The inertial migration of non-neutrally buoyant spherical particles in two-dimensional shear flows

1994 ◽  
Vol 272 ◽  
pp. 285-318 ◽  
Author(s):  
Andrew J. Hogg
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Chemical Engineering ◽  
Practical Importance ◽  
Spherical Particles ◽  
Lateral Motion ◽  
Order Unity ◽  
Inertial Migration ◽  
Induced Motion ◽  
Neutrally Buoyant

The inertial migration of a small rigid spherical particle, suspended in a fluid flowing between two plane boundaries, is investigated theoretically to find the effect on the lateral motion. The channel Reynolds number is of order unity and thus both boundary-induced and Oseen-like inertial migration effects are important. The particle Reynolds number is small but non-zero, and singular perturbation techniques are used to calculate the component of the migration velocity which is directed perpendicular to the boundaries of the channel. The particle is non-neutrally buoyant and thus its buoyancy-induced motion may be either parallel or perpendicular to the channel boundaries, depending on the channel alignment. When the buoyancy results in motion perpendicular to the channel boundaries, the inertial migration is a first-order correction to the magnitude of this lateral motion, which significantly increases near to the boundaries. When the buoyancy produces motion parallel with the channel boundaries, the inertial migration gives the zeroth-order lateral motion either towards or away from the boundaries. It is found that those particles which have a velocity exceeding the undisturbed shear flow will migrate towards the boundaries, whereas those with velocities less than the undisturbed flow migrate towards the channel centreline. This calculation is of practical importance for various chemical engineering devices in which particles must be filtered or separated. It is useful to calculate the forces on a particle moving near to a boundary, through a shear flow. This study may also explain certain migration effects of bubbles and crystals suspended in molten rock flow flowing through volcanic conduits.

scholarly journals Microstructure and rheology of finite inertia neutrally buoyant suspensions

2014 ◽  
Vol 749 ◽  
pp. 431-459 ◽  
Author(s):  
Hamed Haddadi ◽  
Jeffrey F. Morris
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Volume Fraction ◽  
Spherical Particles ◽  
Simple Shear Flow ◽  
Flow Reynolds Number ◽  
Boltzmann Method ◽  
Neutrally Buoyant ◽  
Boltzmann Model ◽  
Scale Shear

AbstractThe microstructure and rheological properties of suspensions of neutrally buoyant hard spherical particles in Newtonian fluid under finite inertia conditions are studied using the lattice-Boltzmann method (LBM), which is based on a discrete Boltzmann model for the fluid and Newtonian dynamics for the particles. The suspensions are subjected to simple-shear flow and the properties are studied as a function of Reynolds number and volume fraction, $\phi $. The inertia is characterized by the particle-scale shear flow Reynolds number $\mathit{Re}= {(\rho \dot{\gamma }a^{2})/\mu }$, where $a$ is the particle radius, $\dot{\gamma }$ is the shear rate and $\rho $ and $\mu $ are the density and viscosity of the fluid, respectively. The influences of inertia and of the volume fraction are investigated for $0.005\leqslant \mathit{Re}\leqslant 5$ and$0.1\leqslant \phi \leqslant 0.35$. The flow-induced microstructure is studied using the pair distribution function $g(\boldsymbol {r})$. Different stress mechanisms, including those due to surface tractions (stresslet), acceleration and the Reynolds stress due to velocity fluctuations are computed and their influence on the first and second normal stress differences, the particle pressure and the viscosity of the suspensions are detailed. The probability density functions (PDFs) of linear and angular accelerations are also presented.

