Tubular Pinch Research Articles

Improving liquid cancer biopsy through computer-aided microfluidic device design based on the tubular pinch effect and Deans flow

A blood test, called a liquid biopsy is being developed by several research groups for the early detection of cancer. The key challenge in this process is enrichment, or isolation, of the small number of cancer cells in blood serum from other cells. This study uses numerical simulations to design devices that advance the state of art in the enrichment of circulating cancer cells. In particular, the lift forces acting on neutrally buoyant small spherical particles in a Poiseuille flow bounded by a wall are modelled using numerical computation. Specifically, we explore the influence of the Reynolds number, distance to the wall, and particle size on these lift forces, building upon previous research on the Segre-Silberberg effect. By validating the mechanism by which particles concentrate at specific positions within the fluid, we confirm the existence of the Segre-Silberberg effect in this system. Subsequently, we apply a customized 3D model of a filtration device to investigate the Segre-Silberberg effect and Deans effect in a non-Poiseuille flow, which can be applied in liquid cancer biopsy. Our results demonstrate that this flow pattern could potentially be used to separate cancer cells from the mean flow. Finally, we apply the flow separation mechanisms in microfluidic design through a rapid 3D design and computational fluid dynamics (CFD) verification cycle. The popular spiral microfluidic design does not demonstrate strong flow separation pattern in our tests, while the sinusoidal design shows much more promise. Further research is needed to empirically evaluate these designs.

Blood cell distribution in small and large vessels: Effects of wall and rotating motion of red blood cells

A two-dimensional computer simulation of blood flow between two parallel plates as the tube was performed to understand the distribution of red blood cells (RBCs) and platelets (PLTs) according to the blood vessel size. The motion of the blood cells (BCs) was directly calculated using the particle method. The tube diameter and hematocrit were set as 20–500 µm and 0–0.4, respectively. In simulations with tank-treading (TT) RBCs under the planar Poiseuille flow, RBCs moved from the tube wall to form a cell-free layer (CFL). Then, the PLTs moved into the CFL, and the RBCs concentrated around the tube center, excluding the PLTs. By comparing the BC distribution between the Couette and Poiseuille flows, the range of the wall effect was estimated to be ≤50–100 µm at the hematocrit of 0.4. Tumbling (TB) RBCs uniformly distributed inside the tube, while forming rouleaux-like aggregates on the wall at 0.4 in hematocrit; at hematocrit ≤0.3, the TB RBCs tended to be excluded from the tube center as known to the tubular pinch effect. The mechanical interaction among the RBCs and tube wall facilitated TT motion even if the apparent shear rate was so small that an RBC in a dilute suspension would exhibit TB motion. These results indicate that the TT motion of RBCs combined with the wall effect plays a major role in forming CFL and avoiding aggregation of BCs and that TB motion helps BCs to distribute uniformly in large vessels where the shear rate is relatively low.

A mathematical model of the erosion process in a channel bend

We develop a mathematical model for the cumulative erosion of a micro-particle in a channel system. At 90∘ bends in the system, a particle may deviate from the flow, impact the wall, and erode material. We highlight the case of the eroded material adhering to the particle, growing in size, and thus demonstrate how the damage accumulates exponentially with time. We describe and quantify the statistical nature of the evolution of particle growth and erosion (mass and location). We perform this analysis according to a number of realistic particle concentration distributions: uniform, Gaussian, and bimodal. A bimodal distribution, corresponding to the tubular pinch effect in suspension flows, results in unequal peak zones of erosion due to the flow characteristics.

