Computation of Stokes Flow due to the Motion or Presence of a Particle in a Tube

The computation of Stokes flow due to the motion or presence of a rigid particle in a fluid-filled tube with arbitrary geometry is discussed with emphasis on the induced upstream to downstream pressure change. It is proposed that expressing the pressure change as an integral over the particle surface involving (a) the a priori unknown traction, and (b) the velocity of the pure-fluid pressure-driven flow, simplifies the numerical implementation and ameliorates the effect of domain truncation. Numerical computations are performed based on the integral formulation in conjunction with a boundary-element method for a particle translating and rotating inside a cylindrical tube with a circular cross-section. The numerical results are consistent with previous asymptotic solutions for small particles, and complement available numerical solutions for particular types of motion     

Interception of two spherical particles with arbitrary radii in simple shear flow

Summary The interception of two force-free and torque-free spherical particles with arbitrary radii freely suspended in simple shear flow is investigated in the limit of vanishing Reynolds number. At any instant, the flow is computed in a frame of reference with origin at the center of one particle using a cylindrical polar coordinate system whose axis of revolution passes through the center of the second particle. The problem is formulated as a decoupled system of integral equations for the zeroth, first, and second Fourier coefficients of the boundary traction with respect to the meridional angle. The derived integral equations are solved with high accuracy using a boundary element method that features adaptive discretization and automatic time-step adjustment according to the inter-particle gap. The results illustrate particle trajectories and describe the particle rotation and evolution of the stress tensor during the interception. The particle interaction is found to always cause a positive shift in the rotational phase angle due to the rolling motion at close contact. As the gap between two particles tends to zero, the shear stress diverges even though the net force and torque exerted on each particle remain zero, independent of the particle relative radius. A frictional force for rough surfaces and small gaps eliminates the slip velocity and promotes the rolling motion.     

Effect of surface slip on Stokes flow past a spherical particle in infinite fluid and near a plane wall

The motion of a spherical particle in infinite linear flow and near a plane wall, subject to the slip boundary condition on both the particle surface and the wall, is studied in the limit of zero Reynolds number. In the case of infinite flow, an exact solution is derived using the singularity representation, and analytical expressions for the force, torque, and stresslet are derived in terms of slip coefficients generalizing the Stokes–Basset–Einstein law. The slip velocity reduces the drag force, torque, and the effective viscosity of a dilute suspension. In the case of wall-bounded flow, advantage is taken of the axial symmetry of the boundaries of the flow with respect to the axis that is normal to the wall and passes through the particle center to formulate the problem in terms of a system of one-dimensional integral equations for the first sine and cosine Fourier coefficients of the unknown traction and velocity along the boundary contour in a meridional plane. Numerical solutions furnish accurate predictions for (a) the force and torque exerted on a particle translating parallel to the wall in a quiescent fluid, (b) the force and torque exerted on a particle rotating about an axis that is parallel to the wall in a quiescent fluid, and (c) the translational and angular velocities of a freely suspended particle in simple shear flow parallel to the wall. For certain combinations of the wall and particle slip coefficients, a particle moving under the influence of a tangential force translates parallel to the wall without rotation, and a particle moving under the influence of a tangential torque rotates about an axis that is parallel to the wall without translation. For a particle convected in simple shear flow, minimum translational velocity is observed for no-slip surfaces. However, allowing for slip may either increase or decrease the particle angular velocity, and the dependence on the wall and particle slip coefficients is not necessarily monotonic.     

