Motion of a rigid cylinder between parallel plates in stokes flow : Part 2: Poiseuille and couette flow

A numerical solution is presented for the motion of a neutrally buoyant circular cylinder in Poiseuille and Couette flows between two plane parallel boundaries. The force and torque on a stationary particle are calculated for a wide range of particle sizes and poisitions across the channel. The resistance matrix calculated in Ref. [1] (henceforth referred to as Part 1) is utilized to find the translational and angular velocity for a drag- and torque-free particle. The results are compared with analytical perturbation solutions for a small cylindrical particle situated on the channel centerline, and for the motion of a spherical particle in a circular tube or between plane parallel boundaries. It is found the behavior of flow around a cylindrical particle in a channel is qualitatively similar to the behavior of flow around a spherical particle in a tube, while the flow around a spherical particle in a channel frequently exhibits different trends from the above two cases.     

Eccentric electrophoretic motion of a sphere in circular cylindrical microchannels

The eccentric electrophoretic motion of a spherical particle in an aqueous electrolyte solution in circular cylindrical microchannels is studied in this paper. The objective is to investigate the influences of separation distance and channel size on particle motion. A theoretical model is developed to describe the electric field, the flow field and the particle motion. A finite element based direct numerical simulation method is employed to solve the model. Numerical results show that, when the particle is eccentrically positioned in the channel, the electric field and the flow field are not symmetric, and the strongest electric field and the highest flow velocity occur in the small gap region. It is shown that the rotational velocity of the particle increases with the decrease of the separation distance. With the decrease of the separation distance, the translational velocity increases in a smaller channel; while it decreases first and then increases in a relatively large channel. When a particle moves eccentrically at a smaller separation distance from the channel wall, both the translational velocity and the rotational velocity increase with the decrease of the channel size.     

Numerical simulations of particle migration in a viscoelastic fluid subjected to Poiseuille flow

In this work we present 2D numerical simulations on the migration of a particle suspended in a viscoelastic fluid under Poiseuille flow. A Giesekus model is chosen as constitutive equation of the suspending liquid. In order to study the sole effect of the fluid viscoelasticity, both fluid and particle inertia are neglected.The governing equations are solved through the finite element method with proper stabilization techniques to get convergent solutions at relatively large flow rates. An Arbitrary Lagrangian–Eulerian (ALE) formulation is adopted to manage the particle motion. The mesh grid is moved along the flow so as to limit particle motion only in the gradient direction to substantially reduce mesh distortion and remeshing.Viscoelasticity of the suspending fluid induces particle cross-streamline migration. Both large Deborah number and shear thinning speed up the migration velocity. When the particle is small compared to the gap (small confinement), the particle migrates towards the channel centerline or the wall depending on its initial position. Above a critical confinement (large particles), the channel centerline is no longer attracting, and the particle is predicted to migrate towards the closest wall when its initial position is not on the channel centerline. As the particle approaches the wall, the translational velocity in the flow direction is found to become equal to the linear velocity corresponding to the rolling motion over the wall without slip.     

Electrophoretic motion of a colloidal cylinder near a plane wall

An analytical study is presented for the electrophoretic motion of a circular cylindrical particle in an electrolyte solution with a transversely imposed electric field near a large plane wall parallel to its axis in the quasisteady limit. The electric double layers at the solid surfaces are assumed to be thin relative to the particle radius and to the particle–wall gap width, but the polarization effect of the diffuse ions in the double layer surrounding the particle is incorporated. The presence of the confining wall causes two basic effects on the particle velocity: first, the local ionic electrochemical potential gradients on the particle surface are altered by the wall, thereby affecting the motion of the particle; secondly, the wall enhances the viscous retardation of the moving particle. Through the use of cylindrical bipolar coordinates, the transport equations governing this problem are solved and the wall effects on the electrophoresis of the cylinder are determined for various cases. The presence of the plane wall prescribed with the ionic electrochemical potentials consistent with the far-field distributions reduces the electrophoretic mobility of the particle, which depends upon the properties of the particle–solution system, the relative particle–wall separation distance, and the direction of the applied electric field relative to the plane wall. The direction of the electrophoretic migration of a cylindrical particle near a plane wall is different from that of the prescribed electric field, except when it is oriented parallel or perpendicular to the wall. The effects of the plane wall on the electrophoresis of a cylinder are found to be much more significant than those for a sphere at the same separation.     

