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.     

Radiation and dynamics of a nanoparticle in equilibrium background radiation upon translational–rotational motion

We have obtained general expressions for the intensity of radiation and tangential force of a small polarizable particle in the process of translational–rotational motion in equilibrium radiation background (thermalized photon gas) of a certain temperature at an arbitrary relative orientation of the linear and angular velocity vectors. It is shown that, in a cold vacuum background, the translational velocity of particle is independent of time and the intensity of its spontaneous emission is determined by the angular velocity and imaginary part of the particle polarizability.     

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.     

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.     

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.     

Sphere rotating in a viscous fluid inside a cylindrical tube

Summary We consider the viscous liquid in Poiseuillian flow in an infinite circular cylinder of radiusR ₀ and a sphere of radiusa which is allowed to rotate with uniform angular velocity about the axis. Using the method of reflections, the effect of the inertia terms on the velocity field and on the force and torque on the body are calculated for small values ofa/R ₀.With 3 Figures     

Singularities of viscous flow

Summary The force and torque on a slender body in Stokes flow is obtained by using a distribution of singularities along the centreline and the required image system to satisfy the no-slip condition and the plane boundary. It is found that the force on a slender body increases rapidly as it approaches within a body length of the boundary, but not the torque. To rule out “wall effects”, experiments on slender bodies, such as flagellated or ciliated micro-organisms, should be carried out a distance of many body lengths (not radii!) from all boundaries. The ratio of normal to tangential resistance coefficients is found to be greater than 2 (the maximum in an infinite fluid) in the presence of walls.     

MHD Flow of an Oldroyd-B fluid between eccentric rotating disks

An exact solution is obtained for the magnetohydrodynamics (MHD) flow of a conducting, incompressible Oldroyd-B fluid between two infinite, parallel, insulated disks rotating about non-coincident axes normal to the disks in the presence of a uniform transverse magnetic field. The effects of the Hartmann number M, the Deborah number D, the Reynolds number R and the elastic number on the velocity field and the force are discussed. It is found that the value of the torque exerted by the fluid on one of the disks is zero.     

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.     

Flow and stability of a second grade fluid between two parallel plates rotating about noncoincident axes

An exact solution is given for the flow of a second grade fluid between two infinite parallel plates rotating about noncoincident axes normal to the plates. A comparison of the torque and force exerted by the fluid on the plates with the corresponding experimental values would enable one to determine the normal stress moduli. Using energy method, the stability of the above flow is investigated for disturbances of finite amplitude. It is found that the region of sure stability decreases with increase in the viscoelastic parameter.     

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.     

Adhesion of a blood platelet to injured tissue

A numerical study of platelet adhesion to injured endothelial wall during blood flow is conducted using the boundary-element method for Stokes flow. An idealised model is presented where the platelet is treated as an elliptical particle carried over a plane wall in simple shear flow. When the platelet is sufficiently close to the wall, adhesive bonds are established tethering the platelet via receptors distributed around its perimeter to ligands sited at specified locations on an injured section of the wall. A generalised boundary-integral equation of the second kind is formulated to determine the translational and angular velocities of the platelet and the force acting on the platelet due to adhesive bonds. Numerical simulations are performed for a small number of bonds behaving as simple springs, and the force required to capture and immobilise an elliptical particle is estimated. Further simulations conducted using an adhesive bond dynamics model show that bond formation operating at realistic biophysical parameter values can lead to platelet capture and arrest.     

Shear flow over a protuberance on a plane wall

Simple shear flow over a protuberance with an axisymmetric shape projecting from a plane wall with its axis normal to the wall is studied by means of a boundary-integral method that is suitable for computing three-dimensional Stokes flow in axisymmetric domains. The problem is formulated in terms of a system of three scalar Fredholm integral equations of the first kind for the distribution of traction over the surface of the protuberance and the wall, and is solved by means of a boundary-element method. Numerical computations are performed for a family of protuberances whose exposed surface is a section of a sphere or of an oblate spheroid with its minor axis normal to the wall, and the results are in agreement with those of previous analytical computations for hemi-spherical and spherical shapes. The numerical computations provide accurate information on the hydrodynamic force and torque exerted on the protuberances due to the shear flow, and the distribution of shear stresses, and illustrate the kinematical structure of the flow with reference to the development of stagnation points and flow reversal.     

A class of exact solutions to the Navier-Stokes equations

Recently, Berker established the existence of an infinite set of solutions to the flow of two infinite parallel plates rotating with the same angular velocity about an axis[4]. In this work we extend Berker's analysis by allowing the plates to be porous. We find that similar to Berker's results an infinite set of solutions is possible.     