Film flow of a suspension down an inclined plane

A method is developed for simulating the film flow of a suspension of rigid particles with arbitrary shapes down an inclined plane in the limit of vanishing Reynolds number. The problem is formulated in terms of a system of integral equations of the first and second kind for the free-surface velocity and the traction distribution along the particle surfaces involving the a priori unknown particle linear velocity of translation and angular velocity of rotation about designated centres. The problem statement is completed by introducing scalar constraints that specify the force and torque exerted on the individual particles. A boundary-element method is implemented for solving the governing equations for the case of a two-dimensional periodic suspension. The system of linear equations arising from numerical discretization is solved using a preconditioner based on a particle-cluster iterative method recently developed by Pozrikidis (2000 Engng Analysis Bound. Elem. 25, 19-30). Numerical investigations show that the generalized minimal residual (GMRES) method with this preconditioner is significantly more efficient than the plain GMRES method used routinely in boundary-element implementations. Extensive numerical simulations for solitary particles and random suspensions illustrate the effect of the particle shape, size and aspect ratio in semi-finite shear flow, and the effect of free-surface deformability in film flow.     

Dynamic simulation of the flow of suspensions of two-dimensional particles with arbitrary shapes

A boundary-element method is implemented for simulating the motion of two-dimensional rigid particles with arbitrary shapes suspended in a viscous fluid in Stokes flow. The numerical implementation results in a system of linear equations for the components of the hydrodynamic traction over boundary elements distributed over the particle surfaces, and for the velocity of translation and angular velocity of rotation of the particles about designated centers. The linear system is solved by the method successive substitutions based on a physically motivated iterative procedure that involves decomposing the influence matrix into diagonal blocks consisting of physical particle clusters, and performing updates by matrix-vector multiplication using the inverses of the diagonal blocks. The iterations are found to converge as long as the particles are separated by a sufficiently large distance that depends on the particle shape and level of discretization. The stiffness of the governing equations due to lubrication forces developing between intercepting particles is removed by preventing the particles from approaching one another by less than a specified distance. Simulations are carried out for doubly-periodic suspensions of circular and elliptical particles in simple shear flow. The simulation algorithm is found to be successful for particles with moderate aspect ratios, and for small and moderate areal fractions up to 0.25. Higher areal fractions require long simulation times due to the large size of spontaneously forming particle clusters. The results illustrate the performance of various aspects of the boundary-element method and provide insights into the effect of the particle areal fraction and aspect ratio on the rheological properties of the suspension and on the geometrical properties of the evolving microstructure.     

Conductive transport through a mixed-matrix membrane

Conductive transport through an infinite homogeneous medium across a layer of finite thickness, a planar array of infinite cylinders, or a planar array of three-dimensional particles with arbitrary conductivity is considered as a model of mixed-matrix membrane separation. The boundary distribution of the transported scalar field on the interior side of the layer, cylinders, or particles is proportional to that on the exterior side according to a linear sorption/desorption kinetics law, while the conductive flux is continuous across the interface. In the case of cylinders and particles, the solution of Laplace’s equation for the transported field is represented by an interfacial distribution of point sources expressed in terms of the periodic Green’s function of Laplace’s equation in two dimensions or the doubly periodic Green’s function of Laplace’s equation in three dimensions. Analytical solutions for small circular cylinders and small spherical particles are derived based on the integral representation, and numerical solution of integral equations arising from the interfacial conditions are computed by boundary-element methods. The results document the displacement of the linear profile of the transported field far from the interfacial layer or array with respect to that prevailing in the absence of membrane. Expressions for the effective diffusivity of the mixed-matrix membrane are derived.     

A boundary-Integral Equation for Two-Dimensional Oscillatory Stokes Flow Past an Arbitrary Body

The fundamental singular velocity and pressure fields generated by the presence of an isolated line force acting at a point in a two-dimensional unbounded viscous incompressible medium executing oscillatory motions are used to formulate an integral equation which governs the flow past an arbitrarily shaped body. The Fredholm integral equation of the first kind is then solved by means of a boundary-element method, for the translational oscillatory flow past circular, elliptic and orthogonally intersecting cylinders. The asymptotic behaviour of the force on the cylinder for large values of the frequency parameter is obtained.     