Electrophoresis of a colloidal sphere with double-layer polarization in a microtube

A theoretical study is presented for the electrophoretic motion of a spherical particle in an electrolyte solution along the axis of a circular microtube, whose wall may be either insulating or prescribed with the linear far-field electric potential distribution. The electric double layers adjoining the charged particle surface and tube wall are finitely thin, and the polarization of the diffuse layer at the particle surface is allowed. The general solutions to the electrostatic and hydrodynamic governing equations are constructed in combined cylindrical and spherical coordinates, and the boundary conditions are enforced on the tube wall by the Fourier transform and along the particle surface by a collocation method. The collocation results for the electrophoretic mobility of the confined particle, which agree well with the asymptotic formulas obtained by using a method of reflections, are obtained for various values of the particle, wall, and solution characteristics. An insulating tube wall and a tube wall with the far-field potential distribution affect the electrophoresis of the particle quite differently. Although the particle mobility in a tube with uncharged wall in general decreases with an increase in the particle-to-tube radius ratio a/b, it can increase with an increase in a/b as this ratio is close to unity for some cases because of the competition between the wall effects of hydrodynamic retardation and possible electrochemical enhancement on the particle migration. When the zeta potential of the tube wall is comparable to that of the particle, the electroosmotic flow of the bulk fluid induced by the tube wall dominates the electrokinetic migration of the particle.     

Slow viscous flow around two particles in a cylinder

This article describes the motion of two arbitrarily located free moving particles in a cylindrical tube with background Poiseuille flow at low Reynolds number. We employ the Lamb’s general solution based on spherical harmonics and construct a framework based on cylindrical harmonics to solve the flow field around the particles and the flow within the tube, respectively. The two solutions are performed in an iterated framework using the method of reflections. We compute the drag force and torque coefficients of the particles which are dependent on the distances among the cylinder wall and the two particles. In addition, we provide detailed flow field in the vicinity of the two particles including streamlines and velocity contour. Our analysis reveals that the particle–particle interaction can be neglected when the separation distance is three times larger than the sum of particles radii when the two particles are identical. Furthermore, the direction of Poiseuille flow, the particle position relative to the axis and the particle size can make the two particles attract or repel. Unlike the single particle case, the two particles can move laterally due to the hydrodynamic interaction. Such analysis can give insights to understand the mechanisms of collision and aggregation of particles in microchannels.     

Modeling complex boundaries using an external force field on fixed Cartesian grids in large-eddy simulations

In the present study a methodology to perform large-eddy simulations around complex boundaries on fixed Cartesian grids is presented. A novel interpolation scheme which is applicable to boundaries of arbitrary shape, does not involve special treatments, and allows the accurate imposition of the desired boundary conditions is introduced. A method to overcome the problems associated with the computation of the subgrid scale terms near solid boundaries is also discussed. A detailed study on the accuracy and efficiency of the method is carried out for the cases of Stokes flow around a cylinder in the vicinity of a moving plate, the three-dimensional flow around a circular cylinder, and fully developed turbulent flow in a plane channel with a wavy wall. It is demonstrated that the method is second-order accurate, and that the solid boundaries are mimicked “exactly” on the Cartesian grid within the overall accuracy of the scheme. For all cases under consideration the results obtained are in very good agreement with analytical and numerical data.     

