Stokes flow past an assemblage of axisymmetric porous spherical shell-in-cell models: effect of stress jump condition

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous concentric spherical shell-in-cell model is studied. Boundary conditions on the cell surface that correspond to the Happel, Kuwabara, Kvashnin and Cunningham/Mehta-Morse models are considered. At the fluid-porous interfaces, the stress jump boundary condition for the tangential stresses along with continuity of normal stress and velocity components are employed. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid are used. The hydrodynamic drag force acting on the porous shell by the external fluid in each of the four boundary conditions on the cell surface is evaluated. It is found that the normalized mobility of the particles (the hydrodynamic interaction among the porous shell particles) depends not only on the permeability of the porous shells and volume fraction of the porous shell particles, but also on the stress jump coefficient. As a limiting case, the drag force or mobility for a suspension of porous spherical shells reduces to those for suspensions of impermeable solid spheres and of porous spheres with jump.     

Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity

This paper concerns the Slow Motion of a Porous Cylindrical Shell in a concentric cylindrical cavity using particle-in-cell method. The Brinkman’s equation in the porous region and the Stokes equation for clear fluid in their stream function formulations are used. The hydrodynamic drag force acting on each porous cylindrical particle in a cell and permeability of membrane built up by cylindrical particles with a porous shell are evaluated. Four known boundary conditions on the hypothetical surface are considered and compared: Happel’s, Kuwabara’s, Kvashnin’s and Cunningham’s (Mehta-Morse’s condition). Some previous results for hydrodynamic drag force and dimensionless hydrodynamic permeability have been verified. Variation of the drag coefficient and dimensionless hydrodynamic permeability with permeability parameter σ, particle volume fraction γ has been studied and some new results are reported. The flow patterns through the regions have been analyzed by stream lines. Effect of particle volume fraction γ and permeability parameter σ on flow pattern is also discussed. In our opinion, these results will have significant contributions in studying, Stokes flow through cylindrical swarms.     

Slow Motion of a Porous Sphere Translating Along the Axis of a Circular Cylindrical Pore Subject to a Stress Jump Condition

The coupled flow problem of an incompressible axisymmetrical quasisteady motion of a porous sphere translating in a viscous fluid along the axis of a circular cylindrical pore is discussed using a combined analytical–numerical technique. At the fluid–porous interface, the stress jump boundary condition for the tangential stress along with continuity of normal stress and velocity components are employed. The flow through the porous particle is governed by the Brinkman model and the flow in the outside porous region is governed by Stokes equations. A general solution for the field equations in the clear region is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The boundary conditions are satisfied first at the cylindrical pore wall by the Fourier transforms and then on the surface of the porous particle by a collocation method. The collocation solutions for the normalized hydrodynamic drag force exerted by the clear fluid on the porous particle is calculated with good convergence for various values of the ratio of radii of the porous sphere and pore, the stress jump coefficient, and a coefficient that is proportional to the permeability. The shape effect of the cylindrical pore on the axial translation of the porous sphere is compared with that of the particle in a spherical cavity; it found that the porous particle in a circular cylindrical pore in general attains a lower hydrodynamic drag than in a spherical envelope.     

Motion through spherical droplet with non-homogenous porous layer in spherical container

The problem of the creeping flow through a spherical droplet with a nonhomogenous porous layer in a spherical container has been studied analytically. Darcy’s model for the flow inside the porous annular region and the Stokes equation for the flow inside the spherical cavity and container are used to analyze the flow. The drag force is exerted on the porous spherical particles enclosing a cavity, and the hydrodynamic permeability of the spherical droplet with a non-homogeneous porous layer is ca...     

Slow Motion of a Porous Eccentric Spherical Particle-in-Cell Models

A combined analytical?Cnumerical method is presented for the quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of porous eccentric spherical particle-in-cell models. The flow inside the porous particle is governed by the Brinkman model and the flow in the fictitious envelope region is governed by Stokes equations. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based on both the porous particle and fictitious spherical envelope. Boundary conditions on the particle??s surface and fictitious spherical envelope that correspond to the Happel, Kuwabara, Kvashnin, and Cunningham/Mehta-Morse models are satisfied by a collocation technique. The drag of these eccentric porous particles relative to the drag experienced by a centered porous particle are investigated as functions of the effective distance between the center of the porous particle and the fictitious envelope, the volume ratio of the porous particle over the surrounding sphere and a coefficient that is proportional to the inverse of the permeability. In the limits of the motions of the porous particle in the concentric position with cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature.     

