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.     

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.     

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.     

Slow motion of a slip spheroid along its axis of revolution

A combined analytical and numerical study of the Stokes flow caused by a rigid spheroidal particle translating along its axis of revolution in a viscous fluid is presented. The fluid is allowed to slip at the surface of the particle. The general solution for the stream function in prolate and oblate spheroidal coordinates can be expressed in an infinite-series form of semi-separation of variables. The slip boundary condition incorporating the shear stress at the particle surface is applied to this general solution to determine its unknown coefficients of the leading orders. The solution of these coefficients can be either numerical results obtained from a boundary-collocation method or explicit formulas derived analytically. The drag force exerted on the spheroidal particle by the fluid is evaluated with good convergence behavior for various values of the slip parameter and aspect ratio of the particle. The agreement between our hydrodynamic drag results and the relevant numerical solutions obtained previously using a singularity method is excellent. Although the drag force acting on the translating spheroid normalized by that on a corresponding sphere with equal equatorial radius increases monotonically with an increase in the axial-to-radial aspect ratio for a no-slip spheroid, it decreases monotonically as this aspect ratio increases for a perfect-slip spheroid. The normalized drag force exerted on a spheroid with a given surface slip coefficient in between the no-slip and perfect-slip limits is not a monotonic function of its aspect ratio. For a spheroid with a fixed aspect ratio, its drag force is a monotonically decreasing function of the slip coefficient of the particle.     

Torque on a slip sphere rotating in a semi-infinite micropolar fluid

In this paper, the steady rotational motion of a slip sphere in a semi-infinite micropolar fluid is investigated. The sphere is assumed to rotate about a diameter perpendicular to an impermeable plane wall. The slip and spin boundary conditions are imposed on the spherical particle surface while on the plane wall surface the classical no-slip and no-spin conditions are utilized. A semi-analytical technique based on the principle of superposition together with a numerical method, called the collocation method, is employed to obtain the hydrodynamic torque acting on the spherical particle. Numerical results for the torque are obtained and illustrated graphically.     

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.     

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.     

The motion of a closely-fitting sphere in a fluid-filled tube

Singular perturbation techniques are used to investigate the slow, asymmetric flow around a sphere positioned eccentrically within a long, circular, cylindrical tube filled with viscous fluid. The results apply to situations in which the sphere occupies virtually the entire cross section of the cylinder, so that the clearance between the particle and tube wall is everywhere small compared with both the sphere and tube radii. The technique is an improvement over conventional “lubrication-theory” analyses.Asymptotic expansions, valid for small dimensionless clearances, are obtained for the hydrodynamic force, torque and pressure drop for flow past a stationary sphere, as well as for the case of a sphere translating or rotating in an otherwise quiescent fluid. These expansions are employed to predict the macroscopic behavior of both a neutrally-buoyant sphere suspended in a Poiseuille flow, and a sedimenting sphere in a vertical tube.The results find application in capillary blood flow, pipeline transport of encapsulated materials, and falling-ball viscometers.     

Hydrodynamic force and torque models for a particle moving near a wall at finite particle Reynolds numbers

This research work is aimed at proposing models for the hydrodynamic force and torque experienced by a spherical particle moving near a solid wall in a viscous fluid at finite particle Reynolds numbers. Conventional lubrication theory was developed based on the theory of Stokes flow around the particle at vanishing particle Reynolds number. In order to account for the effects of finite particle Reynolds number on the models for hydrodynamic force and torque near a wall, we use four types of simple motions at different particle Reynolds numbers. Using the lattice Boltzmann method and considering the moving boundary conditions, we fully resolve the flow field near the particle and obtain the models for hydrodynamic force and torque as functions of particle Reynolds number and the dimensionless gap between the particle and the wall. The resolution is up to 50 grids per particle diameter. After comparing numerical results of the coefficients with conventional results based on Stokes flow, we propose new models for hydrodynamic force and torque at different particle Reynolds numbers. It is shown that the particle Reynolds number has a significant impact on the models for hydrodynamic force and torque. Furthermore, the models are validated against general motions of a particle and available modeling results from literature. The proposed models could be used as sub-grid scale models where the flows between particle and wall can not be fully resolved, or be used in Lagrangian simulations of particle-laden flows when particles are close to a wall instead of the currently used models for an isolated particle.     