A Parallel Overset Grid High-Order Flow Solver for Large Eddy Simulation

This work describes the development and validation of a parallel high-order compact finite difference Navier–Stokes solver for application to large-eddy simulation (LES) and direct numerical simulation. The implicit solver can employ up to sixth-order spatial formulations and tenth-order filtering. The parallelization of the solver is founded on the overset grid technique. LES were then performed for turbulent channel flow with Reynolds numbers ranging from Re _τ=180 to 590, and flow past a circular cylinder with a transitional wake at Re _D =3900. The channel flow solutions were obtained using both an implicit LES (ILES) approach and a dynamic sub-grid scale model. The ILES method obtained virtually identical solutions at half the computational cost. The original vector and new parallel solvers produce indistinguishable mean flow solutions for the circular cylinder. Repeating the cylinder simulation on a much finer mesh resulted in significantly better agreement with experimental data in the near wake than the coarse grid solution and other previous numerical studies.     

A parallel 3D Poisson solver for space charge simulation in cylindrical coordinates

This paper presents the development of a parallel three-dimensional Poisson solver in cylindrical coordinate system for the electrostatic potential of a charged particle beam in a circular tube. The Poisson solver uses Fourier expansions in the longitudinal and azimuthal directions, and Spectral Element discretization in the radial direction. A Dirichlet boundary condition is used on the cylinder wall, a natural boundary condition is used on the cylinder axis and a Dirichlet or periodic boundary condition is used in the longitudinal direction. A parallel 2D domain decomposition was implemented in the (r,θ) plane. This solver was incorporated into the parallel code PTRACK for beam dynamics simulations. Detailed benchmark results for the parallel solver and a beam dynamics simulation in a high-intensity proton LINAC are presented. When the transverse beam size is small relative to the aperture of the accelerator line, using the Poisson solver in a Cartesian coordinate system and a Cylindrical coordinate system produced similar results. When the transverse beam size is large or beam center located off-axis, the result from Poisson solver in Cartesian coordinate system is not accurate because different boundary condition used. While using the new solver, we can apply circular boundary condition easily and accurately for beam dynamic simulations in accelerator devices.     

Inertial effects on cylindrical particle migration in linear shear flow near a wall

Micro-/nanoparticle-based systems are regarded as one of the possible candidates due to the engineerability and multifunctionality to maximize the accumulation of the nano-/microparticle-based drug delivery system on the target. Recent advances in nanotechnology enable the fabrication of diverse particle shapes from simple spherical particles to more complex shapes. The particle dynamics in blood flow and endocytosis characteristics of non-spherical particles change as the non-sphericity effect increases. We used a numerical approach to investigate the particle dynamics in linear shear flow near a wall. We examined the dynamics of slender cylindrical particles with aspect ratio γ = 5.0 in terms of particle trajectory, velocity, and force variation for different Stokes numbers over time. We identified the rotating inertia of particle near a wall as the source of inertial migration toward the wall. The drift velocity of slender cylindrical particles is comparable to that of discoidal particles. We discuss the possibilities and limitations of using slender cylindrical particles as a drug delivery system.     

Numerical Simulation of the Sedimentation of Rigid Bodies in an Incompressible Viscous Fluid by Lagrange Multiplier/Fictitious Domain Methods Combined with the Taylor–Hood Finite Element Approximation

In this work we discuss an application of a distributed Lagrange multiplier based fictitious domain method, to the numerical simulation of the motion of rigid bodies settling in an incompressible viscous fluid. The solution method combines a third order finite element approximation, and time integration by operator splitting. Convergence results are shown for a simple Stokes flow with a circular rigid body that rotates with constant angular velocity. Results of numerical experiments for two sedimenting cylinders in a two-dimensional channel are presented. We present also results for the sedimentation of 100 and 504 cylinders.     

Particle transport in flow through a ratchet-like channel

We present a fluidic device that shows ratchet-like characteristics for particle transport at low Reynolds. The ratchet consists of a two-dimensional saw-tooth channel, within which a laminar flow is generated by imposing a longitudinal pressure gradient. Particle trajectories are calculated by solving the continuity and Navier–Stokes equations for the fluid flow and the equations for particle transport in both flow directions. The ratchet-like effect is connected with a large asymmetry in the mean transit time, with regard to flow direction, due to particle motion within zones of low flow velocity near the asymmetric wall profile. We show how to obtain ratchet of particles with select Stokes under given flow conditions by adjusting the geometry of the ratchet channel.     