Studies on the axisymmetric sphere–sphere interaction problem in Newtonian and non-Newtonian fluids

In this research, experimental studies have been performed on the hydrodynamic interaction between two spheres by using particle image velocimetry and measuring the force between the spheres. To approach the system as a resistance problem, a servo-driving system was set-up by assembling a microstepping motor, a ball screw and a linear motion guide for the particle motion. Glycerin and a dilute solution of polyacrylamide in glycerin were used as Newtonian and non-Newtonian fluids, respectively. The polymer solution behaves like a Boger fluid when the concentration is 1000 ppm or less. The experimental results were compared with the asymptotic solution of Stokes equation. The result shows that fluid inertia and unsteadiness play important roles in the particle–particle interaction in the Newtonian fluid. This implies that the motion of two particles in suspension is not reversible even in the Newtonian fluid. In the non-Newtonian fluid, in addition to inertial effect, normal stress differences and viscoelasticity play important roles as expected. In dilute solutions weak shear thinning and the migration of polymer molecules in the inhomogeneous flow field also appear to affect the physics of the problem.     

Creeping motion of a slip spherical particle in a circular cylindrical pore

A combined analytical–numerical study for the creeping flow caused by a spherical fluid or solid particle with a slip-flow surface translating in a viscous fluid along the centerline of a circular cylindrical pore is presented. To solve the axisymmetric Stokes equations for the fluid velocity field, a general solution is constructed from the superposition of the fundamental solutions in both cylindrical and spherical coordinate systems. The boundary conditions are enforced first at the pore wall by the Fourier transforms and then on the particle surface by a collocation technique. Numerical results for the hydrodynamic drag force acting on the particle are obtained with good convergence for various values of the relative viscosity or slip coefficient of the particle, the slip parameter of the pore wall, and the ratio of radii of the particle and pore. For the motion of a fluid sphere along the axis of a cylindrical pore, our drag results are in good agreement with the available solutions in the literature. As expected, the boundary-corrected drag force for all cases is a monotonic increasing function of the ratio of particle-to-pore radii, and approaches infinity in the limit. Except for the case that the cylindrical pore is hardly slip and the value of the ratio of particle-to-pore radii is close to unity, the drag force exerted on the particle increases monotonically with an increase in its relative viscosity or with a decrease in its slip coefficient for a constant ratio of radii. In a comparison for the pore shape effect on the axial translation of a slip sphere, it is found that the particle in a circular cylindrical pore in general acquires a lower hydrodynamic drag than in a spherical cavity, but this trend can be reversed for the case of highly slippery particles and pore walls.     

Removal of Particle Bridges From a Porous Material by Ultrasonic Irradiation

Particle bridge formation during the flow of a liquid with particles through a porous material is a fouling mechanism that can block the pores and, hence, decrease the permeability of the material. Ultrasonic irradiation of the material is a cleaning method that can restore the permeability. We make a numerical study of this cleaning method using the lattice-Boltzmann method. We start from a pore blocked by two spherical particles attached to the pore wall by colloidal adhesion forces, thus forming a particle bridge. Next we calculate the hydrodynamic force exerted by a high-frequency acoustic wave on the two particles. By comparing the hydrodynamic force and the adhesion force we investigate, whether the particle bridge will be removed by the ultrasonic irradiation. A sensitivity study is carried out to investigate the influence of some relevant parameters, such as the acoustic wave amplitude, the acoustic frequency, the fluid flow velocity and the ratio of particle diameter and pore diameter. An upscaling procedure is applied to translate the microscopic results for the removal of the particles at the pore level to the permeability improvement of the material at the macroscopic level. A comparison is made between numerical results and experimental data. The agreement is reasonable.     

