The effect of slip on the motion of a sphere close to a wall and of two adjacent spheres

Summary The motion of a sphere towards a plane or another sphere is opposed by the fluid between them with a force which is inversely proportional to the gap. In consequence, it is impossible for a constant force to produce contact in a finite time, unless the Stokes equations are modified. When the gap is of the same order as the mean free path of the air molecules, the Stokes theory for the motion of the air must be modified. The Maxwell slip flow approximation is used in this paper to show that, when the gap is small, the resisting force between the approaching surfaces becomes only logarithmically dependent on the gap, and contact can be achieved in a finite time. The difficulty in applying the Stokes theory to the problem of determining collision efficiencies for cloud droplets is thereby removed.The calculated values of the resistance to approach are used to determine the motion of a sphere falling towards a plane. If the motion is compared with the corresponding motion when no allowance is made for slip flow, the sphere without slip would still be at a distance of 1.3 times the mean free path from the plane, when the sphere with slip has made contact.Transverse motion must also be considered if the trajectory of a particle close to a collector is required. The forces and couples on the sphere in that situation have a logarithmic dependence on the gap without slip, but they tend to constant values when the effect of slip is included. Some calculations of collision efficiency of drops falling under gravity (Hocking and Jonas [1]) have been amended to include the effect of slip when the colliding drops are very close together, and show a significant increase in the collision efficiency.     

Dynamic simulation of rod-like and plate-like particle dispersed systems

A particle simulation method (PSM) is presented to simulate the dynamics of rod-like and plate-like particle dispersed systems. In this method, the particle is modeled with arrays of spheres connected by three types of springs. The motion of particles in flow is followed by solving the translational and rotational equations of motion for each constituent sphere. The mobility matrix for each particle is calculated to obtain the hydrodynamic force and torque exerted on each sphere. For the hydrodynamic interaction among particles, the near-field lubrication force is considered. The method was applied to the simulation of the transient behavior of particles in a shear flow by dispersing them into a cell with periodic boundaries. In semi-dilute to concentrated systems, the overshoot of viscosity was observed for rigid rod-like particle dispersed systems, but not for flexible ones. This was due to the transient change of the microstructure from the flow-directional orientation to the planar one of particles. The normal stress appeared in the flexible particle dispersed systems because of the deformation of particles. In the rectangular plate-like particle dispersed system, the planar orientation of particles was observed and furthermore the orientation of the major axis of particles in the shear direction appeared.     

A note on the drag force on a spherical particle in a quadratic flow of a micropolar fluid

The motion of a rigid sphere, suspended in a micropolar fluid which is undergoing a slow unidirectional two-dimensional flow, is studied. The expression for pressure, velocity and the spin in the fluid and for the force on the sphere are obtained in the closed form. It is noted that the consideration of the substructure in the fluid is to increase the drag force on the sphere.     

Direct numerical simulation of two-dimensional channel flows of electro-rheological fluids

Direct numerical simulation (DNS) of electro-rheological (ER) fluid flows in two-dimensional (2D) electrode channel has been performed by adopting a combined finite element method (FEM). Hydrodynamic interactions between the particles and the fluid are described by the Navier-Stokes equations for the fluid in combination with the equations of motion for the particles, while the multi-body electrostatic interaction is represented by the point-dipole model.ER effects on the plane channel flow for a given pressure gradient have been studied by varying the Mason number and volume fraction of the particles, and interrogating the motion of the particles in views of the formation of ER chain structures, the fluid velocity profile in the channel, and the shear stress versus the shear rate. As the Mason number decreases and volume fraction increases, the tendency that particles align to form chain structures becomes stronger. The yield stress of the ER fluid increases with the electric field intensity and the particle concentration. The quadratic correlation between the yield stress and the electric field intensity has been extracted from the present direct numerical simulation. Lastly, it has been shown that the yield stress linearly increases with the volume fraction in the intermediate range.     

