Sedimentation of spheres at small Reynolds number

The effect of fluid inertia on the settling of spheres in a viscous incompressible fluid is studied in the limit of small Reynolds number. The kinetic energy of flow depends on the positions of the spheres, and gives rise to forces on the spheres. In the dilute limit it suffices to study the corresponding pair interaction. The interaction is calculated from the Stokes flow for two spheres settling between plane walls in the point particle limit. The dissipative interaction between a pair of spheres is calculated from the Proudman-Pearson [I. Proudman and J. R. A. Pearson, J. Fluid Mech. 2, 237 (1957)] solution of the Navier-Stokes equations for flow about a sphere in unbounded geometry. The combination of kinetic and dissipative interaction gives rise to a repulsive force of range of the order of the sphere diameter divided by the Reynolds number.     

Creeping motion of an assemblage of composite spheres relative to a fluid

The sedimentation of a homogeneous distribution of spherical composite particles and the fluid flow through a bed of these particles are investigated theoretically. Each composite particle is composed of a spherical solid core and a surrounding porous shell. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. The effect of interactions among the particles is taken into explicit account by employing a fundamental cell-model representation which is known to provide good predictions for the motion of a swarm of nonporous spheres within a fluid. In the limit of a small Reynolds number, the Stokes and Brinkman equations are solved for the flow field in a unit cell, and the drag force exerted by the fluid on the particle is obtained in a closed form. For a distribution of composite spheres, the normalized mobility of the particles decreases or the particle interactions increase monotonically with a decrease in the permeability of their porous shells. The effect of particle interactions on the creeping motion of composite spheres relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solutions describing the drag force or mobility for a suspension of composite spheres reduce to those for suspensions of solid spheres and of porous spheres. The hydrodynamic behavior for composite spheres may be approximated by that for permeable spheres when the porous layer is sufficiently thick, depending on the permeability.     

Numerical Analysis of 3‐D Flow Past a Stationary Sphere with Slip Condition at Low and Moderate Reynolds Numbers

Computations are performed to determine the steady 3‐D viscous fluid flow forces acting on the stationary spherical suspended particle at low and moderate Reynolds numbers in the range of 0.1≤Re≤200. A slip is supposed on the boundary so that the slip velocity becomes proportional to the shear stress. This model possesses a single parameter to account for the slip coefficient λ (Pa.s/m), which is made dimensionless and is called Trostel number (Tr=λ a/μ). Decreasing slip, increases drag in all Reynolds limits, but slip has smaller effects on drag coefficient at lower Reynolds number regimes. Increasing slip at known Reynolds number causes to delay of flow separation and inflect point creation in velocity profiles. At full slip conditions, shear drag coefficient will be zero and radial drag coefficient reaches to its maximum values. Flow around of sphere at full‐slip condition is not equal to potential flow around a sphere. Present numerical results corresponding to full slip (Tr→0) are in complete accord with certain results of flow around of inviscid bubbles, and the results corresponding to no‐slip (Tr→∞) have excellent agreement with the results predicted by the no‐slip boundary condition.     

An experimental study of the drag force on a sphere exposed to an argon thermal plasma flow

Experimental data are presented concerning the drag force on a stationary phere exposed to an argon plasma flow with temperatures about 10⁴ K and velocities about 10² m/s. A novel probe construction has been employed in the drag measurements in order to exclude the effect of the supporting wire on the sphere drag data. By using the new probe construction with a compensating wire, drag forces on an individual steel sphere in the plasma flow have been measured and compared with those obtained by using the probe construction ernployed by a few previous authors. Experimental results show that the measured drag forces are always less than their counterparts obtained from the standard sphere-drag curve under isothermal flow conditions with the same Reynolds numbers based on the oncoming plasma properties. The drag force on a sphere increases only slightly with the increasing surface temperature of the sphere before it melts. Appreciable diference was found between the experimental data and the predicted results of the available expressions for drag on a sphere exposed to a thermal plasma flow. Further research effort is required to build a more suitable drag correlation.     

