The effects of fluid viscoelasticity on the settling behaviour of horizontally aligned spheres

A study on the interaction of particles settling in non-Newtonian fluids of shear-thinning, thixotropic and viscoelastic characteristics has been conducted. Key aspects of the rheological characteristics of the fluids that influence the interaction of the particles were examined by analysing the trajectories of two particles that are initially placed side-by-side in the fluid medium.The interaction of the particles was found to be highly dependent on the separation distance that is initially set between them. If the initial distance is smaller than a critical value, the spheres would tend to attract and converge. In cases where the initial distance is greater than this critical value, the two spheres would tend to diverge, resulting in a slight (∼20%) increase in their separation distance over their course of settling. This tendency to diverge was found to diminish as the initial distance is increased further from the critical value.The magnitude of the critical separation distance was found to be primarily dependent on the normal stresses of the fluids. A correlation was thus proposed based on this observation. In cases where the two spheres do attract and converge, it was found that the spheres tend to follow a non-symmetrical trajectory, where one of the spheres possesses a slightly lower settling velocity than the other. As a result, the spheres appear to re-arrange themselves into a vertically aligned configuration. Once aligned, the shear-thinning and thixotropic characteristics of the fluid causes the lagging sphere to accelerate and collide with the leading sphere.     

CONJUGATE HEAT AND MASS TRANSFER IN CONVECTIVE DRYING OF MULTIPARTICLE SYSTEMS. PART I: THEORETICAL CONSIDERATIONS

Abstract

This is the first of two papers concerning conjugate transport in convective drying of multiparticle systems. In this study, a methodology for the solution of conjugate transport problems is proposed. The theoretical aspects and relevant assumptions are briefly discussed and the results for convective heat and mass transfer in assemblages of spheres are presented. It is shown that both heat and mass transfer rates for an assemblage of two spheres in tandem arragement are significantly different from those of a single sphere case, and that care should be taken when modeling multiparticle systems by means of single-particle analysis.     

Drag on two coaxial rigid spheres moving along the axis of a cylinder filled with Carreau fluid

The boundary effect on the drag on two identical, rigid spheres moving along the axis of a long cylinder filled with a Carreau fluid for Reynolds number ranges from 0.1 to 40 is investigated. The influences of the key parameters of the problem under consideration, including the separation distance between two spheres, the relaxation time constant and the power-law index of a Carreau fluid, the Reynolds number, and the ratios (radius of sphere/radius of cylinder), on the drag acting on two spheres are investigated. We show that the boundary effect for the present case is more significant than that for the corresponding Newtonian fluid. The presence of the cylinder has the effect of enhancing the convective motion in the rear part of a sphere, thereby forming wakes and a reverse flow field, and this phenomenon is enhanced by the shear-thinning nature of a fluid. If the boundary effect is insignificant, the shear-thinning nature of a fluid has the effect of reducing the deviation of the ln(drag coefficient)-ln(Reynolds number) curve from a Stokes'-law-like relation. On the other hand, if it is significant, this deviation has a local minimum as the shear-thinning nature of a fluid varies.     

Cylindrical object contact detection for use in discrete element method simulations,Part II—Experimental validation

This article is the second of two focusing on cylindrical object contact detection. The first part presents algorithms for determining if contact occurs between a cylindrical object and a plane, and between two cylindrical objects, in addition to expressions for contact overlap, contact normal vector, and contact location. This article presents the tests and results used to quantitatively validate the algorithms implemented in three-dimensional discrete element method (DEM) simulations. In particular, four tests are performed. The first consists of a comparison against analytical expressions for post-collision translational and rotational velocities for a single cylinder impacting a flat wall. The second involves comparisons against experimental residence time measurements of a cylindrical particle moving within a baffled, rotating horizontal cylinder. The third test consists of comparisons against experimental bulk density measurements of cylindrical particles dropped into a cylindrical container. The last test compares experimental and computational values of the dynamic angle of repose in a rotating drum. In addition to true cylindrical objects, comparisons are also made in the first three tests against two cylindrical object approximations using glued spheres. The true cylinder model performs well in each of the tests while the glued sphere approximations perform poorly, especially when fewer spheres are used to represent the cylinder shape. These studies demonstrate that the cylinder contact algorithms presented in the first article perform correctly, and more accurately model real cylinder behavior than glued sphere approximations.     