Probabilistic Detection and Tracking of Motion Boundaries

We propose a Bayesian framework for representing and recognizing local image motion in terms of two basic models: translational motion and motion boundaries. Motion boundaries are represented using a non-linear generative model that explicitly encodes the orientation of the boundary, the velocities on either side, the motion of the occluding edge over time, and the appearance/disappearance of pixels at the boundary. We represent the posterior probability distribution over the model parameters given the image data using discrete samples. This distribution is propagated over time using a particle filtering algorithm. To efficiently represent such a high-dimensional space we initialize samples using the responses of a low-level motion discontinuity detector. The formulation and computational model provide a general probabilistic framework for motion estimation with multiple, non-linear, models.     

Fluid rheological effects on particle migration in a straight rectangular microchannel

There has recently been a significantly increasing interest in the passive manipulation of particles in the flow of non-Newtonian fluids through microchannels. However, an accurate and comprehensive understanding of the various fluid rheological effects on particle migration is still largely missing. We present in this work a systematic experimental study of both the individual and the combined effects of fluid inertia, elasticity, and shear thinning on the motion of rigid spherical particles in a straight rectangular microchannel. We first study the sole effect of each of these rheological properties in a Newtonian fluid, purely elastic (i.e., Boger) fluid, and purely shear-thinning (i.e., pseudoplastic) fluid, respectively. We then study the combined effects of two or all of these rheological properties in a pseudoplastic fluid and two types of elastic shear-thinning fluids, respectively. We find that the fluid elasticity effect directs particles toward the centerline of the channel while the fluid shear-thinning effect causes particle migration toward both the centerline and corners. These two effects are combined with the fluid inertial effect to understand the particle migration in inertial pseudoplastic and viscoelastic fluid flows.     

Particle behavior in a vertical channel with thermal convection in the low Grashof number regime

In the present study the motion of isothermal circular particles in a two-dimensional vertical channel with hot and cold isothermal conditions at the left and right walls in the presence of thermal convection was investigated. An isothermal circular particle for a particle to fluid density ratio ρ_r = ρ_p/ρ_f = of 1.00232 where ρ_p and ρ_f denote the particle and the fluid densities, respectively, was considered. Numerical simulations were carried out using the direct forcing/fictitious domain (DF/FD) method to investigate the solid motion in a fluid with a Prandtl number of 0.7 for different Grashof numbers ranging from 0 to 50. Under the conditions of the present problem, the particle motion is mainly governed by the thermal convection between the side walls of the channel and the particle, and by the wall confinement. The results of the present study indicate that three regimes of particle behavior can be identified in the present range of Grashof numbers regardless of the cold and hot thermal boundary conditions of the particle. In the first regime, the particle exhibits steady settling behavior; in the second regime, it undergoes a transient overshoot before the steady settling; in the third regime, the particle motion is submerged in the thermal levitation.     

Efficient computation of micro-particle dynamics including wall effects

This study describes an effective method for one-way coupled Eulerian-Lagrangian simulations of spherical micro-size particles, including particle-wall interactions and the quantification of near-wall stasis at possibly elevated concentrations. The focus is on particle-hemodynamics simulations where particle suspensions are composed of critical blood cells, such as monocytes, and the carrier fluid is non-Newtonian. Issues regarding adaptive time-step integration of the particle motion equation, relevant point-force model terms, and adaptation of surface-induced particle forces to arbitrary three-dimensional geometries are outlined. By comparison to available experimental trajectories, it is shown that fluid-element pathlines may be used to simulate non-interacting blood particles removed from wall boundaries under dilute transient conditions. However, when particle-wall interactions are significant, an extended form of the particle trajectory equation is required which includes terms for Stokes drag, near-wall drag modifications, or lubrication forces, pressure gradients, and near-wall particle lift. Still, additional physical and/or biochemical wall forces in the nano-meter range cannot be readily calculated; hence the near-wall residence time (NWRT) model indicating the probability of blood particle deposition is presented. The theory is applied to a virtual model of a femoral bypass end-to-side anastomosis, where profiles of the Lagrangian-based NWRT parameter are illustrated and convergence is verified. In order to effectively compute the large number of particle trajectories required to resolve regions of particle stasis, the proposed particle tracking algorithm stores all transient velocity field solution data on a shared memory architecture (SGI Origin 2400) and computes particle trajectories using an adaptive parallel approach. Compared to commercially available particle tracking packages, the algorithm presented is capable of reducing computational time by an order of magnitude for typical transient one-way coupled blood particle simulations in complex cyclical flow domains.     