Combined effects of blockage and yield stress on drag and heat transfer from an in-line array of three spheres

This work reports results on the drag and heat transfer from an in-line array of three isothermal spheres falling in a cylindrical confinement filled with Bingham plastic fluids. The effects of dimensionless parameters, such as the Reynolds number (1 ≤ Re?≤?100), Prandtl number (1 ≤ Pr?≤?100), Bingham number (0 ≤ Bn?≤?100), blockage ratio (2 ≤ β?≤?4) and sphere-to-sphere distance (1.5 ≤ t?≤?6) have been elucidated. The flow and heat transfer characteristics were analysed in terms of yielded/unyielded regions, streamline and isotherm contours, drag coefficient, pressure coefficient, and local and average Nusselt number. Broadly, the drag coefficient shows a positive dependence on Bn and sphere-to-sphere distance (t) while it exhibits an inverse dependence on Re and β. On the other hand, the Nusselt number shows a positive dependence on Re, Pr, Bn and β; and a complex dependence on t for each sphere. Simple predictive expressions for the average Nusselt number for each sphere are formulated, thereby enabling its prediction in a new application.     

Hydrodynamic drag force exerted on activated sludge floc at intermediate Reynolds number

We hung the activated sludge flocs on an elastic nylon stick and then subjected it to a uniform water flow and measured its displacement. The hydrodynamic drag force exerted on the floc was subsequently estimated, both for cationic flocculated flocs and for flocculated and then frozen/thawed flocs. A confocal laser scanning microscope (CLSM) was employed to probe the interior structure of flocs. Polyelectrolyte flocculation leads to a compact global structure, and hence high drag force exerted on the floc by water. The corresponding C(D)Omega value at Re=12-27 for flocs ranges from 1.58 to 3.61. Fast freezing would little affect the hydrodynamic drag force. Slow freezing, in contrast, considerably consolidated the floc structure and hence presented impermeable sphere-like behavior of the slowly frozen/thawed flocs.     

Mobility of permeable fractal agglomerates in slip regime

Hydrodynamic drag and mobility of fractal aggregates in the slip creeping flow regime are calculated. A theoretical continuum model of the gas slip flow past and within agglomerates is developed. It accounts for effects of flow rarefaction and porous fractal structure upon the molecular mean free path, apparent viscosity, and effective permeability of agglomerates. It is shown that flow rarefaction significantly diminishes the aggregates' drag to an extent that cannot be predicted by the Cunningham's drag correction factor. The developed model allows calculation the agglomerates' transport properties in a wide range of fractal dimensions. For low D(f) agglomerates the drag force agrees with the Friedlander's expression based on the Epstein's single sphere drag in the free molecular regime.     

Critical flux of hard sphere suspensions in crossflow filtration: Hydrodynamic force bias Monte Carlo simulations

A Monte Carlo method is developed for crossflow membrane filtration to determine the critical flux of hard sphere suspensions. Brownian and shear-induced diffusion are incorporated into an effective hydrodynamic force exerted on the hard spheres in a concentrated shear flow. Effects of shear rate and particle size on the critical flux are investigated using hydrodynamic force bias Monte Carlo simulations, providing a baseline of the critical flux.     

Suspension viscosity of colloidal crystals and liquids in exhaustively deionized aqueous suspensions coexisting with ion-exchange resins

 Viscosities of exhaustively deionized aqueous suspensions of colloidal silica spheres are measured with coexisting ion-exchange resins using an Ubbelohde-type viscometer. The reduced viscosities of small silica spheres (56.3 nm in diameter) with and without resins decrease as the sphere concentration increases. However, the former are larger than the latter especially at low sphere concentrations. The reduced viscosities of other silica spheres, 81.2, 103, 110 and 136 nm in diameter, with resins decrease as the sphere concentration increases, whereas those without resins increase especially at low sphere concentrations. The significant effect of the extent of deionization upon the viscometric properties supports the important role of the extended electrical double layers formed around the colloidal spheres. Received: 28 October 1999 Accepted: 24 December 1999     

Stokes flow in periodic systems of parallel cylinders with porous permeable shells

The problem solved in this study concerns steady-state flow of a viscous incompressible liquid at low Reynolds numbers in a model filter consisting of parallel cylinders with porous permeable shells. Both a separate row and lattices (square and hexagonal) of cylinders directed perpendicularly to the flow are considered. The flow field outside and inside the porous shell is found from the solutions to the Stokes and Brinkman equations. The drag force and the filtration efficiency are determined both as functions of the ratio between the cylinder diameter and the distance between the axes of adjacent cylinders and as functions of the thickness and permeability of the shells. The cell model is shown to be applicable for describing the flow field in a hexagonal lattice of cylinders with porous shells within a wide range of packing densities. Original Russian Text ? V.A. Kirsh, 2006, published in Kolloidnyi Zhurnal, 2006, Vol. 68, No. 2, pp. 198–206.     