Single-Fluid Model of a Mixture for Laminar Flows of Highly Concentrated Suspensions

A model of laminar flow of a highly concentrated suspension is proposed. The model includes the equation of motion for the mixture as a whole and the transport equation for the particle concentration, taking into account a phase slip velocity. The suspension is treated as a Newtonian fluid with an effective viscosity depending on the local particle concentration. The pressure of the solid phase induced by particle-particle interactions and the hydrodynamic drag force with account of the hindering effect are described using empirical formulas. The partial-slip boundary condition for the mixture velocity on the wall models the formation of a slip layer near the wall. The model is validated against experimental data for rotational Couette flow, a plane-channel flow with neutrally buoyant particles, and a fully developed flow with heavy particles in a horizontal pipe. Based on the comparison with the experimental data, it is shown that the model predicts well the dependence of the pressure difference on the mixture velocity and satisfactorily describes the dependence of the delivered particle concentration on the flow velocity.     

CFD-DEM simulation of the hole cleaning process in a deviated well drilling: The effects of particle shape

We investigate the effect of particle shape on the transportation mechanism in well-drilling using a three-dimensional model that couples computational fluid dynamics (CFD) with the discrete element method (DEM). This numerical method allows us to incorporate the fluid–particle interactions (drag force, contact force, Saffman lift force, Magnus lift force, buoyancy force) using momentum exchange and the non-Newtonian behavior of the fluid. The interactions of particle−particle, particle−wall, and particle−drill pipe are taken into account with the Hertz–Mindlin model. We compare the transport of spheres with non-spherical particles (non-smooth sphere, disc, and cubic) constructed via the multi-sphere method for a range of fluid inlet velocities and drill pipe inclination angles. The simulations are carried out for laboratory-scale drilling configurations. Our results demonstrate good agreement with published experimental data. We evaluate the fluid–particle flow patterns, the particle velocities, and the particle concentration profiles. The results reveal that particle sphericity plays a major role in the fluid–solid interaction. The traditional assumption of an ideal spherical particle may cause inaccurate results.     

Drag forces of interacting spheres in power-law fluids

Drag forces of interacting particles suspended in power-law fluid flows were investigated in this study. The drag forces of interacting spheres were directly measured by using a micro-force measuring system. The tested particles include a pair of interacting spheres in tandem and individual spheres in a cubic matrix of multi-sphere in flows with the particle Reynolds number from 0.7 to 23. Aqueous carboxymethycellulose (CMC) solutions and glycerin solutions were used as the fluid media in which the interacting spheres were suspended. The range of power-law index varied from 0.6 to 1.0. In conjunction to the drag force measurements, the flow patterns and velocity fields of power-law flows over a pair of interacting spheres were also obtained from the laser assisted flow visualization and numerical simulation.

Both experimental and computational results suggest that, while the drag force of an isolated sphere depends on the power-index, the drag coefficient ratio of an interacting sphere is independent from the power-law index but strongly depends on the separation distance and the particle Reynolds number. Our study also shows that the drag force of a particle in an assemblage is strongly positions dependent, with a maximum difference up to 38%.     

Slow motions of a circular cylinder experiencing slip near a plane wall

A theoretical study is presented for the two-dimensional creeping flow caused by a long circular cylindrical particle translating and rotating in a viscous fluid near a large plane wall parallel to its axis. The fluid is allowed to slip at the surface of the particle. The Stokes equations for the fluid velocity field are solved in the quasi-steady limit using cylindrical bipolar coordinates. Semi-analytical solutions for the drag force and torque acting on the particle by the fluid are obtained for various values of the slip coefficient associated with the particle surface and of the relative separation distance between the particle and the wall. The results indicate that the translation and rotation of the confined cylinder are not coupled with each other. For the motion of a no-slip cylinder near a plane wall, our hydrodynamic drag force and torque results reduce to the closed-form solutions available in the literature. The boundary-corrected drag force and torque acting on the particle decrease with an increase in the slip coefficient for an otherwise specified condition. The plane wall exerts the greatest drag on the particle when its migration occurs normal to it, and the least in the case of motion parallel to it. The enhancement in the hydrodynamic drag force and torque on a translating and rotating particle caused by a nearby plane wall is much more significant for a cylinder than for a sphere.     