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.     

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.     

Motion of a circular cylinder in a viscous liquid between parallel plates

The two-dimensional motion of a cylinder in a viscous fluid between two parallel walls of a vertical channel is studied. It is found that when the cylinder moves very closely along one of the channel walls, it always rotates in the direction opposite to that of contact rolling along the nearest wall. When the cylinder is away from the walls, its rotation depends on the Reynolds number of the flow. In this study two numerical methods were used. One is for the unsteady motion of a sedimenting cylinder initially released from a position close to one of the channel walls, where the Navier-Stokes equations are solved for the fluid and Newton's equations of motion are solved for the rigid cylinder. The other method is for the steady flow in which a cylinder is fixed in a uniform flow field where the channel walls are sliding past the cylinder at the speed of the approaching flow, or equivalently a cylinder is moving with a constant velocity in a quiescent fluid. The flow field, the drag, the side force (lift), and the torque experienced by the cylinder are studied in detail. The effects of the cylinder location in the channel, the size of the channel relative to the cylinder diameter, and the Reynolds number of the flow are examined. In the limit when the cylinder is translating very closely along one of the walls, the flow in the gap between the cylinder and the wall is solved analytically using lubrication theory, and the numerical solution in the other region is used to piece together the whole flow field.This research was supported by NSF DMR91-20668 through the Laboratory for Research on the Structure of Matter at the University of Pennsylvania and from the Research Foundation of the University of Pennsylvania.     

Two-dimensional approach to the motion of a red blood cell in a plane couette flow of plasma

In relation to microrheology of blood, a theoretical approach to the motion of a red blood cell in a plane Couette flow between two parallel plates is made with emphasis on effects of wall. The red blood cell is assumed to be an elliptic cylindrical particle with a thin, inextensible membrane moving like a tank-tread along its perimeter and to contain a Newtonian fluid inside. Fluid motions are analysed numerically both inside and outside the particle on the basis of the Stokes equations, using the finite element method.A quasi-static equilibrium condition leads to the solution for the motion of the particle. It is shown that two types of motion exist (a stationary orientation motion and a flipping motion), depending on the viscosity ratio of inner to outer fluid, the axis ratio of the elliptic cylinder and the ratio of particle size to channel width. The results are applied to capillary blood flow.     

Transverse radiation pressure and torque exerted by a plane electromagnetic wave on a rotating circular cylinder of isotropic material

Relativistically correct expressions are derived for the time-averaged total force and torque acting on a rotating conducting circular cylinder, immersed in an incident plane electromagnetic wave. With the help of Maxwell's stress tensor analytical expressions can be obtained for the time-averaged force and torque.     

Creeping-flow rotation of a slip spheroid about its axis of revolution

The problem of rotation of a rigid spheroidal particle about its axis of revolution in a viscous fluid is studied analytically and numerically in the steady limit of negligible Reynolds number. The fluid is allowed to slip at the surface of the particle. The general solution for the fluid velocity in prolate and oblate spheroidal coordinates can be expressed in an infinite-series form of separation of variables. The slip boundary condition on the surface of the rotating particle is applied to this general solution to determine the unknown coefficients of the leading orders, which can be numerical results obtained from a boundary collocation method or explicit formulas derived analytically. The torque exerted on the spheroidal particle by the fluid is evaluated for various values of the slip parameter and aspect ratio of the particle. The agreement between our hydrodynamic torque results and the available analytical solutions in the limiting cases is good. It is found that the torque exerted on the rotating spheroid normalized by that on a sphere with radius equal to the equatorial radius of the spheroid increases monotonically with an increase in the axial-to-radial aspect ratio for a no-slip or finite-slip spheroid and vanishes for a perfectly slip spheroid. For a spheroid with a specified aspect ratio, the torque is a monotonically decreasing function of the slip capability of the particle.     