Tangential velocity profiles of granular material within horizontal rotating cylinders modelled using the DEM

Flow regimes of granular materials in horizontal rotating cylinders are industrially important since they have a strong influence on the rates of heat and mass transfer within these systems. The tangential velocity profile, which describes how the average particle velocity in the direction parallel to the surface of the bed varies along a radius perpendicular to the surface of the bed, has been examined for many experimental and simulated systems. This paper is concerned with tangential velocity profiles within rotating cylinders simulated using the discrete element method. For high fill levels good agreement is found between the simulated velocity profiles and the equation proposed by Nakagawa et al. (Exp Fluids 16:54–60, 1993) based on magnetic resonance measurements. At lower fill levels slip is observed between the cylinder wall and the particles in contact with it and also between the outer layer of particles and the bulk of the bed. It is demonstrated that this slip occurs when the particles in contact with the wall are able to rotate and that it may be prevented either by using non-spherical particles or by attaching “lifters” to the cylinder wall.     

On the steady flow produced in fluid-particle suspension by an infinite rotating disk with surface suction

The steady flow of a fluid-particle suspension over an infinite rotating disk with uniform suction is considered. The equations of motion are reduced to ordinary differential equations by similarity transformations and solved numerically by using a least-squares finite-element method. Some typical results for both fluid and particle phases and density distributions of the particles are presented graphically for the suction parameter A = 3.0 in order to illustrate some interesting features of the solutions. It is observed that the radial velocity of the particle attains its maximum on the surface of the disk and the particles slip in the tangential direction on the disk. The magnitudes of the radial velocity components of both the fluid and particle phases are found to decrease rapidly as suction increases.     

平行圆盘间电流变液的挤压流研究

采用双粘度本构模型研究了两平行圆盘间电流变液的挤压流特性,壁面边界上采用Navier滑移模型.流场根据其特性被分为两个区域:对称轴附近的牛顿区以及远离对称轴的双粘区;双粘区存在屈服面.本文在牛顿区和双粘区分别求解出其速度场和压力梯度场.壁面上的滑移速度与当地的压力梯度成正比;而压力梯度在牛顿区与r成正比,在双粘区r值较大的地方与r近似成线性关系.通过将压力梯度在双粘区近似为r的线性函数,可积分出流场的压力分布与作用在圆盘上的挤压力.此外,本文还通过计算,考察了速度场的分布特点,分析了滑移系数对速度场、压力梯度场、屈服面位置以及挤压力的影响.     

Boundary element solution of thermal creep flow in microfluidic devices

Flow in rarefied gases can be caused by a tangential temperature gradient along the contour boundaries (tangential heat flux), without the presence of any other external driven force, inducing a fluid motion from colder to hotter regions. This phenomenon is known as thermal creep and has gained importance in recent years in connection with micro-scale gas flow systems. Prediction of the flow field in micro-systems can be obtained by using continuum based models under appropriate boundary conditions accounting for the slip velocity due to tangential shear rate and heat flux. In this work a boundary integral equation formulation for Stokes slip flow, based on the normal and tangential projection of the Green's integral representational formulae for the velocity field is presented. The tangential heat flux is evaluated in terms of the tangential gradient of the temperature integral representational formulae presenting singularities of the Cauchy type, which are removed by the use of an auxiliary potential field. These formulations are used to evaluate the performance of different microfluidic devices.     

Hydromagnetic flow past a rotating porous plate in a conducting fluid rotating about a noncoincident parallel axis

Summary The hydromagnetic flow of an incompressible viscous electrically conducting fluid past a porous plate is investigated when the plate rotates with a uniform angular velocity about an axis normal to the plate and the fluid at infinity rotates with the same angular velocity about a non-coincident parallel axis. It is shown that in the presence of a uniform magnetic field parallel to the axis of rotation the boundary layer thickness decreases with an increase in either the suction at the plate or the magnetic parameter M. In the presence of suction at the plate, the velocity component u in the direction normal to the plane containing the axis of rotation of the plate and that of the fluid increases with an increase in M, while the velocity component v in the transverse direction parallel to the plane of the plate decreases with an increase in M. For a fixed value of M, at a given location u increases with an increase in the suction parameter S while v decreases with increasing S. For a fixed value of M, at a given location both u and v decrease with an increase in the blowing parameter S₁. Further, for a fixed value of S₁, at a given position u increases with an increase in M but v decreases with increasing M. It is shown that no torque is exerted by the fluid on the plate, and non-coaxial rotations of the plate and the fluid at infinity have no influence on the torque.The solution of the heat transfer equation reveals that for given values of the suction parameter S, Prandtl number P and Eckert number E, the temperature at a given point in the flow increases with increasing M. On the other hand, for fixed values of M, P and E, the temperature at a given point decreases with increasing S. No steady distribution of temperature exists when there is blowing at the plate.