Revisiting the Homogenized Lattice Boltzmann Method with Applications on Particulate Flows

The simulation of surface resolved particles is a valuable tool to gain more insights in the behaviour of particulate flows in engineering processes. In this work the homogenized lattice Boltzmann method as one approach for such direct numerical simulations is revisited and validated for different scenarios. Those include a 3D case of a settling sphere for various Reynolds numbers. On the basis of this dynamic case, different algorithms for the calculation of the momentum exchange between fluid and particle are evaluated along with different forcing schemes. The result is an updated version of the method, which is in good agreement with the benchmark values based on simulations and experiments. The method is then applied for the investigation of the tubular pinch effect discovered by Segré and Silberberg and the simulation of hindered settling. For the latter, the computational domain is equipped with periodic boundaries for both fluid and particles. The results are compared to the model by Richardson and Zaki and are found to be in good agreement. As no explicit contact treatment is applied, this leads to the assumption of sufficient momentum transfer between particles via the surrounding fluid. The implementations are based on the open-source C++ lattice Boltzmann library OpenLB.

Spatiotemporal Dynamics of Dilute Red Blood Cell Suspensions in Low-Inertia Microchannel Flow

Microfluidic technologies are commonly used for the manipulation of red blood cell (RBC) suspensions and analyses of flow-mediated biomechanics. To enhance the performance of microfluidic devices, understanding the dynamics of the suspensions processed within is crucial. We report novel, to our knowledge, aspects of the spatiotemporal dynamics of RBC suspensions flowing through a typical microchannel at low Reynolds number. Through experiments with dilute RBC suspensions, we find an off-center two-peak (OCTP) profile of cells contrary to the centralized distribution commonly reported for low-inertia flows. This is reminiscent of the well-known “tubular pinch effect,” which arises from inertial effects. However, given the conditions of negligible inertia in our experiments, an alternative explanation is needed for this OCTP profile. Our massively parallel simulations of RBC flow in real-size microfluidic dimensions using the immersed-boundary-lattice-Boltzmann method confirm the experimental findings and elucidate the underlying mechanism for the counterintuitive RBC pattern. By analyzing the RBC migration and cell-free layer development within a high-aspect-ratio channel, we show that such a distribution is co-determined by the spatial decay of hydrodynamic lift and the global deficiency of cell dispersion in dilute suspensions. We find a cell-free layer development length greater than 46 and 28 hydraulic diameters in the experiment and simulation, respectively, exceeding typical lengths of microfluidic designs. Our work highlights the key role of transient cell distribution in dilute suspensions, which may negatively affect the reliability of experimental results if not taken into account.

A Minireview on Inertial Microfluidics Fundamentals: Inertial Particle Focusing and Secondary Flow

In 1961, Segre and Silberberg first reported the tubular pinch effect and numerous theoretical studies were subsequently published to explain the inertial particle migration phenomenon. Presently, as fluid mechanics meets micro- and nanotechnology, theoretical studies on intrinsic particle migration and flow phenomena associated with inertia are being experimentally tested and validated. This collective study on the fluid-particle-structure phenomena in microchannels involving fluid inertia is called, “inertial microfluidics”. Beyond theoretical studies, now inertial microfluidics has been gaining much attention from various research fields ranging from biomedicine to industry. Despite the positive contributions, there is still a lack of clear understanding of intrinsic inertial effects in microchannels. Therefore, this minireview introduces the mechanisms and underlying physics in inertial microfluidic systems with specific focuses on inertial particle migration and secondary flow, and outlines the opportunities provided by inertial microfluidics, along with an outlook on the field.

Thermodynamics of two-phase granular fluids

Starting from basic thermodynamic principles, we derive equations for a two-phase fluid. The first phase is the micropolar fluid and the second phase is the viscous Newtonian fluid. The phases differ in velocities, densities and viscosities. The first granular phase is described with the use of notion of the Cosserat continuum. To illustrate the model, we consider steady pressure driven flows between two parallel planes. Particularly, we address the “tubular pinch effect” of Ségre and Silberberg.

Twin tubular pinch effect in curving confined flows.