Convectional, sedimentation, and drying dissipative patterns of colloidal silica (183 nm in diameter) suspensions in a glass dish and a watch glass

Convectional, sedimentation, and drying dissipative structural patterns formed during the course of drying aqueous colloidal crystals of silica spheres (183 nm in diameter) have been studied in a glass dish and a watch glass. Spoke-like convectional patterns were observed in a watch glass. The broad ring sedimentation patterns formed especially in a glass dish within 30–40 min in suspension state by the convectional flow of water and colloidal spheres. The macroscopic broad ring drying patterns formed both in a glass dish and a watch glass. The ratio of the broad ring size in a glass dish against the initial size of suspension, i.e., inner diameter of the glass dish, d _f/d _i, in this work, were compared with previous work of other silica spheres having sizes of 305 and 560 nm and 1.2 μm in diameter. The d _f/d _i values in a glass dish increased as sphere concentration increased, but were rather insensitive to colloidal size. The d _f/d _i values on a watch glass also increased as sphere concentration increased, and further increased as sphere size decreased. Segregation effect by sphere size in a watch glass takes place by the balancing between the upward convectional flow of spheres in the lower layers of the liquid and the downward sedimentation of spheres. Colorful microscopic drying patterns formed both in a glass dish and a watch glass.     

Giant colloidal crystals of fluorine-containing polymer spheres in exhaustively deionized aqueous suspensions and in the presence of sodium chloride

 Gigantic colloidal single crystals (2–6 mm) are formed for fluorine-containing polymer spheres (120–210 nm in diameter) in exhaustively deionized aqueous suspensions. The spheres used are poly(tetrafluoroethylene) (PTFEA and PTFEB), copolymer of tetrafluoroethylene and perfluorovinylether (PFA) and copolymer of tetrafluoroethylene and perfluoropropylene (PTP). The phase diagrams of these spheres are obtained in the deionized suspensions and also in the presence of sodium chloride for PFA. The critical sphere concentrations of crystal melting (φ _c) for these spheres are around 0.0006 in volume fraction, which are close to, but slightly larger than, those of monodispersed polystyrene spheres (φ _c ≈ 0.00015) and colloidal silica spheres(φ _c = 0.0002–0.0004) reported previously. The crystals are largest when the sphere concentrations are a bit higher than the φ _c value and their size decreases as the sphere concentration increases. Reflection spectra are taken in sedimentation equilibrium as a function of the height from the bottom of the suspension. The static elastic modulus is estimated to be 10.8 and 28.7 Pa for PTFEA and PTP spheres at the sphere concentrations 0.00325 and 0.00322 in volume fraction, respectively. Received: 27 October 1999 Accepted in revised form: 16 November 1999     

Sedimentation of a circular disk in power law fluids

The continuity and momentum equations have been solved numerically for the two-dimensional steady flow of power law fluids over a thin circular disk oriented normal to the direction of flow. Extensive results on the individual and total drag coefficients are obtained as functions of the power law flow behavior index (0.4 < or = n < or = 1.0), Reynolds number (1 < or = Re < or = 100) and the blockage ratio, disk-to-cylinder diameter ratio (0.02 < or = e < or = 0.5), which can be used to estimate the settling velocity of a circular disk. The numerical predictions of drag are consistent with the scant experimental results available in the literature.     

Forces on a porous particle in an oscillating flow

We investigate theoretically forces acting on a porous particle in an oscillating viscous incompressible flow. We use the unsteady equations describing the creeping flow, namely the Stokes equations exterior to the particle and the Darcy or Brinkman equations inside it. The effect of particle permeability and oscillation frequency on the flow and forces is expressed via the Brinkman parameter beta = a/square root(k) and the frequency parameter Y = square root(a(2)omega/2nu) = a/delta, respectively. Here a is particle radius, k is its permeability, omega is the angular frequency, delta is the thickness of Stokes layer (penetration depth) and nu is the fluid kinematic viscosity. It is shown that the oscillations interact with permeable structure of the particle and affect both the Stokes-like viscous drag and the added mass force components.     

