Creeping flow over a composite sphere: Solid core with porous shell

Creeping flow past a solid sphere with a porous shell has been solved using the Stokes and Brinkman equations. The dimensionless solid core and shell radii, normalized by the square root of the shell permeability, are the two parameters that govern the flow. In the limiting cases, the analytical solution describing the flow past the composite sphere reduces to that for flow past a solid sphere and a homogeneous porous sphere. The settling rates of a solid sphere with attached threads are measured experimentally. This system can be considered a model for rigid linear molecules anchored or adsorbed onto a colloidal particle. The analytical solution for the composite sphere is in excellent agreement with the experimental results.     

Motions of a porous particle in stokes flow part 2. Linear flows near a fluid interface

Analytical results are presented for the motion of a porous sphere in the vicinity of a plane fluid-fluid interface. The fluids are assumed to undergo a linear undisturbed flow and the viscosily ratio of the two fluids is assumed to be arbitrary. The analysis consists of the method of reflections, coupled with an application of fundamental singularity solutions for Stokes flow to calculate the hydrodynamic force and torque on the particle. The fundamental relationships for Ihe force and torque are then applied, in combination with Ihe corresponding solutions obtained in earlier publications for the translation and rotalion through a quiescent fluid, to determine the motion of a neutrally buoyant particle freely suspended in the flow.     

Fluidization of elongated particles—Effect of multi-particle correlations for drag,lift, and torque in CFD-DEM

Having proper correlations for hydrodynamic forces is essential for successful CFD-DEM simulations of a fluidized bed. For spherical particles in a fluidized bed, efficient correlations for predicting the drag force, including the crowding effect caused by surrounding particles, are already available and well tested. However, for elongated particles, next to the drag force, the lift force, and hydrodynamic torque also gain importance. In this work, we apply recently developed multi-particle correlations for drag, lift and torque in CFD-DEM simulations of a fluidized bed with spherocylindrical particles of aspect ratio 4 and compare them to simulations with widely used single-particle correlations for elongated particles. Simulation results are compared with previous magnetic particle tracking experimental results. We show that multi-particle correlations improve the prediction of particle orientation and vertical velocity. We also show the importance of including hydrodynamic torque.     

On the unsteady forces during the motion of a sedimenting particle

We consider the unsteady motion of a sedimenting rigid spherical particle in order to examine the relative strength of the hydrodynamical forces acting on particles in fluid flows. The relative strength of the forces on all stages of the particle motion is a major concern for closing constitutive equations describing the more complex motion of particulate flows such as fluidised beds. The formulation results in a first order nonlinear integro-differential equation in terms of the instantaneous velocity of the sphere. This equation is made dimensionless and the particle Reynolds number and the fluid-particle density ratio are identified as the relevant physical parameters describing the particle motion. We obtain analytical solutions for the limits of small density ratios and small Reynolds number. In addition, a numerical solution is used for arbitrary values of the density ratio. The results show that the motion of spherical particles is significantly affected by the unsteady drag dominated by the memory Basset force on the early stages of the motion and on the approach to the steady state (terminal velocity). The present calculations indicate that the unsteady hydrodynamic drags might become of the same order of magnitude of the dominant viscous drag for flows with moderate particle-fluid density ratio. Therefore, unsteady drags should be taken into account on modelling multiphase particulate flows with moderate density ratio.     

Viscous flow past a porous sphere with an impermeable core : effect of stress jump condition

An arbitrary flow of a viscous, incompressible fluid past a porous sphere of radius `a' with an impermeable core of radius `b', using Brinkman's equation in the porous region is discussed. At the interface of the clear fluid and porous region, stress jump boundary condition for the tangential stresses along with the continuity of normal stresses and the velocity components are used. On the surface of the impermeable core no slip condition is used. The corresponding Faxen's laws are derived to compute the drag and torque acting on the surface r=a. It is found that the drag and torque not only change with the change of the permeability, but also a significant effect of the stress jump co-efficient is observed. The variation of drag and torque with permeability for different thickness (a-b) of the porous region as well as for different values of stress jump coefficient is discussed when the basic flow is due to uniform flow, two dimensional irrotational flow, doublet in a uniform flow, stokeslet, rotlet. In case of uniform flow the flow field has been plotted. In all the cases, a significant effect of the stress jump coefficient has been realized.     