Viscous fluid flow around a semipermeable particle

The problem of hydrodynamic interaction between a laminar flow of a viscous fluid and a partially permeable spherical particle is formulated and solved analytically. The filtration flow inside the particle is assumed to obey the Darcy law. Expressions for the filtration flow velocity, drag, sedimentation velocity, and stream functions are obtained. The effect of the permeability of the particle on the flow characteristics is studied. Stream functions of the flow are constructed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 50, No. 4, pp. 48–53, July–August, 2009.     

气固两相流介尺度LBM-DEM模型

LBM-DEM耦合方法通常是指一种颗粒流体系统直接数值模拟算法,即是一种不引入经验曳力模型的计算方法,颗粒尺寸通常比计算网格的长度大一个量级,颗粒的受力通过表面的粘性力与压力积分获得,其优点是能描述每个颗粒周围的详细流场,产生详细的颗粒-流体相互作用的动力学信息,可以探索颗粒流体界面的流动、传递和反应的详细信息及两相相互作用的本构关系,但其缺点是计算量巨大,无法应用于真实流化床过程模拟。本文针对气固流化床中的流体以及固体颗粒间的多相流体力学行为,建立了一种稠密气固两相流的介尺度LBMDEM模型,即LBM-DEM耦合的离散颗粒模型,实现在颗粒尺度上流化床的快速离散模拟。该耦合模型采用格子玻尔兹曼方法(LBM)描述气相的流动和传递行为,离散单元法(DEM)用于描述颗粒相的运动,并利用能量最小多尺度(EMMS)曳力解决气固耦合不成熟问题,以提高其模拟精度。通过经典快速流态化的模拟,验证了介尺度LBM-DEM耦合模型的有效性。模拟结果表明介尺度LBM-DEM模型是一种探索实验室规模气固系统的有力手段。     

Motion of a porous sphere in a spherical container

The creeping motion of a porous sphere at the instant it passes the center of a spherical container has been investigated. The Brinkman's model for the flow inside the porous sphere and the Stokes equation for the flow in the spherical container were used to study the motion. The stream function (and thus the velocity) and pressure (both for the flow inside the porous sphere and inside the spherical container) are calculated. The drag force experienced by the porous spherical particle and wall correction factor is determined. To cite this article: D. Srinivasacharya, C. R. Mecanique 333 (2005).     

Slow motion of a slip spherical particle perpendicular to two plane walls

The problem of the quasisteady motion of a spherical fluid or solid particle with a slip-flow surface in a viscous fluid perpendicular to two parallel plane walls at an arbitrary position between them is investigated theoretically in the limit of small Reynolds number. To solve the axisymmetric Stokes equation for the fluid velocity field, a general solution is constructed from the superposition of the fundamental solutions in both circular cylindrical and spherical coordinate systems. The boundary conditions are enforced first at the plane walls by the Hankel transform and then on the particle surface by a collocation technique. Numerical results for the hydrodynamic drag force exerted on the particle are obtained with good convergence for various values of the relative viscosity or slip coefficient of the particle and of the relative separation distances between the particle and the confining walls. For the motions of a spherical particle normal to a single plane wall and of a no-slip sphere perpendicular to two plane walls, our drag results are in good agreement with the available solutions in the literature for all relative particle-to-wall spacings. The boundary-corrected drag force acting on the particle in general increases with an increase in its relative viscosity or with a decrease in its slip coefficient for a given geometry, but there are exceptions. For a specified wall-to-wall spacing, the drag force is minimal when the particle is situated midway between the two plane walls and increases monotonically when it approaches either of the walls. The boundary effect on the particle motion normal to two plane walls is found to be significant and much stronger than that parallel to them.     