The hydrodynamic lift of a slender, neutrally buoyant fibre in a wall-bounded shear flow at small Reynolds number

2019 ◽  
Vol 879 ◽  
pp. 121-146 ◽  
Author(s):  
Johnson Dhanasekaran ◽  
Donald L. Koch
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Solid Body ◽  
Rotational Motion ◽  
Fibre Orientation ◽  
Reciprocal Theorem ◽  
Microfluidic Channels ◽  
Slender Body Theory ◽  
Cross Flow Filtration ◽  
Neutrally Buoyant

The hydrodynamic lift velocity of a neutrally buoyant fibre in a simple shear flow near a wall is determined for small, but non-zero, fibre Reynolds number, illustrating the role of non-sphericity in lift. The rotational motion and effects of fibre orientation on lift are treated for fibre positions that induce and do not induce solid-body wall contacts. When the fibre does not contact the wall its lift velocity can be obtained in terms of the Stokes flow field by using a generalized reciprocal theorem. The Stokes velocity field is determined using slender-body theory with the no-slip velocity at the wall enforced using the method of images. To leading order the lift velocity at distances large compared with the fibre length and small compared with the Oseen length is found to be $0.0303\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{2}a/(\unicode[STIX]{x1D707}\ln [2l/a])$, where $l$ and $a$ are the fibre half-length and radius, $\unicode[STIX]{x1D70C}$ is the density, $\dot{\unicode[STIX]{x1D6FE}}$ is the shear rate and $\unicode[STIX]{x1D707}$ is the viscosity of the fluid. When the fibre is close enough to the wall to make solid-body contact during its rotational motion, a process known as pole vaulting coupled with inertially induced changes of fibre orientation determines the lift velocity. The order of magnitude of the lift in this case is larger by a factor of $l/a$ than when the fibre does not contact the wall and it reaches a maximum of $0.013\unicode[STIX]{x1D70C}\dot{\unicode[STIX]{x1D6FE}}^{2}l^{3}/(\unicode[STIX]{x1D707}\ln [l/a])$ for the case of a highly frictional contact and about half that value for a frictionless contact. These results are used to illustrate how particle shape can contribute to separation methods such as those in microfluidic channels or cross-flow filtration processes.

Equilibrium radial positions of neutrally buoyant spherical particles over the circular cross-section in Poiseuille flow

2017 ◽  
Vol 813 ◽  
pp. 750-767 ◽  
Author(s):  
Yusuke Morita ◽  
Tomoaki Itano ◽  
Masako Sugihara-Seki
Keyword(s):  
Experimental Study ◽  
Cross Section ◽  
Poiseuille Flow ◽  
Equilibrium Position ◽  
Probability Function ◽  
Circular Cross Section ◽  
Spherical Particles ◽  
Inertial Migration ◽  
Entry Length ◽  
Neutrally Buoyant

An experimental study of the inertial migration of neutrally buoyant spherical particles suspended in the Poiseuille flow through circular tubes has been conducted at Reynolds numbers $(Re)$ from 100 to 1100 for particle-to-tube diameter ratios of ${\sim}$0.1. The distributions of particles in the tube cross-section were measured at various distances from the tube inlet and the radial probability function of particles was calculated. At relatively high $Re$, the radial probability function was found to have two peaks, corresponding to the so-called Segre–Silberberg annulus and the inner annulus, the latter of which was first reported experimentally by Matas et al. (J. Fluid Mech. vol. 515, 2004, pp. 171–195) to represent accumulation of particles at smaller radial positions than the Segre–Silberberg annulus. They assumed that the inner annulus would be an equilibrium position of particles, where the resultant lateral force on the particles disappears, similar to the Segre–Silberberg annulus. The present experimental study showed that the fraction of particles observed on the Segre–Silberberg annulus increased and the fraction on the inner annulus decreased further downstream, accompanying an outward shift of the inner annulus towards the Segre–Silberberg annulus and a decrease in its width. These results suggested that if the tubes were long enough, the inner annulus would disappear such that all particles would be focused on the Segre–Silberberg annulus for $Re<1000$. At the cross-section nearest to the tube inlet, particles were absent in the peripheral region close to the tube wall including the expected Segre–Silberberg annulus position for $Re>700$. In addition, the entry length after which radial migration has fully developed was found to increase with increasing $Re$, in contrast to the conventional estimate. These results may be related to the developing flow in the tube entrance region where the radial force profile would be different from that of the fully developed Poiseuille flow and there may not be an equilibrium position corresponding to the Segre–Silberberg annulus.