Drag force acting on an evaporating particle immersed in a rarefied plasma flow

Analytical expressions are presented for the drag force acting on an evaporating or nonevaporating particle immersed in a plasma flow for the extreme case of free-molecule flow regime and thin plasma .sheath. It is shown that the drag force on a spherical particle is proportional to the square of the particle radius and to the relative velocity between the particle and the bulk plasma at low speed ratios. The existence of a relative velocity between the particle and the plasma results in a nonuniform heat flux distribution with its rnaximum value at the frontal stagnation point of tire sphere. This nonuniform distribution of the local heat fux density causes a nonuniforrn distribution of the local evaporated-mass flux and vapor reaction force around the surface of an evaporating particle, and thus induces an additional force on the particle. Consequently, the drag force acting on art evaporating particle is always greater than that on a nonevaporating one. This additional drag force due to particle evaporation is more significant for nonmetallic particles and for particle materials with lower latent heat of evaporation and lower vapor molecular mass. It increases with increasing plasma temperature and with decreasing gas pressure at the high plasma temperatures associated with appreciable gas ionization. The drag ratio increases with increasing electron/heavy-particle temperature ratio at high electron temperatures for a two-temperature plasma.     

Rigidity of colloidal alloys as studied by reflection spectroscopy in sedimentation equilibrium

Rigidities of colloidal alloys of binary mixtures of colloidal silica spheres (CS82; 103 nm in diameter) with larger silica spheres (CS91; 110 nm, CS121; 136 nm and CS161; 184 nm) have been measured by reflection spectroscopy in sedimentation equilibrium. Substitutional-solid-solution-type alloy structures are formed for mixtures of CS82 and CS91 and for CS82 and CS121. A superlattice, probably MgCu₂ type, is formed for CS82 and CS161 mixtures. The rigidities of the colloidal crystals of the single component of the spheres increase as the sphere size increases at the same number density of spheres. The rigidities of the colloidal alloys decrease when a comparatively small number of the larger spheres are mixed with the small spheres at the same total sphere number density. Received: 14 June 2000 Accepted: 3 November 2000     

A Physical Method of Fabricating Hollow Polymer Spheres Directly from Oil/Water Emulsions of Solutions of Polymers

Summary: A new physical method of fabricating hollow spheres from different polymers has been developed. In this method, emulsions were prepared by mixing organic solutions of polystyrene, poly(D ,L ‐lactide‐co‐glycolide) (PLGA), and bacterial poly(3‐hydroxybutyrate‐co‐3‐hydroxyvalerate) (PHBV), water, and surfactants. The evaporation of solvents at room temperature caused a phase separation that eventually yielded hollow spheres. Molecular weights, concentrations of polymers, and the natures of surfactant and solvent were important aspects of hollow sphere formation and structure. A mechanism for the formation of hollow spheres is proposed based on observations made using an optical microscope equipped with a digital camcorder and using scanning electron microscopy images of hollow spheres obtained under different conditions.

A scanning electron microscopy image of a broken smaller hollow sphere prepared using a 7 wt.‐% polystyrene solution (diameter of the sphere ∼10 micrometers).     

Strong magnetic field effects on solid-liquid and particle-particle interactions during the processing of a conducting liquid containing non-conducting particles