Drag and Separation Flow Past Solid Sphere with Porous Shell at Moderate Reynolds Number

Axisymmetric viscous, two-dimensional steady and incompressible fluid flow past a solid sphere with porous shell at moderate Reynolds numbers is investigated numerically. There are two dimensionless parameters that govern the flow in this study: the Reynolds number based on the free stream fluid velocity and the diameter of the solid core, and the ratio of the porous shell thickness to the square root of its permeability. The flow in the free fluid region outside the shell is governed by the Navier–Stokes equation. The flow within the porous annulus region of the shell is governed by a Darcy model. Using a commercially available computational fluid dynamics (CFD) package, drag coefficient and separation angle have been computed for flow past a solid sphere with a porous shell for Reynolds numbers of 50, 100, and 200, and for porous parameter in the range of 0.025–2.5. In all simulation cases, the ratio of b/a was fixed at 1.5; i.e., the ratio of outer shell radius to the inner core radius. A parametric equation relating the drag coefficient and separation point with the Reynolds number and porosity parameter were obtained by multiple linear regression. In the limit of very high permeability, the computed drag coefficient as well as the separation angle approaches that for a solid sphere of radius a, as expected. In the limit of very low permeability, the computed total drag coefficient approaches that for a solid sphere of radius b, as expected. The simulation results are presented in terms of viscous drag coefficient, separation angles and total drag coefficient. It was found that the total drag coefficient around the solid sphere as well as the separation angle are strongly governed by the porous shell permeability as well as the Reynolds number. The separation point shifts toward the rear stagnation point as the shell permeability is increased. Separation angle and drag coefficient for the special case of a solid sphere of radius r = a was found to be in good agreement with previous experimental results and with the standard drag curve.     

Validation of fluid–particle interaction force relationships in binary-solid suspensions

In this work several relationships governing solid–fluid dynamic interaction forces were validated against experimental data for a single particle settling in a suspension of other smaller particles. It was observed that force relationships based on Lattice-Boltzmann simulations did not perform as well as other interaction types tested. Nonetheless, it is apparent that, in the case of a suspension of different particle types, it is important that the correct choice is made as to how the contribution to the overall fluid–particle interaction force is split between buoyancy and drag. Experimental evidence clearly suggests that the “generalized” Archimedes’ principle (where the foreign particle is considered to displace the whole suspension and not just the fluid) provides the best result.     

Axisymmetric creeping motion of a slip spherical particle in a nonconcentric spherical cavity

A semianalytical study of the creeping flow caused by a spherical fluid or solid particle with a slip surface translating in a viscous fluid within a spherical cavity along the line connecting their centers is presented in the quasisteady limit of small Reynolds number. In order to solve the Stokes equations for the flow field, a general solution is constructed from the superposition of the fundamental solutions in the two spherical coordinate systems based on both the particle and cavity. The boundary conditions on the particle surface and cavity wall are satisfied by a collocation technique. Numerical results for the hydrodynamic drag force exerted on the particle are obtained with good convergence for various values of the ratio of particle-to-cavity radii, the relative distance between the centers of the particle and cavity, the relative viscosity or slip coefficient of the particle, and the slip coefficient of the cavity wall. In the limits of the motions of a spherical particle in a concentric cavity and near a cavity wall with a small curvature, our drag results are in good agreement with the available solutions in the literature. As expected, the boundary-corrected drag force exerted on the particle for all cases is a monotonic increasing function of the ratio of particle-to-cavity radii, and becomes infinite in the touching limit. For a specified ratio of particle-to-cavity radii, the drag force is minimal when the particle is situated at the cavity center and increases monotonically with its relative distance from the cavity center to infinity in the limit as it is located extremely away from the cavity center. The drag force acting on the particle, in general, increases with an increase in its relative viscosity or with a decrease in its slip coefficient for a given configuration, but surprisingly, there are exceptions when the ratio of particle-to-cavity radii is large.     

Fundamental solutions for axi-symmetric translational motion of a microstretch fluid

The fundamental solution for the axi-symmetrictranslational motion of a microstretch fluid due to a concentrated point body force is obtained.A general formula for thedrag force exerted by the fluid on an axi-symmetric rigid particle translating in it is then deduced.As an application to theobtained drag formula,this paper has discussed the problemof creeping translational motion of a rigid sphere in a microstretch fluid.The slip boundary condition on the surfaceof the spherical particle is applied.The drag force and theother physical quantities are obtained and represented graphically for various values of the micropolarity and slip parameters.