focusing is exploited in biomedical and industrial technologies to separate particles by size. In curved ducts, the particle focusing is a result of balance between two dominant forces on the particle: (i) inertial lift arising from small inertia of the fluid, and (ii) drag arising from cross-sectional vortices induced by the centrifugal force on the fluid. Bifurcations of particle equi…
Particles traveling at high velocities through microfluidic channels migrate from their starting streamlines due to inertial lift forces. Theories predict different scaling laws for these forces and there is little experimental evidence by which to validate theory. Here we experimentally measure the three dimensional positions and migration velocities of particles. Our experimental method relies on a combination of sub-pixel accurate particle tracking and velocimetric reconstruction of the depth dimension to track thousands of individual particles in three dimensions. We show that there is no …
We discuss an inertial migration of oblate spheroids in a plane channel, where steady laminar flow is generated by a pressure gradient. Our lattice Boltzmann simulations show that spheroids orient in the flow, so that their minor axis coincides with the vorticity direction (a log-rolling motion). Interestingly, for spheroids of moderate aspect ratios, the equilibrium positions relative to the channel walls depend only on their equatorial radius $a$. By analysing the inertial lift force we argue that this force is proportional to $a^3b$, where $b$ is the polar radius, and conclude that the dime…
We examine theoretically the inertial migration of a neutrally buoyant rigid sphere in pressure-driven channel flow, accounting for its finite size relative to the channel width (the confinement ratio). For sufficiently large channel Reynolds numbers\,($Re_c$), a small but finite confinement ratio qualitatively alters the inertial lift velocity profiles obtained using a point-particle formulation. Finite size effects are shown to lead to new equilibria, in addition to the well known Segre-Silberberg pinch locations. Consequently, a sphere can migrate to either the near-wall Segre-Silberberg eq…
Within microcentrifuge devices, a microfluidic vortex separates larger particles from a heterogeneous suspension using inertial migration, a phenomenon that causes particles to migrate across streamlines. The ability to selectively capture particles based on size differences of a few microns makes microcentrifuges useful diagnostic tools for trapping rare cells within blood samples. However, rational design of microcentrifuges has been held back from its full potential by a lack of quantitative modeling of particle capture mechanics. Here we use an asymptotic method, in which particles are acc…
Inertial lift forces are exploited within inertial microfluidic devices to position, segregate, and sort particles or droplets. However the forces and their focusing positions can currently only be predicted by numerical simulations, making rational device design very difficult. Here we develop theory for the forces on particles in microchannel geometries. We use numerical experiments to dissect the dominant balances within the Navier-Stokes equations and derive an asymptotic model to predict the lateral force on the particle as a function of particle size. Our asymptotic model is valid for a …
We perform fully Eulerian numerical simulations of an initially spherical hyperelastic particle suspended in a Newtonian pressure-driven flow in a cylindrical straight pipe. We study the full particle migration and deformation for different Reynolds numbers and for various levels of particle elasticity, to disentangle the interplay of inertia and elasticity on the particle focusing. We observe that the particle deforms and undergoes a lateral displacement while traveling downstream through the pipe, finally focusing at the pipe centerline. We note that the migration dynamics and the final equi…
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…
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…
At finite Reynolds numbers particles migrate across flow streamlines to their equilibrium positions in microchannels. Such a migration is attributed to an inertial lift force, and it is well-known that the equilibrium location of neutrally-buoyant particles is determined only by their size and the Reynolds number. Here we demonstrate that the decoration of a bottom wall of the channel by superhydrophobic grooves provides additional possibilities for manipulation of neutrally-buoyant particles. It is shown that the effective anisotropic hydrodynamic slip of such a bottom wall can be readily use…