The behavior of micrometer-sized weak magnetic insulating particles migrating in a conductive liquid metal is of broad interest during strong magnetic field processing of materials. In the present paper, we develop a numerical method to investigate the solid-liquid and particle-particle interactions by using a computational fluid dynamics (CFDs) modeling. By applying a strong magnetic field, for example, 10 Tesla, the drag forces of a single spherical particle can be increased up to around 15% at a creeping flow limit. However, magnetic field effects are reduced when the Reynolds number becomes higher. For two identical particles migrating along their centerline in a conductive liquid, both the drag forces and the magnetic interaction will be influenced. Factors such as interparticle distance, Reynolds number and magnetic flux density are investigated. Shielding effects are found from the leading particle, which will subsequently induce a hydrodynamic interaction between two particles. Strong magnetic fields however do not appear to have a significant influence on the shielding effects. In addition, the magnetic interaction forces of magnetic dipole-dipole interaction and induced magneto-hydrodynamic interaction are considered. It can be found that the induced magneto-hydrodynamic interaction force highly depends on the flow field and magnetic flux density. Therefore, the interaction between insulating particles can be controlled by applying a strong magnetic field and modifying the flow field. The present research provides a better understanding of the magnetic field induced interaction during liquid metal processing, and a method of non-metallic particles manipulation for metal/ceramic based materials preparation may be proposed.     

Density-induced coupling effects on the dispersivity of a flexible chain particle

A model is introduced to investigate the transport properties of an inhomogeneously dense flexible chain particle. The specific model used is a sedimenting non-neutrally buoyant inhomogenously weighted flexible Brownian dumbbell, and it is shown that density inhomogeneity gives rise to a novel coupling effect between the "shape-fluctuation" and "size-fluctuation" dispersion mechanisms. The previously reported shape-fluctuation dispersion term stems from the dumbbell's nonspherical shape and the ensuing anisotropic mobility tensor, while the already investigated size fluctuation term is the result of the dependence of the overall dumbbell translational mobility on the separation distance between the constitutive spheres. Because the density of the constitutive spheres is unequal, the external force simultaneously reorients and deforms the flexible dumbbell, and it is this mutual dependence between dumbbell orientation and size that induces the coupling. Numerical results are presented for the case of a tethered dumbbell composed of two spheres, identical in size but differing in density. The "weak-field" limit is addressed, where the externally applied torque and particle deformation forces are dominated by the thermal fluctuations associated with rotational and deformation Brownian motion. This numerical solution, obtained by including a large number of higher order hydrodynamic interactions (120 terms), describes the Brownian particle's long-time transport without resorting to ad hoc approximations, such as preaveraging the hydrodynamic force or incorporating only first-order hydrodynamic interaction effects (such as employing the Burgers-Oseen tensor). Separate analytical solutions, based on these respective approximations, are also presented and it is concluded that in the limit of "long tethers," where the ratio of tether length to sphere size is greater than seven, no more than 15% error is introduced by neglecting higher-order hydrodynamic interactions. Similarly, the preaveraging approximation introduces no more than a few percent error in the limit of "almost-rigid" dumbbells, where the ratio of tether length to sphere size is less than three. For tethers of "intermediate" length, the full numerical solution must be employed.     

Colloidal crystallization of poly(n-butyl acrylate) spheres in deionized aqueous suspension and the melting during dryness

Colloidal crystallization of poly(n-butyl acrylate) spheres (ammonium persulfate-poly(n-butyl acrylate) (APS-PBA), 320?±?50 nm in diameter) was studied in deionized aqueous suspension. Coexistence of the crystal and distorted crystal structures was observed by the reflection spectroscopy. The critical concentrations of melting were ca. 0.01 and 0.03 in volume fraction in the presence of ion-exchange resins and in their absence, respectively. Crystal structures melted away during dryness by fusion of each spheres on the substrates, i.e., cover glass, watch glass, and Petri glass dish. Thickness profiles of the dried film changed sharply from the broad ring to the round hill as sphere concentration increased. The sharpness parameter S was evaluated from the ratio of the film size (diameter) against the full width at half maximum in the thickness profiles of the ring and/or the round hill. The S values decreased sharply from 30 to 1.2 as initial volume fraction of the spheres increased from 0.0005 to 0.1. The S values were significantly low compared with those of typical colloidal spheres, which supports the aggregate and/or fusion of the spheres resulting in their low convectional flow during dryness. The round hill profile at the high sphere concentration also supports that the fusion takes place easier during dryness. Microscopic observation of the dried film supports the formation of the homogeneous fused structures. It was clarified that colloidal crystallization of APS-PBA spheres takes place by the extended electrical double layers around the spheres like typical colloidal crystals of hard spheres. However, APS-PBA spheres are not so stable by the fusion especially at the high sphere concentrations and on the substrates.