Gravitational and zero-drag motion of a spheroid adjacent to an inclined plane at low Reynolds number

1994 ◽  
Vol 268 ◽  
pp. 267-292 ◽  
Author(s):  
Richard Hsu ◽  
Peter Ganatos
Keyword(s):  
Reynolds Number ◽  
Angular Velocity ◽  
Shear Flow ◽  
Equilibrium Position ◽  
Prolate Spheroid ◽  
Inclined Plane ◽  
Gravitational Settling ◽  
Angular Velocities ◽  
Planar Wall ◽  
Neutrally Buoyant

The first highly accurate solutions for the resistance tensor of an oblate or prolate spheroid moving near a planar wall obtained by Hsu & Ganatos (1989) are used to compute the translational and angular velocities and trajectories of a neutrally buoyant spheroid in shear flow and the gravitational settling motion of a non-neutrally buoyant spheroid adjacent to an inclined plane. The neutrally buoyant spheroid in shear flow undergoes a periodical motion toward and away from the wall as it continually tumbles forward. For some orientation angles it is found that the wall actually enhances the angular velocity of the particle. For certain inclinations a spheroid settling under gravity near an inclined plane reaches an equilibrium position, after which it translates parallel to the wall without rotation.

Inertial migration of neutrally buoyant spheres in a pressure-driven flow through square channels

2014 ◽  
Vol 749 ◽  
pp. 320-330 ◽  
Author(s):  
Kazuma Miura ◽  
Tomoaki Itano ◽  
Masako Sugihara-Seki
Keyword(s):  
Cross Sections ◽  
Side Wall ◽  
Reynolds Numbers ◽  
Spherical Particles ◽  
Lateral Migration ◽  
Inertial Migration ◽  
Square Channel ◽  
Circular Pipes ◽  
Pressure Driven Flow ◽  
Neutrally Buoyant

AbstractThe inertial migration of neutrally buoyant spherical particles in square channel flows was investigated experimentally in the range of Reynolds numbers ($\mathit{Re}$) from 100 to 1200. The observation of particle positions at several cross-sections downstream from the channel entrance revealed unique patterns of particle distribution which reflects the presence of eight equilibrium positions in the cross-section, located at the centres of the channel faces and at the corners, except for low $\mathit{Re}$. At $\mathit{Re}$ smaller than approximately 250, equilibrium positions at the corners are absent. The corner equilibrium positions were found to arise initially in the band formed along the channel face, followed by a progressive shift almost parallel to the side wall up to the diagonal line with increasing $\mathit{Re}$. Further increase in $\mathit{Re}$ moves the corner equilibrium positions slightly toward the channel corner, whereas the equilibrium positions at the channel face centres are shifted toward the channel centre. As the observation sites become downstream, the particles were found to be more focused near the equilibrium positions keeping their positions almost unchanged. These lateral migration behaviours and focusing properties of particles in square channels are different to that observed in microchannels at lower $\mathit{Re}$ and to what would be expected from extrapolating from the results for circular pipes at comparable $\mathit{Re}$.

The inertial lift on a spherical particle in a plane Poiseuille flow at large channel Reynolds number

1999 ◽  
Vol 381 ◽  
pp. 63-87 ◽  
Author(s):  
EVGENY S. ASMOLOV
Keyword(s):  
Reynolds Number ◽  
Shear Flow ◽  
Reynolds Numbers ◽  
Thin Layers ◽  
Rigid Sphere ◽  
Outer Flow ◽  
Inertial Migration ◽  
Flow Past ◽  
Large Channel ◽  
Linear Shear

The inertial migration of a small rigid sphere translating parallel to the walls within a channel flow at large channel Reynolds numbers is investigated. The method of matched asymptotic expansions is used to solve the equations governing the disturbance flow past a particle at small particle Reynolds number and to evaluate the lift. Both neutrally and non-neutrally buoyant particles are considered. The wall-induced inertia is significant in the thin layers near the walls where the lift is close to that calculated for linear shear flow, bounded by a single wall. In the major portion of the flow, excluding near-wall layers, the wall effect can be neglected, and the outer flow past a sphere can be treated as unbounded parabolic shear flow. The effect of the curvature of the unperturbed velocity profile is significant, and the lift differs from the values corresponding to a linear shear flow even at large Reynolds numbers.