Modeling of transverse self-oscillations of a circular cylinder in an incompressible fluid flow in a plane channel with circulation

Experimentally observed self-oscillations of a cylinder in a plane channel whose width is slightly greater than the cylinder diameter under the impact of the incoming fluid flow are modeled. Within the model of a nonseparated potential flow around the cylinder, the coefficients of added mass of the cylinder are calculated with the help of the generalized method of images. When the cylinder touches the channel wall, the circulation sign changes, and its value is determined by the boundary element method and the no-slip condition for the fluid at the contact point. The Joukowski force and the drag force proportional to the square of velocity are taken into account in the equations of motion of cylinder.     

On the orbital motion of a rotating inner cylinder in annular flow

In this paper, numerical calculations have been performed to analyse the influence of the orbital motion of an inner cylinder on annular flow and the forces exerted by the fluid on the inner cylinder when it is rotating eccentrically. The flow considered is fully developed laminar flow driven by axial pressure gradient. It is shown that the drag of the annular flow decreases initially and then increases with the enhancement of orbital motion, when it has the same direction as the inner cylinder rotation. If the eccentricity and rotation speed of the inner cylinder keep unchanged (with respect to the absolute frame of reference), and the orbital motion is strong enough that the azimuthal component (with respect to the orbit of the orbital motion) of the flow‐induced force on the inner cylinder goes to zero, the flow drag nearly reaches its minimum value. When only an external torque is imposed to drive the eccentric rotation of the inner cylinder, orbital motion may occur and, in general, has the same direction as the inner cylinder rotation. Under this condition, whether the inner cylinder can have a steady motion state with force equilibrium, and even what type of motion state it can have, is related to the linear density of the inner cylinder. Copyright © 2006 John Wiley & Sons, Ltd.     

Slow Motion of an Assemblage of Porous Spherical Shells Relative to a Fluid

The body-force-driven motion of a homogeneous distribution of spherically symmetric porous shells in an incompressible Newtonian fluid and the fluid flow through a bed of these shell particles are investigated analytically. The effect of the hydrodynamic interaction among the porous shell particles is taken into account by employing a cell-model representation. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field around a single particle in a unit cell, and the drag force acting on the particle by the fluid is obtained in closed forms. For a suspension of porous spherical shells, the mobility of the particles decreases or the hydrodynamic interaction among the particles increases monotonically with a decrease in the permeability of the porous shells. The effect of particle interactions on the creeping motion of porous spherical shells relative to a fluid can be quite significant in some situations. In the limiting cases, the analytical solution describing 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. The particle-interaction behavior for a suspension of porous spherical shells with a relatively low permeability may be approximated by that of permeable spheres when the porous shells are sufficiently thick.     

A unified view of energetic efficiency in active drag reduction,thrust generation and self-propulsion through a loss coefficient with some applications

An analysis of the energy budget for the general case of a body translating in a stationary fluid under the action of an external force is used to define a power loss coefficient. This universal definition of power loss coefficient gives a measure of the energy lost in the wake of the translating body and, in general, is applicable to a variety of flow configurations including active drag reduction, self-propulsion and thrust generation. The utility of the power loss coefficient is demonstrated on a model bluff body flow problem concerning a two-dimensional elliptical cylinder in a uniform cross-flow. The upper and lower boundaries of the elliptic cylinder undergo continuous motion due to a prescribed reflectionally symmetric constant tangential surface velocity. It is shown that a decrease in drag resulting from an increase in the strength of tangential surface velocity leads to an initial reduction and eventual rise in the power loss coefficient. A maximum in energetic efficiency is attained for a drag reducing tangential surface velocity which minimizes the power loss coefficient. The effect of the tangential surface velocity on drag reduction and self-propulsion of both bluff and streamlined bodies is explored through a variation in the thickness ratio (ratio of the minor and major axes) of the elliptical cylinders.     

Inertia and drag of elliptic cylinders oscillating in a fluid

The results of an experimental investigation of the hydrodynamic forces acting on elliptic cylinders oscillating in water at rest are presented. The coefficients of the inertia force and the drag are determined as functions of the ratio of the amplitude of the oscillations to the length of the axis of the cylindrical section in a plane normal to the direction of oscillation. These forces are also shown to depend significantly on the cylinder thickness. deceased Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 112–117, January–February, 1998.