High Order Computation of the History Term in the Equation of Motion for a Spherical Particle in a Fluid

The historical evolution of the equation of motion for a spherical particle in a fluid and the search for its general solution are recalled. The presence of an integral term that is nonzero under unsteady motion and viscous conditions allowed simple analytical or numerical solutions for the particle dynamics to be found only in a few particular cases. A general solution to the equation of motion seems to require the use of computational methods. Numerical schemes to handle the integral term of the equation of motion have already been developed. We present here adaptations of a first order method for the implementation at high order, which may employ either fixed or variable computation time steps. Some examples are shown to establish comparisons between diverse numerical methods.     

Simulation of vorticity dominated flows using a hybrid approach. I: Formulation

The formulation of a numerical solution method for the Navier–Stokes equations in domains with cylindrical boundaries based on a hybrid spectral – finite difference approach is presented. The main contributions of the present paper are complete proofs for the pole conditions for scalar and vector fields and the modified pdes for the streamfunction modes that are shown to possess solutions satisfying the pole conditions.     

A study of the Brownian motion of the non-spherical microparticles on fluctuating lattice Boltzmann method

In this paper, the fluctuating lattice Boltzmann (FLB) model is used to simulate the Brownian motion of spherical and non-spherical particles directly on 3D. The FLB model is validated by comparing the simulation results with theoretical and experimental results. And then the Brownian motion characteristics of different shapes of non-spherical particles are analyzed, which includes some ellipsoidal and cylindrical particles with different aspect ratios. It is found that, although the ellipsoidal and cylindrical particles are anisotropic, they still obey the energy equalization theorem. The velocity and angular velocity autocorrelation functions of the particles still have the “long-time tail” effect. Moreover, the velocity and angular velocity autocorrelation functions are independent of the particle shape, and the diffusion coefficient is insensitive to the shape under a low aspect ratio. This work is a fundamental study for the further research of the motion of microparticles based on Brownian motion.     

Ambiguity in Structure from Motion: Sphere versus Plane

If 3D rigid motion can be correctly estimated from image sequences, the structure of the scene can be correctly derived using the equations for image formation. However, an error in the estimation of 3D motion will result in the computation of a distorted version of the scene structure. Of computational interest are these regions in space where the distortions are such that the depths become negative, because in order for the scene to be visible it has to lie in front of the image, and thus the corresponding depth estimates have to be positive. The stability analysis for the structure from motion problem presented in this paper investigates the optimal relationship between the errors in the estimated translational and rotational parameters of a rigid motion that results in the estimation of a minimum number of negative depth values. The input used is the value of the flow along some direction, which is more general than optic flow or correspondence. For a planar retina it is shown that the optimal configuration is achieved when the projections of the translational and rotational errors on the image plane are perpendicular. Furthermore, the projection of the actual and the estimated translation lie on a line through the center. For a spherical retina, given a rotational error, the optimal translation is the correct one; given a translational error, the optimal rotational negative deptherror depends both in direction and value on the actual and estimated translation as well as the scene in view. The proofs, besides illuminating the confounding of translation and rotation in structure from motion, have an important application to ecological optics. The same analysis provides a computational explanation of why it is easier to estimate self-motion in the case of a spherical retina and why shape can be estimated easily in the case of a planar retina, thus suggesting that nature's design of compound eyes (or panoramic vision) for flying systems and camera-type eyes for primates (and other systems that perform manipulation) is optimal.