Colloidal suspensions of buoyancy neutral particles flowing in circular pipes focus into narrow distributions near the wall due to lateral migration effects associated with fluid inertia. In curving flows, these distributions are altered by Dean currents and the interplay between Reynolds and Dean numbers is used to predict equilibrium positions. Here, we propose a new description of inertial lateral migration in curving flows that expands current understanding of both focusing dynamics and equilibrium distributions. We find that at low Reynolds numbers, the ratio δ between lateral inertial migration and Dean forces scales simply with the particle radius, coil curvature and pipe radius as . A critical value δc = 0.148 of this parameter is identified along with two related inertial focusing mechanisms. In the regime below δc, coined subcritical, Dean forces generate permanently circulating, twinned annuli, each with intricate equilibrium particle distributions including eyes and trailing arms. At δ > δc (supercritical regime) inertial lateral migration forces are dominant and particles focus to a single stable equilibrium position.

Lateral and cross-lateral focusing of spherical particles in a square microchannel

The inertial migration of particles in micro-scale flows has received much attention due to its promising applications, such as the membrane-free passive separation of particles or cells. The particles suspended in rectangular channels are known to be focused near the center of each channel face as the channel Reynolds number (R(C)) increases due to the lift force balance and the hydrodynamic interactions of the particles with the wall. In this study, the three-dimensional positions of neutrally buoyant spherical particles inside a square microchannel are measured using the digital holographic microscopy technique, and a transition from the lateral tubular pinch to the cross-lateral focusing with increasing R(C) is reported. The particles are found to migrate first in the lateral direction and then cross-laterally toward the four equilibrium positions. A general criterion that can be used to secure the fully developed state of particle focusing in Lab-on-a-Chip applications is also derived. This criterion could be helpful for the accurate estimation of the design parameters of inertial microfluidic devices, such as R(C), channel length and width, and particle diameter.

Particle Segregation and Dynamics in Confined Flows

Nonlinearity in finite-Reynolds-number flow results in particle migration transverse to fluid streamlines, producing the well-known "tubular pinch effect" in cylindrical pipes. Here we investigate these nonlinear effects in highly confined systems where the particle size approaches the channel dimensions. Experimental and numerical results reveal distinctive dynamics, including complex scaling of lift forces with channel and particle geometry. The unique behavior described in this Letter has broad implications for confined particulate flows.

Asymptotic model of the inertial migration of particles in a dilute suspension flow through the entry region of a channel

The inertial migration of particles in a dilute suspension flow through the entry region of a plane channel (or a circular pipe) is considered. Within the two-fluid approach, an asymptotic one-way coupling model of the dilute suspension flow in the entry region of a channel is constructed. The carrier phase is a viscous incompressible Newtonian fluid, and the dispersed phase consists of identical noncolloidal rigid spheres. In the interphase momentum exchange, we take into account the drag force, the virtual mass force, the Archimedes force, and the inertial lift force with a correction factor due to the wall effect and an arbitrary particle slip velocity. The channel Reynolds number is high and the particle-to-fluid density ratio is of order unity or significantly larger unity. The solution is constructed using the matched asymptotic expansion method. The problem of finding the far-downstream cross-channel profile of particle number concentration is reduced to solving the equations of the two-phase boundary layer developing on the channel walls. The full Lagrangian approach is used to study the evolution of the cross-flow particle concentration profile. The inertial migration results in particle accumulation on two symmetric planes (an annulus) distanced from the walls, with a nonuniform concentration profile between the planes (inside the annulus) and particle-free layers near the walls. When the particle-to-fluid density ratio is of order unity, an additional local maximum of the particle concentration on inner planes (an inner annulus) is revealed. The inclusion of the corrected lift force makes it possible to resolve the nonintegrable singularity in the concentration profile on the wall, which persisted in all previously published solutions for the dilute suspension flow in a boundary layer. The numerical results are compared to the tubular pinch effect observed in experiments, and a qualitative analogy is found.