On the magneto-hydrodynamic flow past a rotating disk in a fluid-particle suspension with a circular magnetic field

The flow of an incompressible, viscous, electrically conducting fluid with a suspension of inert particles over a rotating disk in the presence of a circular magnetic field is investigated. The governing equations of motion are reduced to a set of nonlinear ordinary differential equations by similarity transformations, and solved numerically by using least squares finite element method. The radial velocity of the panicles attains its maximum on the surface of the disk and the particles slip in the tangential direction. The flow boundary layer is thickened and the axial flow field is reduced as a result of the magnetic field. The particle density is maximum near the surface of the disk.     

Accurate treatment of lubrication forces between rigid spheres in viscous fluids using a traction-corrected boundary element method

Traditional boundary element methods cannot accurately resolve lubrication forces in the interstitial regions between nearly touching particles in viscous flows. In many cases, the interstitial tractions are underestimated and the relative particle velocities are overestimated resulting in significant errors in predicting particle trajectories. In order to accurately treat the lubrication forces between nearly touching particles, a traction-corrected boundary element method (TC-BEM) for multiple particles is developed by combining the analytical asymptotic solution for the tractions in the interstitial regions with the boundary element method. An adaptive meshing algorithm is developed to provide appropriate meshes on surfaces of particles with close interactions. The numerical method also employs an efficient parallelization scheme to make possible prediction of long-time behavior of particles suspended in viscous flow fields. The results of the TC-BEM are benchmarked by comparisons to analytical results for two particles in a linear shear flow and by considering the reversibility of three particles in a circular Couette flow. It is shown that the TC-BEM is able to correctly resolve the lubrication forces between nearly touching particles, thus enabling the accurate analysis of particles suspended in nonlinear shear flows.     

High-velocity transport of nanoparticles through 1-D nanochannels at very large particle to channel diameter ratios

We explore the possibility of generating high-velocity flows of nanoparticles through flat-rectangular nanochannels, which are only 50% deeper than the diameter of the particles. Using the shear-driven flow principle, 200-nm particles can, for example, be transported through a 300-nm-deep channel at velocities up to 35 mm/s (upper limit of our current setup). Working under high-pH conditions, the velocity of the carboxylated nanoparticles still respects the small-molecule velocity law, despite the high degree of confinement to which the particles are subjected. The high degree of confinement is also found to lead to a reduced band broadening. When injecting sharply delimited particle plugs, the plate heights observed for the flow of 0.2-microm particles through a 0.3-microm channel (with plate heights of the order of 1-2 microm) are, for example, approximately 1 order of magnitude smaller than for the flow of 1.0-microm particles through a 1.4-microm channel. It is also found that the band broadening is, within its statistical variation, independent of the fluid velocity over a large range of particle velocities (5-35 mm/s). The flow method distinguishes itself from pressure-driven field-flow fractionation and hydrodynamic chromatography in that the mean particle velocity is independent of the particle size over the entire range of possible particle to channel diameter ratios.     

A parametric study of dispersed laminar gas-particle flows through vertical and horizontal channels

Particle diameter, particle phase material density and inlet particle volume fraction are three important parameters governing the flow physics of dispersed gas-particle flows. In this work, an inhouse numerical solver is developed to investigate the effects of particle diameter (Stokes number), particle phase material density, inlet particle volume fraction and inlet phase velocities in the flow characteristics of gas-particle flows through vertical and horizontal channels and also in open domains. It is found that, for a constant inlet particle volume fraction, lower diameter particles attain a higher steady state velocity at any section inside the channel than the higher diameter particles; while the corresponding steady state gas velocity at any section increases with increase in particle diameter. On the other hand, for a constant particle diameter, the steady state gas phase velocity at any section decreases with increase in inlet particle volume fraction. Significant changes in both gas and particle velocity and volume fraction profiles have also been observed with inlet slip, i.e., when the velocities of both the phases at inlet are distinct as opposed to being equal, keeping all other flow and physical parameters invariant.     