Simulations of solid–liquid scalar transfer for a spherical particle in laminar and turbulent flow

Scalar transfer from a solid sphere to a surrounding liquid has been studied numerically. The simulation procedure involves full hydrodynamic resolution of the solid–liquid interaction and the flow (laminar and turbulent) of the carrier fluid by means of the lattice‐Boltzmann method. Scalar transport is solved with a finite volume method on coupled overlapping domains (COD): an outer domain discretized with a cubic grid and a shell around the solid sphere with a spherical grid with fine spacing in the radial direction. The shell is needed given the thin scalar boundary layer around the sphere that is the result of high Schmidt numbers (up to Sc = 1000). After assessing the COD approach for laminar benchmark cases, it is applied to a sphere moving through homogeneous isotropic turbulence with the sphere radius larger (by typically a factor of 10) than the Kolmogorov length scale so that it experiences an inhomogeneous hydrodynamic environment. This translates in pronounced scalar concentration variations and transfer rates over the sphere's surface. Overall scalar‐transfer coefficients are compared to those derived from classical Sherwood number correlations. © 2014 American Institute of Chemical Engineers AIChE J, 60: 1202–1215, 2014     

Motions of a sphere in a time-dependent stokes flow: A generalization of Faxen’s law

A general solution of the unsteady Stokes equation in spherical coordinates is derived for flow in the exterior of a sphere, and then applied to study the arbitrary unsteady motion of a rigid sphere in an unbounded single fluid domain which is undergoing a time-dependent mean flow. Calculation of the hydrodynamic force and torque on the sphere leads to a generalization of the Faxen’s law to time-dependent flow fields which satisfy the unsteady Stokes equation. For illustrative purposes, we consider the relative motion of gas bubbles which undergo very rapid oscillations so that the generalized Faxen’s law derived for a solid sphere can be applied. We also demonstrate that our results reduce to those of Faxen for the steady flow limit.     

Low-Reynolds-number motion of a heavy sphere between two parallel plane walls

A novel boundary-integral algorithm [Staben, M.E., Zinchenko, A.Z., Davis, R.H., 2003. Motion of a particle between two parallel plane walls in low-Reynolds-number Poiseuille flow. Physics of Fluids 15, 1711-1733; Erratum: Phys. Fluids 16, 4206] is used to obtain O(1)-nonsingular terms that are combined with two-wall lubrication asymptotic terms to give resistance coefficients for near-contact or contact motion of a heavy sphere translating and rotating between two parallel plane walls in a Poiseuille flow. These resistance coefficients are used to describe the sphere's motion for two cases: a heavy sphere driven by a Poiseuille flow in a horizontal channel and a heavy sphere settling due to gravity through a quiescent fluid in an inclined channel. When the heavy sphere contacts a wall in either system, which occurs when the gap between the sphere and the wall becomes equal to the surface roughness of the sphere (or plane), a contact-force model using the two-wall resistance coefficients is employed. For a heavy sphere in a Poiseuille flow, experiments were performed using polystyrene particles with diameters 10%-60% of the channel depth, driven through a glass microchannel using a syringe pump. The measured translational velocities for these particles show good agreement with theoretical results. The predicted translational velocity increases for increasing particle diameter, as the spheres extend further into the Poiseuille flow, except for particles that are so large (diameters of 80%-85% of the channel depth) that the upper wall has a dominant influence on the particle velocity. For a heavy sphere settling in a quiescent fluid in an inclined channel, the transition from the no-slip regime to slipping motion occurs for a larger inclination angle of the channel with respect to the horizontal for an increase in particle diameter, since the larger particles are more slowed by the second wall. Limited experiments were performed for Teflon spheres with diameters 64%-95% of the channel depth settling in a very viscous fluid along the lower wall of an inclined acrylic channel. The measured translational velocities, which are only about 15%-25% of the tangential component of the undisturbed Stokes settling velocity, are in close agreement with theory using physical parameters obtained from similar experiments with a single wall [Galvin, K.P., Zhao, Y., Davis, R.H., 2001. Time-averaged hydrodynamic roughness of a noncolloidal sphere in low Reynolds number motion down an inclined plane. Physics of Fluids 13, 3108-3119].     