Axial particle displacements in fluid slugs after passing a simple serpentiform microchannel

In this work, we designed a simple microchannel to separate particles in fluids by size. We found that mixtures of polystyrene 2 and 0.5 μm particles in a fluid slug can be differentiated by size after passing a long serpentiform microchannel. In contrast to the tubular pinch effect, which separates the particle in radial direction, we found that the particle suspensions in fluid slugs are displaced along the flow directions. The separation performance increases with the increasing flow velocity. The feature of differentiation along the flow directions in our device leads to the result that the separation process can be easily improved by stretching the slug before cutting the slug into pieces. Furthermore, the alternative air slugs between working fluid slugs can also prevent clogging inside the microchannels.

Membrane-free microfiltration by asymmetric inertial migration

Membrane-free microfiltration by asymmetric inertial migration is studied and evidence of the filtration capability is presented. Centrifugal force induced by flow in spiral channel geometry modifies the lateral symmetry of straight-channel tubular pinch equilibrium resulting in a focused particle band nearer to the inner sidewall. Bifurcated outlets separately collect the concentrated particle band and remaining effluent. The spiral continuous flow filtration relies solely on internal fluidic shear characteristics, eliminating the need for membrane filters or external force fields. This device has the desirable combinations of high throughput and low cost, making it inherently suited for preparative filtration in the range of micro- to macroscale applications.

Membraneless microseparation by asymmetry in curvilinear laminar flows

Membraneless microseparation by asymmetric inertial migration is studied in curvilinear laminar flows and evidence of the microseparation is presented. Along a curvilinear laminar flow, transverse particle migration involves competition between three shear-flow effects; the tubular pinch effect, centrifugal force, and Dean's vortex. Equilibrating control of migration allows for particle separation to different outlets. No filter-media or external force is necessary for the microseparation utilizing only shear-flow characteristics. A double-spiral design effectively controls the migration to optimize microseparation. The concentration ratio of 10 μm beads from the two different outlets was 660 times at 92 mm/s of flow velocity. This new technology has great potential for high-throughput and low cost in bio-agent and particulate separation at both macro and micro scales.

Direct numerical simulation of fluid flow laden with many particles

Two-dimensional channel flow of fluid laden with many particles is studied by direct numerical simulation using the Navier–Stokes equation coupled with the equation of motion for respective particles. Fractional four-step method with Crank–Nicolson scheme and ALE technique is adopted for P2P1 mixed finite element formulation of the governing equations for fluid motion. The motion and distribution of particles in the fluid is virtually described and the calculated relative viscosity is compared with previous results within the limits of possibility. The effect of the ratio of channel gap to particle diameter on the relative viscosity and the tubular pinch effect are also delineated.

입자가 포함된 유체의 유동특성에 대한 직접수치해석

Flow characteristics of the fluid laden with many particles in the two-dimensional channel are investigated using the Navier-Stokes equations coupled with the equation of motion of particles by direct numerical simulation. A four-step fractional step method with Crank-Nicolson scheme and ALE technique is used for P2P I mixed finite element method. The motion and distribution of particles in the fluid is virtually described as a result of direct numerical simulation and the increase of viscosity is compared with theoretical equations. The effect of channel height on the relative viscosity and the tubular pinch effect are discussed.

Lateral migration of blood cells and microspheres in two-dimensional Poiseuille flow: A laser-Doppler study

The inertia-induced lateral migration of rigid microspheres, platelets and erythrocytes is studied experimentally. The concentration and velocity profiles of the particles have been determined with a laser-Doppler anemometer designed for high resolution measurements. Data are compared with empirical and analytical models on inertia-induced lateral migration of rigid spheres. Experiments done in rectangular flow channels of high aspect ratio reveal that at a sufficiently high particle Reynolds number, platelets exhibit tubular pinch effects comparable with those of rigid polystyrene microspheres. Erythrocytes also exhibit inertia-induced lateral migration at high particle Reynolds number and low medium viscosity. At a higher medium viscosity, erythrocytes show deformation-induced lateral migration towards the center of the flow channel.