Stokes flow with slip caused by the axisymmetric motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid

The first part of this paper investigates the motion of a solid spherical particle in an incompressible axisymmetric micropolar Stokes flow. A linear slip, Basset-type, boundary condition has been used. Expressions for the drag force and terminal velocity has been obtained in terms of the parameter characterizing the slip friction. In the second part, we consider the flow of an incompressible axisymmetrical steady semi-infinite micropolar fluid arising from the motion of a sphere bisected by a free surface bounding a semi-infinite micropolar fluid. Two cases are considered for the motion of the sphere: perpendicular translation to the free surface and rotation about a diameter which is also perpendicular to the free surface. The speed of the translational motion and the angular speed for the rotational motion of the sphere are assumed to be small so that the nonlinear terms in the equations of motion can be neglected under the usual Stokesian approximation. Also a linear slip, Basset-type, has been used. The analytical expressions for velocity and microrotation components are determined in terms of modified Bessel functions of second kind and Legendre polynomials. The drag for the translation case and the couple for the rotational motion on the submerged half sphere are calculated and expressed in terms of nondimensional coefficients whose variation is studied numerically. The variations of the drag and couple coefficients with respect to the micropolarity parameter and slip parameter are tabulated and displayed graphically.     

Slow flow of a non-Newtonian liquid past a fluid sphere

Summary The motion of a non-Newtonian fluid past a Newtonian fluid sphere has been investigated using the Stokes approximation. The stream functions characterizing the internal and external flow fields have been determined and the special case of flow past a solid sphere is deduced. The drag experienced by the fluid sphere has been evaluated and found to be greater than the classical counterpart.     

Coupling of membrane element with material point method for fluid–membrane interaction problems

This paper proposes a coupled particle–finite element method for fluid–membrane structure interaction problems. The material point method (MPM) is employed to model the fluid flow and the membrane element is used to model the membrane structure. The interaction between the fluid and the membrane structure is handled by a contact method, which is implemented on an Eulerian background grid. Several numerical examples, including membrane sphere interaction, water sphere impact and gas expansion problems, are studied to validate the proposed method. The numerical results show that the proposed method offers advantages of both MPM and finite element method, and it can be used to simulate fluid–membrane interaction problems.     

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.     

Motion of a rigid sphere in a shear field in a micropolar fluid

The motion of a rigid sphere, suspended in a micropolar fluid which is undergoing a shearing motion, is discussed. The expressions for the pressure, velocity and spin in the fluid and those for the force and torque on the sphere are obtained. A compromise boundary condition, relating the spin of the particle with the vorticity vector at the boundary, is employed. The results are compared with the classical values and apart from other interesting observations, it is noted that the torque on the sphere depends upon the various parameters in a complicated manner. By extending the definitions of the effective viscosity for the viscous fluids, an expression for the viscosity of the suspension in the micropolar fluid is derived.     

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.     

On a local predictor of shear instability and the initiation of local turbulence

The stability of a local laminar shear flow and its transition into turbulent flow is considered as a local phenomenon. This transition may remain local, in which case the flow field is partially laminar and partially turbulent, or it may spread and make the whole field turbulent. One of the applications of this analysis is the prediction of local heat-convection rates, which are enhanced by local turbulence. Another application is in heart-lung blood pumps, where excessive shear rates are detrimental to red blood cells.The analysis is Lagrangian, which concentrates on the stability of a fluid particle in maintaining its position in a laminar shear flow. This stability is shown to depend on the magnitude of a non-dimensional parameter, namely the local Reynolds numberRe _L =ha ²/v whereh is the local shear rate,a is the particle radius andv is the fluid's kinematic viscosity. It is shown that when, locally,Re _L > 530, the flow is, locally, unstable. The application of this criterion is simple and direct, and in certain cases it can be shown that the resulting unstable flow is indeed turbulent.Because the analysis relies on an experimental coefficient which has been obtained for a rigid sphere, rather than for a fluid particle, the criterion is introduced at this stage as a conjecture. Several examples are presented which demonstrate the criterion's ability to yield correct predictions for instability.     