Effect of fluid inertia on the dynamics and scaling of neutrally buoyant particles in shear flow

2013 ◽  
Vol 738 ◽  
pp. 563-590 ◽  
Author(s):  
T. Rosén ◽  
F. Lundell ◽  
C. K. Aidun
Keyword(s):  
Reynolds Number ◽  
Steady State ◽  
Aspect Ratio ◽  
Shear Flow ◽  
Stokes Number ◽  
Spheroidal Particle ◽  
Fluid Inertia ◽  
Prolate Spheroidal ◽  
Particle Inertia ◽  
Neutrally Buoyant

AbstractThe basic dynamics of a prolate spheroidal particle suspended in shear flow is studied using lattice Boltzmann simulations. The spheroid motion is determined by the particle Reynolds number (${\mathit{Re}}_{p} $) and Stokes number ($\mathit{St}$), estimating the effects of fluid and particle inertia, respectively, compared with viscous forces on the particle. The particle Reynolds number is defined by ${\mathit{Re}}_{p} = 4G{a}^{2} / \nu $, where $G$ is the shear rate, $a$ is the length of the spheroid major semi-axis and $\nu $ is the kinematic viscosity. The Stokes number is defined as $\mathit{St}= \alpha \boldsymbol{\cdot} {\mathit{Re}}_{p} $, where $\alpha $ is the solid-to-fluid density ratio. Here, a neutrally buoyant prolate spheroidal particle ($\mathit{St}= {\mathit{Re}}_{p} $) of aspect ratio (major axis/minor axis) ${r}_{p} = 4$ is considered. The long-term rotational motion for different initial orientations and ${\mathit{Re}}_{p} $ is explained by the dominant inertial effect on the particle. The transitions between rotational states are subsequently studied in detail in terms of nonlinear dynamics. Fluid inertia is seen to cause several bifurcations typical for a nonlinear system with odd symmetry around a double zero eigenvalue. Particle inertia gives rise to centrifugal forces which drives the particle to rotate with the symmetry axis in the flow-gradient plane (tumbling). At high ${\mathit{Re}}_{p} $, the motion is constrained to this planar motion regardless of initial orientation. At a certain critical Reynolds number, ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $, a motionless (steady) state is created through an infinite-period saddle-node bifurcation and consequently the tumbling period near the transition is scaled as $\vert {\mathit{Re}}_{p} - {\mathit{Re}}_{c} {\vert }^{- 1/ 2} $. Analyses in this paper show that if a transition from tumbling to steady state occurs at ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $, then any parameter $\beta $ (e.g. confinement or particle spacing) that influences the value of ${\mathit{Re}}_{c} $, such that ${\mathit{Re}}_{p} = {\mathit{Re}}_{c} $ as $\beta = {\beta }_{c} $, will lead to a period that scales as $\vert \beta - {\beta }_{c} {\vert }^{- 1/ 2} $ and is independent of particle shape or any geometric aspect ratio in the flow.

Inertial migration of a sphere in Poiseuille flow

1989 ◽  
Vol 203 ◽  
pp. 517-524 ◽  
Author(s):  
Jeffrey A. Schonberg ◽  
E. J. Hinch
Keyword(s):  
Reynolds Number ◽  
Poiseuille Flow ◽  
Equilibrium Position ◽  
Migration Velocity ◽  
Small Sphere ◽  
Order Unity ◽  
Inertial Migration

The inertial migration of a small sphere in a Poiseuille flow is calculated for the case when the channel Reynolds number is of order unity. The equilibrium position is found to move towards the wall as the Reynolds number increases. The migration velocity is found to increase more slowly than quadratically. These results are compared with the experiments of Segré & Silberberg (1962 a, b).

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