Drag on Freely Falling Cones in Newtonian and in Power Law Fluids

New extensive data on the terminal falling velocities of conical shaped bodies in scores of Newtonian and power law fluids are reported. Altogether, 11 Newtonian and 11 non‐Newtonian test liquids together with 33 cones made from four different materials and 14 spheres of three different materials have been used to gather 486 individual data points covering wide ranges of conditions as follows: Reynolds number 0.0019 to 507; power law flow behaviour index 0.4 to 1, the value of sphericity 0.59 to 0.79; and the cone‐to‐fall tube diameter ratios up to 0.264 to assess the extent of wall effects. A simple expression is developed to estimate the terminal falling velocity of a cone from a knowledge of its dimensions, and the terminal velocity of an equivalent sphere. A generalized drag equation applicable to both Newtonian and power law liquids is also presented.     

A numerical study of the accelerating motion of a dense rigid sphere in non-newtonian power law fluids

The equations of motion of an accelerating sphere falling through non-Newtonian fluids with power law index n in the range 0.2 ≤ n ≤ 1.8 were integrated numerically using the assumption that the drag on the sphere was a function of both power law index and terminal Reynolds number, Re_t For 10^?2 ≤ Re_t ≤ 10³ both dimensionless time and distance travelled by the sphere under transient conditions showed a much stronger dependence on the flow behaviour index, n, for shear-thinning than for shear-thickening fluids. The form of this dependence is investigated here. Furthermore, results in four typical shear-thinning fluids suggested a strong correlation between the distance and time travelled by the sphere under transient conditions and the value of the fluid consistency index. The analysis reported herein is, however, restricted to dense spheres falling in less dense fluids, when additional effects arising from the Basset forces can be neelected.     

COMPARISON OF THE VELOCITIES AND THE WALL EFFECT BETWEEN SPHERES AND CUBES IN THE ACCELERATING REGION

The velocities and the wall effect for spheres and cubes in the accelerating region were compared experimentally for very high Reynolds number (18000 < N _Re  < 77000 for spheres and 12000 < N _Re  < 49000 for cubes). Experiments were conducted with 0.012, 0.015, 0.018, 0.021, 0.025, and 0.03175 m spheres and their respective cubes of equal volume falling in water inside cylinders of diameters equal to 0.034, 0.049, 0.07, 0.10, 0.14, and 0.19 m. The velocities of the spheres were always greater than the velocities for the respective cubes of equal volume. The wall effect increases with the particle size and fall distance. The relative wall effect between the spheres and the cubes varies with the fall distance and the ratio of the particle to tube diameter. The wall factor as a function of particle-to-tube ratio for the spheres is described well by the Munroe equation (5.7% absolute error), while for the cubes the points are more dispersed.     

An experimental study of motion of cylinders in newtonian fluids: Wall effects and drag coefficient

The terminal velocity of several cylinders (of glass, perspex and stainless steel) falling with their axis parallel to the direction of motion has been measured in a series of Newtonian fluids embracing a 40-fold variation in liquid viscosity. The measurements have been carried out in fall tubes of four different diameters to elucidate the importance of wall effects. The experimental results encompass the following ranges of conditions: cylinder to fall tube diameter ratio: 0.08 to 0.4; length to diameter ratio: 0.05 to 2 and Reynolds number varied from 0.2 to 180. The wall effects are discussed in a manner analogous to those for spherical particles. Terminal velocity data are analysed using two approaches, namely, drag coefficient-Reynolds number relationship and a dimensionless velocity ratio denoting the departure from the behaviour of an equivalent sphere. Predictive equations have been developed using both schemes.     

CONJUGATE HEAT AND MASS TRANSFER IN CONVECTIVE DRYING OF MULTIPARTICLE SYSTEMS. PART I: THEORETICAL CONSIDERATIONS

This is the first of two papers concerning conjugate transport in convective drying of multiparticle systems. In this study, a methodology for the solution of conjugate transport problems is proposed. The theoretical aspects and relevant assumptions are briefly discussed and the results for convective heat and mass transfer in assemblages of spheres are presented. It is shown that both heat and mass transfer rates for an assemblage of two spheres in tandem arragement are significantly different from those of a single sphere case, and that care should be taken when modeling multiparticle systems by means of single-particle analysis.     