On the slow motion of a sphere in fluids with non-constant viscosities

We study a slowly moving sphere in fluids where the viscosity depends upon factors such as shear-rate, temperature and pressure, with the flow field approximated by the Stokes flow past a sphere. We derive an expression for the stresses generated in the fluid due to these various factors. This gives us information about both, the force imposed by the fluid upon the sphere and also the reaction force due to the sphere upon the fluid, referred to as the stress density. The values of the force and stress density are numerically computed in each of the cases and analyzed for various values of the flow and material parameters. Our computations show interesting variations in the distribution of stress density in the fluid for the various cases and also give us valuable information about the effect of walls. Our calculations also indicate that particle heating or cooling can serve as a significant control parameter since the drag force upon the sphere increases dramatically for a cold particle and can be reduced considerably upon heating it.     

Unsteady MHD rotating flow over a rotating sphere near the equator

The unsteady rotating flow of a laminar incompressible viscous electrically conducting fluid over a rotating sphere in the vicinity of the equator has been studied. The fluid and the body rotate either in the same direction or in opposite directions. The effects of surface suction and magnetic field have been included in the analysis. There is an initial steady state that is perturbed by a sudden change in the rotational velocity of the sphere, and this causes unsteadiness in the flow field. The nonlinear coupled parabolic partial differential equations governing the boundary-layer flow have been solved numerically by using an implicit finite-difference scheme. For large suction or magnetic field, analytical solutions have also been obtained. The magnitude of the radial, meridional and rotational velocity components is found to be higher when the fluid and the body rotate in opposite directions than when they rotate in the same direction. The surface shear stresses in the meridional and rotational directions change sign when the ratio of the angular velocities of the sphere and the fluid ₀. The final (new) steady state is reached rather quickly which implies that the spin-up time is small. The magnetic field and surface suction reduce the meridional shear stress, but increase the surface shear stress in the rotational direction.     

Hydrodynamic interaction between two red blood cells in simple shear flow: its impact on the rheology of a semi-dilute suspension

Blood is a suspension of red blood cells (RBCs) and its rheology is important when discussing the physiology of the cardiovascular system. In this study, we performed a numerical investigation of the rheological properties of an RBC suspension from the dilute to semi-dilute regime. RBCs were modelled as a capsule with a two-dimensional hyperelastic membrane. Large deformation of the thin membrane was calculated by a finite element method. Due to the small size of the RBC, fluid motion around the RBC was assumed to follow Stokes flow and was solved by a boundary element method. In the dilute limit, cell–cell interactions were omitted and the bulk stress of the suspension was calculated by the stresslet generated on a single RBC. Interestingly, the effective shear viscosity of the dilute suspension decreased with increasing viscosity of the internal liquid. In the semi-dilute regime, cells can be considered as showing pairwise interactions. The effective shear viscosity of the semi-dilute suspension shows a quadratic increase with respect to the volume fraction. These findings are important for understanding the complex phenomena of blood rheology.     

On some similarity solutions to shear flows of non-Newtonian power law fluids

Summary The nonlinear rheological effects of non-Newtonian power law fluids on some shear flows are addressed. Exact similarity solutions to Stokes' first problem for unsteady flow generated by the vertical motion of a slender cylinder in an unbounded fluid are presented. The nonlinear effects on the velocity and shear stress distributions are shown and discussed. These reveal the existence of traveling wave characteristics for a shear thickening fluid, which determine a moving shear front for which the shear disturbances propagate with a finite velocity.     

PARTICULATE MOTION AND CONCENTRATION FIELDS IN CENTRIFUGAL SLURRY PUMPS

Erosion wear of centrifugal slurry pumps is primarily governed by the particulate motion and concentration as well as their physical properties. This paper presents a quasi-3D approach to predict particulate phase motion and concentration in an arbitrary radial section of a centrifugal slurry pump. A brief discussion of the fully developed turbulent flow solution of the carrier fluid is followed by a computation of the particulate phase velocity resulting from a force balance between the pressure, gravity, viscous and inertial effects. The concentration distribution is obtained by invoking the convection-diffusion equation. The governing partial differential equation is cast into a weak Galerkin finite element form. The system of algebraic equations is solved by a Newton-Raphson scheme via a frontal solver. An iterative solution scheme is employed to alternate between the fields of particle motion and concentration. Numerical solutions are examined in light of their applicability to the solution of the pump wear problem.     