This work focuses on the dynamics of a train of unconfined bubbles flowing in microchan- nels. We investigate the transverse position of a train of bubbles, its velocity and the associated pressure drop when flowing in a microchannel depending on the internal forces due to viscosity, inertia and capillarity. Despite the small scales of the system, inertia, referred to as inertial migration force, play a crucial role in determining the transverse equilibrium position of the bubbles. Beside inertia and viscosity, other effects may also affect the transverse migration of bubbles such as the Maran…
KEY WORDS suspensions, inertial focusing, particle-laden flows, high Reynolds numbers SHORT SUMMARY The inertial migration of particles in square channel flows at the micro-scale has been deeply investigated in the last two decades. The well-known four equilibrium positions are located near the center of each channel face at moderate Reynolds numbers [1]. More recently, Miura et al. [2] revealed experimentally the presence of eight equilibrium positions in millimetric square channels for Reynolds numbers higher than 250. The aim of the present work is to extend these results obtained at millim…
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…
We carry out direct numerical simulation of compressible square duct flow in the range of bulk Mach numbers M_b = 0.2-3, and up to friction Reynolds number Re_τ = 500. The effects of flow compressibility on the secondary motions are found to be negligible, as the typical Mach number associated with the cross-stream flow is always less than 0.1. As in the incompressible case, we find that the wall law for the mean streamwise velocity applies with good approximation with respect to the nearest wall, upon suitable compressibility transformation. The same conclusion also applies to a passive scala…
Inertial migration of slender prolate and thin oblate spheroids in plane Poiseuille flow
International Centre for Theoretical Sciences（国际理论科学中心）；Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research（贾瓦哈拉尔·尼赫鲁高级科学研究工程力学组）
The effects of cell size and deformability on the lateral migration and deformation of living cells flowing through a rectangular microchannel has been numerically investigated and compared with the inertial-microfluidics data on detection and separation of cells. The results of this work indicate that the cells move closer to the centerline if they are bigger and/or more deformable and that their equilibrium position is largely determined by the solvent (cytosol) viscosity, which is much less than the polymer (cytoskeleton) viscosity measured in most rheological systems. Simulations also sugg…
We present a numerical analysis of the lateral movement and equilibrium radial positions of red blood cells (RBCs) with major diameter of 8 $μ$m under a Newtonian fluid in a circular channel with 50-$μ$m diameter. Each RBC, modelled as a biconcave capsule whose membrane satisfies strain-hardening characteristics, is simulated for different Reynolds numbers $Re$ and capillary numbers $Ca$, the latter of which indicate the ratio of the fluid viscous force to the membrane elastic force. The effects of initial orientation angles and positions on the equilibrium radial position of an RBC centroid a…
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 such particles is contingent upon the presence of lift forces in microchannels. This importance initiated several works to analyze and quantify such forces acting on particles. However, since most of such attempts have focused on solid and non-deformable particles, we extend the lift force analysis for the case of deformable droplets and study the effects of Capillary numbers on their dynamics in this paper. Furt…
The mechanical deformability of single cells is an important indicator for various diseases such as cancer, blood diseases and inflammation. Lab-on-a-chip devices allow to separate such cells from healthy cells using hydrodynamic forces. We perform hydrodynamic simulations based on the lattice-Boltzmann method and study the behavior of an elastic capsule in a microfluidic channel flow in the inertial regime. While inertial lift forces drive the capsule away from the channel center, its deformability favors migration in the opposite direction. Balancing both migration mechanisms, a deformable c…
We study the inertial migration of a torque-free neutrally buoyant sphere in wall-bounded plane Couette flow over a wide range of channel Reynolds numbers, $Re_c$, in the limit of small particle Reynolds number\,($Re_p\ll1$) and confinement ratio\,($λ\ll1$). Here, $Re_c = V_\text{wall}H/ν$ where $H$ denotes the separation between the channel walls, $V_\text{wall}$ denotes the speed of the moving wall, and $ν$ is the kinematic viscosity of the Newtonian suspending fluid; $λ= a/H$, $a$ being the sphere radius, with $Re_p=λ^2 Re_c$. The channel centerline is found to be the only (stable)\,equilib…