Motion of a sphere parallel to plane walls in a Poiseuille flow. Application to field-flow fractionation and hydrodynamic chromatography

The motion of a solid spherical particle entrained in a Poiseuille flow between parallel plane walls has various applications to separation methods, like field-flow fractionation and hydrodynamic chromatography. Various handy formulae are presented here to describe the particle motion, with these applications in mind. Based on the assumption of a low Reynolds number, the multipole expansion method coupled to a Cartesian representation is applied to provide accurate results for various friction factors in the motion of a solid spherical particle embedded in a viscous fluid between parallel planes. Accurate results for the velocity of a freely moving solid spherical particle are then obtained. These data are fitted so as to obtain handy formulae, providing e.g. the velocity of the freely moving sphere with a 1% error. For cases where the interaction with a single wall is sufficient, simpler fitting formulae are proposed, based on earlier results using the bispherical coordinates method. It appears that the formulae considering only the interaction with a nearest wall are applicable for a surprisingly wide range of particle positions and channel widths. As an example of application, it is shown how in hydrodynamic chromatography earlier models ignoring the particle-wall hydrodynamic interactions fail to predict the proper choice of channel width for a selective separation. The presented formulae may also be used for modeling the transport of macromolecular or colloidal objects in microfluidic systems.     

Lift and drag forces on a sphere attached to a wall in a Blasius boundary layer

Lift and drag forces on a sphere attached to a planar wall, over which a laminar flat plat boundary layer flows, are examined numerically in this study. Particle Reynolds number ranged from 0.1–250, which represents steady, laminar flow about the sphere, and the plate Reynolds number was held constant at 32 400. A finite-volume computational fluid dynamics program was utilised. Simulation results were validated against analytical results for drag and lift in creeping flow and against experimental results available in the literature for lift at higher particle Reynolds number. The model results were curve-fitted and interpolating drag and lift coefficient functions are reported. The lift and drag results are shown to be weakly dependent upon plate Reynolds number. The resulting correlations are expected to be useful in the development of particle impending motion and aerosol entrainment predictions of particles adhering to planar walls.     

Measurement of the Aerodynamic Drag Force on Single Aerosol Particles

The “picobalance” (quadrupole) was used to measure the aerodynamic drag force on individual solid particles and droplets by suspending the object in a laminar jet of gas introduced through the bottom electrode. Particles ranging in diameters from about 1 to 150 μm can be studied in this manner. The DC voltage required to maintain the particle position against the opposing forces of aerodynamic drag and gravity was measured to determine the drag force. The flow velocity at which the aerodynamic drag force balances the gravitational force yields information on the aerodynamic size, and the DC voltage required to suspend the particle against gravity with no flow provides a measure of the particle mass. Particle mobilities for spherical and irregularly shaped solids are presented. Light-scattering measurements for spherical particles provide an independent determination of size; the results are generally in good agreement with the aerodynamic size. It is shown that the electrodynamic balance can be used to measure drag forces much larger than the particle weight.     

Modelling the motion of cylindrical particles in a nonuniform flow

The models currently used in computational fluid dynamics codes to predict solid fuel combustion rely on a spherical shape assumption. Cylinders and disks represent a much better geometrical approximation to the shape of bio-fuels such as straws and woods chips. A sphere gives an extreme in terms of the volume-to-surface-area ratio, which impacts both motion and reaction of a particle. For a nonspherical particle, an additional lift force becomes important, and generally hydrodynamic forces introduce a torque on the particle as the centre of pressure does not coincide with the centre of mass. Therefore, rotation of a nonspherical particle needs to be considered. This paper derives a model for tracking nonspherical particles in a nonuniform flow field, which is validated by a preliminary experimental study: the calculated results agree well with measurements in both translation and rotation aspects. The model allows to take into account shape details of nonspherical particles so that both the motion and the chemical reaction of particles can be modelled more reasonably. The ultimate goal of such a study is to simulate flow and combustion in biomass-fired furnaces using nonspherical particle tracking model instead of traditional sphere assumption, and thus improve the design of biomass-fired boilers.     

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.     

Direct numerical simulation of moderate-Reynolds-number flow past arrays of ellipsoids

Through particle-resolved direct numerical simulations of flow past arrays of ellipsoids, the hydrodynamic force on ellipsoids depends on the particle orientation, aspect ratio, particle Reynolds number, and solid volume fraction is revealed at moderate Reynolds numbers. The results show that the mean drag force on arrays of prolate/oblate ellipsoids decreases/increases as the Hermans orientation factor increases when flows are in the reference direction defined by the average symmetric axis of particles. The individual drag force on a prolate/oblate ellipsoid increases/decreases with the increase of incidence angle, and it is also affected by the orientation of surrounding particles. The individual lift force is also significant when the aspect ratio is away from unity at large particle Reynolds numbers. Based on simulation results, correlations for the hydrodynamic force on ellipsoids at arbitrary particle Reynolds numbers, solid volume fractions, Hermans orientation factors, incidence angles, and aspect ratios are formulated.     