Radial porosity distribution in cylindrical beds packed with spheres

Experimental determinations of radial porosity for cylindrical beds packed with spheres are reported. The data indicate that, for a wide range of bed and sphere sizes, porosity varies significantly and regularly near the container wall. For uniformly sized spheres, the oscillations in porosity can be detected up to a distance of about 5 particle diameters from the wall. For mixtures of spheres of two sizes, regular oscillations are detected only up to 2 or 3 diameters from the wall and for three sizes the effect of the wall is observed only within a distance of 1 particle diameter.     

Drag on two co-axial rigid spheres moving normal to a plane: Newtonian and Carreau fluids

The boundary effect and the presence of a nearby entity on the drag of a rigid entity is investigated by considering the movement of two identical, rigid, coaxial spheres normal to a plane in both a Newtonian and a Carreau fluid at a low to medium large Reynolds number. The parameters key to the phenomenon under consideration, including the nature of the fluid, the separation distance between two spheres, the distance between the near sphere and the plane, and the Reynolds number, on the drag coefficient are discussed. We show that the influence of a boundary on the drag coefficient is more important than that of the nature of a fluid and that of the separation distance between two spheres. The variation of the drag coefficient as a function of Reynolds number for a Carreau fluid is similar to that for a Newtonian fluid. Due to the shear-thinning nature of the former the drag coefficient in the former is smaller than that in the latter. The influence of the index parameter of a Carreau fluid becomes appreciable only if the Carreau number is sufficiently large. Correlations between the drag coefficient and the key parameters of a system are developed for the case when the Reynolds number is smaller than l.     

MEASUREMENTS OF INERTIAL DEPOSITION OF AEROSOL PARTICLES IN REGULAR ARRAYS OF SPHERES

Inertial deposition of aerosol particles on single spheres and regular assemblies of spheres was investigated in the range of Stokes numbers from 0.03 to 5. Measurements were carried out on steel spheres suspended from wires in a 100 mm wide flow channel at air velocities from 5 to 28 m s^-1, using monodisperse DES (Di-II-ethyl-hexyl-sebacate) test aerosol in the size range from 1.5 to 15 μm. Experimentally determined deposition efficiencies on single spheres are in very good agreement with theoretically predicted results based on a potential flow field as well as a numerical flow simulation. Results in various types of arrays consisting of up to 12 equal-sized spheres show substantial variation of deposition efficiency from sphere to sphere which are strongly influenced by Stokes number and geometry.     

用落球法测量悬浮体表观粘度

采用落球法测量了密相气固流化床表观粘度,发现流化床的流变特性可用Bingham粘塑性模型描述,床层的有效粘度可由下式计算 μ     

Adherence and Rolling Kinetics of a Rigid Cylinder in Contact with a Natural Rubber Surface

Equilibrium conditions of a rigid cylinder in contact with the flat and smooth surface of a natural rubber sample are studied using concepts of fracture mechanics, such as the strain energy release rate or the stress intensity factor. It is shown that an equilibrium contact area exists if the applied force per unit axial length is greater than a negative critical value, closely related to the cylinder radius and mechanical and superficial properties of the elastic solid. Due to the intervention of molecular attraction forces, of van der Waals type, a light cylinder rolls under an inclined rubber surface and it is displayed that the rolling speed is the same when the cylinder rolls upon the same inclined surface. It has been verified that if a flat rubber substrate, with an adequate length, is rotated at constant angular velocity, a steel cylinder rolls alternately upon and under the surface, unceasingly without falling down.     

AN ANALYTICAL STUDY OF THE TRANSIENT MOTION OF A DENSE RIGID SPHERE IN AN INCOMPRESSIBLE NEWTONIAN FLUID

A theoretical study of the transient sphere motion (under the influence of gravity) through an incompressible Newtonian fluid subject to an Oseen-type drag relationship has been carried out. Exact closed form expressions for the instantaneous position, velocity and acceleration of the sphere are presented. An analytical expression developed herein also enables the delineation of the “best” sphere-fluid combination for the experimental observations of transient effects and these provide useful guidelines for designing laboratory experiments. However, this study is restricted to dense spheres falling in light liquids when the additional effects arising from the added mass and the Basset forces are negligible. Also, the boundary effects are altogether neglected.     