We present an experimental investigation of turbulent drag reduction with flexible and rigid polymer solutions. The flexible polymer is partially hydrolyzed polyacrylamide (HPAM) and the rigid polymer is xanthan gum (XG). The experiments are carried out at low drag reduction ($\%DR < 40$), high drag reduction ($\%DR > 40$) and maximum drag reduction (where the velocity profile $U^+$ roughly matches Virk's asymptote). We compare velocity profiles, streamwise and wall-normal Reynolds stresses and power spectra of streamwise velocity fluctuations measured by Laser Doppler Anemometry (LDA). Our re…
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 eq…
We study the cross-stream inertial migration of a torque-free neutrally buoyant spheroid, of an arbitrary aspect ratio $κ$, in wall-bounded plane Poiseuille flow for small particle Reynolds numbers\,($Re_p\ll1$) and confinement ratios\,($λ\ll1$), with the channel Reynolds number, $Re_c = Re_p/λ^2$, assumed to be arbitrary; here, $λ=L/H$ where $L$ is the semi-major axis of the spheroid and $H$ denotes the separation between the channel walls. In the Stokes limit\,($Re_p =0)$ and for $λ\ll 1$, a spheroid rotates along any of an infinite number of Jeffery orbits parameterized by an orbit constant…
In recent years, manipulation of particles by inertial microfluidics has attracted significant attention. Most studies focused on inertial focusing of particles suspended within liquid phase, in which the ratio of the density of the particle to that of the medium is O(1). the investigation on manipulation of aerosol particles in an inertial microfluidics is very limited. In this study, we numerically investigate the aerosol particle motion in a 3D straight microchannel with rectangular cross section by fully resolved simulation of the particle-air flow based on the contiuum model. The air flow…
The dynamics of dilute micron-sized spherical inertial particles in turbulent duct flows is studied by means of direct numerical simulations of the carrier phase turbulence with one-way coupled Lagrangian particles. The geometries are a square and a rectangular duct with width-to-height aspect ratio $AR$ of 3 operating at $Re_{τ,c}=360$ . The present study is designed to determine the effect of turbulence-driven secondary motion on the particle dynamics. Our results show that a weak cross-flow secondary motion significantly changes the cross-sectional map of the particle concentration, mean ve…
We experimentally investigate the influence of finite-size spherical particles in turbulent flows of a Newtonian and a drag reducing viscoelastic fluid at varying particle volume fractions and fixed Reynolds number. Experiments are performed in a square duct at a Reynolds number $Re_{2H}$ of nearly $1.1\times10^4$, Weissenberg number $Wi$ for single-phase flow is between 1-2 and results in a drag-reduction of 43% compared to a Newtonian flow (at the same $Re_{2H}$). Particles are almost neutrally-buoyant hydrogel spheres having a density ratio of 1.0035$\pm$0.0003 and a duct height ${2H}$ to p…
At finite Reynolds numbers, Re, particles migrate across laminar flow streamlines to their equilibrium positions in microchannels. This migration is attributed to a lift force, and the balance between this lift and gravity determines the location of particles in channels. Here we demonstrate that velocity of finite-size particles located near a channel wall differs significantly from that of an undisturbed flow, and that their equilibrium position depends on this, referred to as slip velocity, difference. We then present theoretical arguments, which allow us to generalize expressions for a lif…
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…
In this paper, we present the results of a numerical study of air-water turbulent bubbly flow in a periodic vertical square duct. The study is conducted using a novel numerical technique which leverages Volume of Fluid method for interface capturing and Sharp Surface Force method for accurate representation of the surface tension forces. A three-dimensional geometry construction method is employed during solution of interface equation which gives absolute conservation of mass and sharp interface between gas and liquid phases. The entire algorithm has been implemented on a data parallel mode on…