Electrophoresis of a soft charged particle in a sparsely packed bed

In this study, the migration of a single charged spherical hydrogel (polyelectrolyte) micro/nanoparticle is considered in an electrolyte solvent medium where several identical micro/nanoparticles are suspended. Each particle can be modeled as a charged linearly elastic fibrous spherical solid saturated with a Newtonian electrolyte. In this study, a new analytical approach is proposed for finding the steady response of the soft charged hydrogel sphere to a DC electric field using the perturbation method. It is possible to obtain a closed form for the electrophoretic mobility of a single sphere within a sparsely packed sphere bed for a wide range of electric double layer thickness. The response of each hydrogel sphere toward the external fields includes both the particle translation and infinitesimal deformation in the polymeric skeleton of a hydrogel sphere. The hydrodynamics inside a sphere and the corresponding strain of the solid phase are studied by adopting field equations from the theory of biphasic mixtures. This study highlights that factors like fixed charge density, dielectric constants, elasticity coeffcients of polymer skeleton influence the mobility and deformation of the particle. This study has a close relevance to an electric field induced drug delivery in biological media.     

CFD study of solids concentration in a fluidised-bed coater with variation of atomisation air pressure

The solids volume fraction inside a tapered fluidised bed coater was simulated with the use of an Eulerian computational fluid dynamics (CFD) model with atomisation nozzle sub-model. The drag force, describing momentum transfer between the gas and solid phases was calculated using the drag model proposed by [1]. In order to account for the particle size distribution of the fluidised solid materials, a 4-phase Eulerian model was used. The model-predicted results for different atomisation air pressures were verified using published experimental data [2]. It was shown that the model proved to be highly sensitive to changes in the fluidisation air flow rate with regard to the model-predicted solids volume distribution.     

Hydrodynamic force between two hard spheres tangentially translating in a power-law fluid

The hydrodynamic interaction between two hard spheres tangentially translating in a power-law fluid is investigated. By considering the gap between the two spheres being sufficiently small such that the Reynolds’ lubrication theory applies, an analytical equation to the pressure in the gap is obtained using truncated Fourier series. To a good approximation, the pressure equation can be further simplified. The simplified approximate equation over-predicts the pressure for shear thickening fluid (n>1) but under-predicts the pressure for shear-thinning fluid (n<1). However, the errors in the predicted tangential force and moment are relatively small. In particular, for a Newtonian fluid, the accurate solution and the simplified approximate solution degenerate to the asymptotic solution of Goldman et al. [1967. Slow viscous motion of a sphere parallel to a plane wall-motion through a quiescent fluid. Chemical Engineering Science 22, 637-651.] and O’Neill and Stewartson [1967. On the slow motion of a sphere parallel to a nearby plane wall. Journal of Fluid Mechanics 27, 705-724.]. Both solutions predict that for shear thickening fluid (n>1), the hydrodynamic force converged in the inner region of the gap between the two spheres and the contribution from the outer region is sufficiently small. For shear thinning fluid (n<1), the contribution from the outer region is also significant.     

THE NORMAL FORCE EXERTED BY A SECOND ORDER FLUID ON A SMALL SPHERE TOUCHING A PLANE

The hydrodynamic force experienced by a small solid sphere of radius a_p resting on a solid plane wall in an axisymmetric stagnation creeping flow of a viscoelastic fluid is computed at order one in Weissenberg number. The solution utilises the method of the reciprocal theorem of Lorentz and the corresponding solution of the Newtonian problem to get the normal force without the need of solving for the velocity field. The resulting expression is a volume integral which has to be solved numerically. The result obtained shows that elastic effect of the constant viscosity second order fluid causes a reduction in the normal force beyond that of a Newtonian fluid of the same viscosity.     

Modeling of unsteady flow around accelerating sphere at moderate reynolds numbers

The flow around an accelerating spherical particle of diameter ranging from 50 to 200 m?m is studied in the range of Reynolds number between 0.1 and 100. The flow around the sphere is assumed to be laminar and two-dimensional axisymmetric. The calculated drag coefficient is compared with the theoretical predictions of added mass term and Basset history term. Appropriate corrections for those two terms are proposed as function of the acceleration rate and the particle diameter.