ON THE MODELLING OF PARTICLE-BODY INTERACTIONS IN STOKES FLOWS INVOLVING A SPHERE AND CIRCULAR DISC OR A TORUS AND CIRCULAR CYLINDER USING POINT SINGULARITIES

Few exact solutions of the Stokes equations are known, even for steady or quasi-steady flows, involving finite sized bodies, and numerical techniques generally have to be resorted to for finding solutions. However, quite effective modelling of flows involving complicated boundary geometries is possible using the three-dimensional Stokeslet and rotlet point singularities. Two problems are studied in detail. In the first example, exact solutions for the three-dimensional Stokeslet and rotlet placed axisymmetrically along the axis of a circular disc are found and combined with Brenner's first order interaction formulae to determine the effect of the presence of the disc on the force and torque acting on a particle whose dimensions are small compared with its distance from the disc. The results are compared with those of a full numerical integration of the Stokes equations for a sphere translating towards a disc. In the second example, Brenner's first order wall correction theory is applied to the motion of a particle in a circular cylinder using the exact solutions for a torus translating or rotating in isolation. The theoretical predictions for the drag on a torus settling symmetrically in a circular cylinder are compared with those determined experimentally.     

PRESSURE DROP IN FLOW OF A NEARLY CRITICAL FLUID THROUGH PACKED BEDS OF SPHERES

The flow of nearly critical carbon dioxide through packed beds of glass beads and steel spheres has been investigated. Experiments have been carried out in the intermediate range of sphere Reynolds number. Results show that around the critical point effects of fluid compressibility are negligible, and, therefore, that the friction factor can be expressed as a function of Reynolds number only. Carman [1] and Dolejs and Lecjaks [2] equations show good agreement with the experimental data.     

Hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline (sphere #1 and sphere #2) through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping flow limit, the drag forces exerted on both spheres were identical. At higher Reynolds numbers the drag force on sphere #1 was higher than sphere #2, revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter (β, =d_f/2k^0.5, d_f and k are floc diameter and interior permeability, respectively) >20, the spheres can be regarded nonporous. At β<20, the drag forces dropped. At β<2, the drag forces approached “no-spheres” limit. An increased size ratio of two spheres (d_f1/d_f2) would increase the drag force on sphere #1 and reduce that on sphere #2. At increasing β for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on β value. These effects were minimal when β became low. Two identical spheres could move steadily along their centerline. At higher Reynolds number, the two spheres would move closer because of the incorporation of inertia force. For spheres of different diameters, the sphere # 2 would move faster than sphere #1 regardless of their size ratio and β value. This occurrence yielded efficient coagulation when two porous spheres were moving in-line.     

The development of viscous wakes in a stokes flow when a particle is near a large obstacle

If the axisymmetric Stokes streaming flow past a small spherical particle close to a large spherical obstacle is modelled by a sphere and plane in an axisymmetric stagnation point flow, it is shown that the flow separates from the boundaries if the centre of the sphere is less than about 9.09 times its radius from the plane. Separation first occurs from the plane, followed by separation from the sphere when its centre is 6.57 times its radius from the plane. At the onset of separation, wakes form on the boundaries, these wakes coalescing when the sphere is about 5.37 times its radius from the plane. For smaller clearances, a cylinder of fluid links the sphere and the plane, the fluid in this cylinder rotating in one or more ring vortices, the number of these vortices increasing as the distance between the sphere and the plane is decreased. When the sphere touches the plane, there is then an infinite set of nested ring vortices enclosing the contact point.     

Creation of Controlled Defects Inside Colloidal Crystal Arrays with a Focused Ion Beam

In this work the reliability of the focused-ion-beam (FIB) patterning on polystyrene (PS) colloidal crystals at different scales is determined. Ordered arrays of PS spheres (465 nm) are successfully modified by selectively removing a single sphere. The water-vapor assisted FIB milling is crucial to obtain this result. Furthermore, isolated PS spheres are FIB drilled with or without chemically enhanced milling aiming at the exploration of the limits of such a technique. These controlled defects created using the FIB-assisted techniques may be helpful in preparing mockups of photonic crystals, sensors or as colloidal masks for diverse lithographic processes.