Capillary attraction of floating rods

The capillary attraction of two parallel cylinders with circular cross-section representing slender particles floating at the interface between two immiscible fluids is considered. Given the particle separation, the elevation of the particle centers in hydrostatics is computed to satisfy the vertical force balance involving the buoyancy force, the capillary force, and the particle weight. A numerical procedure is developed for calculating the horizontal force exerted on a pair of cylinders in solitary or periodic arrangement. The results confirm that the particles attract each other under the conditions considered. The particle motion and transient flow due to the particle attraction are computed using a boundary-integral method for Stokes flow. In the algorithm, the particle center velocity of translation and angular velocity of rotation are calculated to satisfy force and torque balances. Numerical simulations using a boundary-element method subject to an initial state provided by hydrostatics illustrate the nature of the motion and furnish estimates for the particle velocity induced by capillarity.     

Using the annular shear cell as a rheometer for rapidly sheared granular materials: a DEM study

Discrete element simulations are performed to examine the kinematics of granular shear flows in an annular shear cell at high shearing rates. The interstitial fluid is absent and gravity is included. To investigate the feasibility of using annular shear cells as rheometers for rapidly sheared dense granular materials, this study focuses on the coupled effect of boundary conditions and the relative particle to shear cell size. Four different particle diameters and three different boundary types are used in the same annular shear cell. These cases correspond to physical experiments reported earlier by the authors. For many cases both shearing and non-shearing regions coexist. The transition from partially to fully shearing flow is shown to depend on the particle diameter, solids concentration, and the boundary conditions. The particles form layers at high solids concentration and with larger particles, as evidenced by the reduction of the flow diffusivity. The slip velocity at the bottom boundary is absent; at the top it varies. This variation is sensitive to the type of boundaries but insensitive to bulk solids concentration. This study shows the interconnectivity of the boundary, the particle to shear cell size, and the flow condition in an annular shear cell. Prior to using these cells as rheometers, a thorough understanding of this interconnectivity needs to be developed.     

Influence of parallelogram cells in the axial behaviour of fibrous composite

Effective longitudinal shear moduli closed-form analytical expressions of two-phase fibrous periodic composites are obtained by means of the asymptotic homogenization method (AHM) for a parallelogram array of circular cylinders. This work is an extension of previous reported results, where elastic, piezoelectric and magneto-electro-elastic composites for square and hexagonal arrays with perfect contact were considered. The constituents exhibit transversely isotropic properties. A doubly period-parallelogram array of cylindrical inclusions under longitudinal shear is studied. The behaviour of the anisotropic shear elastic coefficients is studied for several cell geometry arrays. Numerical examples and comparisons with other theoretical results demonstrate that the present model is efficient for the analysis of composites in which the periodic cell is rectangular, rhombic or a parallelogram. The effect of the arrangement of the cells on the shear effective property is discussed. The present method can provide benchmark results for other numerical and approximate methods.     

Wall slip phenomena in talc-filled polypropylene compounds

Slip velocities of unfilled and talc-filled polypropylene (PP) compounds, detectable at the die wall during pressure driven shear flow, have been determined using capillary rheometry. The presence of low molar mass, polar additives is responsible for the detection of wall slip in unmodified PP. Slip velocity increases with shear stress, beyond the critical onset condition. Increasing talc concentration in the PP compounds reduces slip velocity systematically, according to the talc volume fraction, whilst talc particle morphology appears to modify the wall slip behaviour to a greater extent than particle size. In comparison to PP-talc composites based on untreated filler, the presence of surface coatings tends to increase wall slip velocity, at any given shear stress, when the coating concentration exceeds monolayer level. These observations are explained in terms of a mechanism for wall slip in a low cohesive strength interphase, rich in low molar mass amide species, close to the flow boundary. This behaviour has also been modelled using a power law, to define wall slip parameters as a function of shear stress and talc concentration that can be used to enhance process simulation. It is demonstrated that the onset and magnitude of wall slip may be controllable by compound formulation and process conditions, creating exploitation potential to enhance process control and product properties of particle-modified PP composites.     