On the slow motion of a solid submerged in a fluid with a surfactant surface film

Summary This paper continues the work of Shail and Gooden [1–4] on the motion generated in a semi-infinite fluid by a singularity or submerged solid moving particle when the surface of the fluid is contaminated with a surfactant film. The fluid motion is assumed to be slow and quasi-steady, but the restriction to axially symmetric flows of earlier investigations is removed. The various linearised models of Shail and Gooden [3,4] governing the variation of film concentration are discussed, the constitutive properties of the film being expressed in terms of Boussinesq coefficients of surface shear and dilatational viscosities. The resulting film boundary conditions are applied to solve the non-axially symmetric problem of a Stokeslet placed in the bulk fluid with its axis parallel to the surface (assumed planar throughout the motion), and the results used to calculate approximate expressions for the resistive force on a particle which translates far from and parallel to the surface. A similar analysis is given for the case of a rotelet whose axis is parallel to the surface.     

PARTICULATE MOTION AND CONCENTRATION FIELDS IN CENTRIFUGAL SLURRY PUMPS

ABSTRACT

Erosion wear of centrifugal slurry pumps is primarily governed by the particulate motion and concentration as well as their physical properties. This paper presents a quasi-3D approach to predict particulate phase motion and concentration in an arbitrary radial section of a centrifugal slurry pump. A brief discussion of the fully developed turbulent flow solution of the carrier fluid is followed by a computation of the particulate phase velocity resulting from a force balance between the pressure, gravity, viscous and inertial effects. The concentration distribution is obtained by invoking the convection-diffusion equation. The governing partial differential equation is cast into a weak Galerkin finite element form. The system of algebraic equations is solved by a Newton-Raphson scheme via a frontal solver. An iterative solution scheme is employed to alternate between the fields of particle motion and concentration. Numerical solutions are examined in light of their applicability to the solution of the pump wear problem.     

Experimental investigation of bubble and particle motion behaviors in a gas-solid fluidized bed with side wall liquid spray

Bubble and particle motion behaviors are investigated experimentally in a gas solid fluidized bed with liquid spray on the side wall. The particles used in the experiment are classified as Geldart B particles. The results reveal that when the fluid drag force is less than the liquid bridge force between particles, liquid distribute all over the bed. Bubble size increases as the increase of inter-particle force, then decreases owing to the increase of particle weight with increasing liquid flow rate. When the fluid drag force is greater than the liquid bridge force, liquid mainly distribute in the upper part of the bed. And it is difficult for the wet particles to form agglomerates. Bubble size decreases with increasing liquid flow rate due to the increasing of minimum fluidization velocity. Besides, the acoustic emission (AE) measurements illustrate that the liquid adhesion and evaporation on particles could enhance the particles motion intensity. Consequently, the bubble and particle behaviors change due to the variation in fluidized gas velocity and liquid flow rate should be seriously considered when attempting to successfully design and operate the side wall liquid spray gas solid fluidized bed.     

Swirl-flow effects in a duct bending through a substantial angle

The vortical three-dimensional flow of an inviscid incompressible fluid through a bend in a slender duct is investigated with nonlinear coupling between the centre-line and cross-sectional velocities. The bend, although gradual, turns the duct flow through a finite angle typically. The theoretical work is motivated directly by the industrial application to food sorting, involving the rapid firing of impulsive ejectors and sprinklers which combine short time scales and flow turbulence, and indirectly by other applications. The flow response depends largely on the relative radius of curvature in the bend along with the cross-sectional swirl and the shear in the perturbed streamwise velocity profile at the entrance. Solutions for the motion within and beyond the bend are determined computationally through a fourth-order compact scheme, for both rectangular and smooth cross-sections, and analytically for a general cross-section, under varying entrance conditions (especially for increasing swirl there, although a long bend for example also enhances swirl downstream). The total streamwise vorticity is found analytically to increase in direct proportion to the distance along the bend, whether the inertial effects are weak or strong. The smallness of the pressure loss also contrasts with that in viscous flows.