On transport of suspended particulates in tube flow.

Blood cells suspended in shear flows exhibit much larger dispersive motions than those predicted by the Stokes-Einstein formula for Brownian diffusion. The lateral migration and the erratic motions of the 8 microns red blood cells (RBC) is thought to be analogous to a diffusive process. It is shown that the often cited convective-diffusion theory may not be an adequate model for describing the transverse migration of suspended cells in blood flow. A comprehensive review of both the classical theory and of contemporary work in particle transport is presented, with particular emphasis on low Reynolds number tube flows. The mechanisms of Taylor dispersion, the effects of Brownian perturbations on translational and rotational motions of the suspended particles in shear fields, and the influence of integratable and chaotic advections, are individually examined. The classical experiment by Segre and Silberberg (1962) lead us to believe that particle hydrodynamics may play an important role in transverse migrations. In this light, we have further examined the hydrodynamic aspects of the so-called "tubular pinch" effect, the lateral migration of rigid spheres. We have also discussed the transverse motions of liquid drops, and the reversibility of the organization of suspensions in transport. The convective accelerations in the entrance region of a tube can produce relative velocities between fluid medium and various type of particulates if there is a difference in density. The deformable RBC, an "active-type" particle, can provide feedback to the flow from both mass and momentum considerations; the more rigid platelet, a "passive-type" particle, will experience a much smaller relative velocity as compared to the RBC. We may expect that particles of different densities are transported to different equilibrium annular positions before entering the fully developed flow region. The erratic, lateral movement of suspended particulates in steady laminar tube flow can be described by the usual Lagrangian coordinates.

Blood platelets are concentrated near the wall and red blood cells, in the center in flowing blood.

Hematocrit and vessel wall shear rate are important factors in the transport and subsequent adherence of platelets to vessel wall subendothelium. When mass transport theory is applied to platelets in flowing blood, the blood is usually considered to be a fluid with platelet and red cell wall concentrations similar to the average tube concentration. With the laser-Doppler technique, we found how red blood cell ghosts and platelets were distributed radially for various hematocrits and wall shear rates. Red cell ghosts are crowded near the axis of the tube, with a local hematocrit higher than the average tube hematocrit, and they decrease steadily toward the wall. In the absence of ghosts, platelets exhibit the 'tubular pinch' effect (rigid particles crowding at 0.6 x tube radius). In the presence of ghosts, the platelets are expelled toward the wall region. This high concentration at the wall increases with higher average tube hematocrit and wall shear rates. Increasing the average tube platelet concentration 10 times causes the wall concentration to increase only three times. The increase in platelet adherence observed with increasing hematocrit and increasing wall shear rate can be partially ascribed to increased platelet concentration near the wall. The observation that the increased platelet concentration does not fully explain the platelet adherence data suggests that platelet transport may also be enhanced by a shear rate-dependent rotary motion.

Lateral migration mechanisms in capillary hydrodynamic chromatography

In all chromatographic systems that achieve separation of colloidal particles based on the particles' hydrodynamic behavior, there are partitioning mechanisms that promote lateral migration of particles across solvent streamlines. In this work, solvent inertia and particle electrostatics are incorporated in Brenner & Gaydos' general diffusive transport theory for particles and solvent flowing in a capillary, yielding the mean axial velocity of particles as a function of particle size. A comparison is made with capillary hydrodynamic chromatography results. The lateral migration of small particles is primarily due to diffusion, while large particles are focused by the inertial force at one equilibrium radial position, as observed in “tubular pinch” experiments. The transition from diffusion- to inertia-controlled lateral migration can be tuned to specific particle size ranges through variation of solvent ionic strength, flowrate and capillary radius. Poor prediction of the separation behavior of large particles is attributed to inaccuracy in the calculation of the inertial radial velocity, suggesting the need for further theoretical analysis and experimental study of inertial migration.