大悬浮轻颗粒固液两相流中的有序结构

应用三维颗粒图像跟踪技术,对竖直管内向上大悬浮轻颗粒固液两相流中分散相即颗粒相瞬时分布进行非接触测量,由此对顺流方向颗粒串组成的有序相分布结构进行观察研究。实验发现,当液体流动速度大于某一确定值时会有明显的颗粒串出现,此时颗粒由于受液体速度梯度诱导的强升力作用而紧贴管壁运动;随着液体流动速度的降低,颗粒串逐渐消失而颗粒沿管径向的分布会向着管中心方向发展;当液流速度进一步降低,颗粒开始在水平方向团聚。分析表明液体流动的剪切作用是颗粒串生成和稳定的机制。实验还显示,随着颗粒相平均份额的增加,流动中串间颗粒的相互作用加强,颗粒分布结构也随之受到影响。     

Interception of two spherical drops in linear Stokes flow

A boundary-integral method is developed for computing the interception of two spherical drops with arbitrary radii and viscosities in infinite linear Stokes flow. At any instant, the flow is computed in a frame of reference with origin at the center of one drop, using a cylindrical polar coordinate system whose axis of revolution passes through the center of the second drop. Taking advantage of the axial symmetry of the interfaces in the drop coordinates, the problem is formulated as a system of integral equations for the zeroth, first, and second Fourier coefficients of the normal component of the jump in the interfacial traction and for the meridional and azimuthal components of the interfacial velocity with respect to the meridional angle. The integral equations are solved with high accuracy using a boundary-element method featuring adaptive boundary-element distribution and automatic time-step adjustment according to the interfacial gap. Simulations of two drops intercepting in uniaxial straining flow provide accurate data on the drop collision velocity and particle stress tensor for gaps as small as 10⁻⁴ times the drop radius. Simulations of two drops intercepting in simple shear flow confirm that slightly offset drops collide during the interception. Accurate data are presented for Batchelor’s relative mobility functions in linear Stokes flow used to describe the relative droplet motion.     

Motion of a spherical particle inside a liquid film

The motion of a spherical particle inside a liquid film coated on a plane wall is considered under conditions of Stokes flow in the limit of vanishing capillary number where the interfacial deformation is infinitesimal. The problem is formulated in terms of a system of one-dimensional integral equations for the velocity and traction Fourier coefficients along the trace of the interface, wall, and particle contour in a meridional plane, and the solution is found using a boundary-element method. Comprehensive results for the force and torque resistance coefficients are presented in the case of particle rotation and translation in quiescent fluids. The velocity of translation and angular velocity or rotation of a freely suspended particle in simple shear flow are computed and discussed over a broad range of conditions.     

A fast integral approach for drag force calculation due to oscillatory slip stokes flows

In this paper, we describe an efficient numerical method for modelling oscillatory incompressible slip Stokes flows in three dimensions. The efficiency is achieved by employing an integral approach combined with an accelerated boundary‐element‐method (BEM) solver. First the integral representations for slip flows with two different slip models are formulated. The resulting integral equations are then solved using the BEM combined with the precorrected‐FFT accelerated technique. 3D numerical codes have been developed based on the method described above. These codes are then used to calculate the drag forces on oscillating objects immersed in an unbounded slip flow. Three objects are considered, namely a sphere, a pair of plates and a comb structure. The simulated drag forces on these objects obtained from the two slip models are compared. In the sphere case, the simulated results are also compared with the analytical solutions for both the steady‐state case and the no‐slip oscillatory case and are found to be in good agreement. In addition, qualitative comparison of the simulation results with the experimental results in the plate problem is also presented in this paper. Copyright © 2004 John Wiley & Sons, Ltd.