Recent experimental studies have shown that particle transfer across streamlines can be controlled passively using stratified flows of co-flowing streams at a finite Reynolds number. The stratification modifies the forces acting on particles through the curvature of the undisturbed velocity profile. In this study, we numerically analyze the particle migration in stratified flow of two liquids of different viscosities flowing parallel to each other between two infinite parallel plates. Particle migration in two different flow fields is considered: (i) Couette flow and (ii) Poiseuille flow. A nu…
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…
The sedimentation of a spherical particle in an elastoviscoplastic fluid in proximity of a flat wall is investigated by direct numerical simulations. The governing equations under inertialess conditions are solved by the finite element method with an Arbitrary Lagrangian-Eulerian formulation to manage the particle motion. The fluid is modeled with the Giesekus constitutive equation modified as proposed by Saramito (2007). The sedimentation, migration, and angular velocities are computed as a function of the particle-wall distance for various Weissenberg and Bingham numbers. The presence of a y…
Inertial microfluidic is able to focus and separate particles in microchannels based on the characteristic geometry and intrinsic hydrodynamic effect. Yet, the vertical position of suspended particles in the microchannel cannot be manipulated in real time. In this study, we utilize the boundary slip effect to regulate the parabolic velocity distribution of fluid in the microchannel and present a scheme to active control the vertical position of particles in inertial microfluidics. The flow field of a microchannel with a unilateral slip boundary is equivalent to that of the microchannel widened…
Particles traveling through inertial microfluidic devices migrate to focusing streamlines. We present a numerical method that calculates migration velocities of particles in inertial microfluidic channels of arbitrary cross section by representing particles by singularities. Refinements to asymptotic analysis are given that improve the regularity of the singularity approximation, making finite element approximations of flow and pressure fields more effective. Sample results demonstrate that the method is computationally efficient and able to capture bifurcations in particle focusing positions …
Understanding particle transport and localisation in porous channels, especially at moderate Reynolds numbers, is relevant for many applications ranging from water reclamation to biological studies. Recently, researchers experimentally demonstrated that the interplay between axial and permeate flow in a porous microchannel results in a wide range of focussing positions of finite sized particles (Garcia & Pennathur 2017) . We numerically explore this interplay by computing the lateral forces on a neutrally buoyant spherical particle that is subject to both inertial and permeate forces over a ra…
Geometry-free prediction of inertial lift forces in microfluidic devices using deep learning
Department of Mechanical & Industrial Engineering, University of Toronto（机械与工业工程系，多伦多大学）；Mechanical Engineering, Purdue University（机械工程，普渡大学）；Department of Materials Science & Engineering, University of Toronto（材料科学与工程系，多伦多大学）；Institute of Biomedical Engineering, University of Toronto（生物医学工程研究所，多伦多大学）
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. Due to the Saffman effect an external force applied along the microchannel induces a cross-streamline migration to a new equilibrium position. 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 an…
We consider suspensions of neutrally-buoyant finite-size rigid spherical particles in channel flow and investigate the relation between the particle dynamics and the mean bulk behavior of the mixture for Reynolds numbers $500 \le Re \le 5000$ and particle volume fraction $0\le Φ\le 0.3$, via fully resolved numerical simulations. Analysis of the momentum balance reveals the existence of three different regimes: laminar, turbulent and inertial shear-thickening depending on which of the stress terms, viscous, Reynolds or particle stress, is the major responsible for the momentum transfer across t…
Inertial microfluidics is a promising tool for many lab-on-a-chip applications. Particles in channel flows with Reynolds numbers above one undergo cross-streamline migration to a discrete set of equilibrium positions in square and rectangular channel cross sections. This effect has been used extensively for particle sorting and the analysis of particle properties. Using the lattice Boltzmann method, we determine equilibrium positions in square and rectangular cross sections and classify their types of stability for different Reynolds numbers, particle sizes, and channel aspect ratios. Our find…
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…
