Exact and approximate expressions for resistance coefficients of a colloidal sphere approaching a solid surface at intermediate Reynolds numbers

A mathematical model has been developed to describe the force of liquid flow acting on a colloidal spherical particle as it approaches a solid surface at intermediate-Reynolds-number-flow regime. The model has incorporated bispherical coordinates to determine a stream function for the flow disturbed by the sphere. The stream function was then used to derive the flow force on the particle as a function of the inter-surface separation distance. The force equation was related to the modified Stokes equation to obtain an exact analytical expression for the correction factor to the Stokes law. Finally, a rational approximation is presented, which is in good agreement with the exact numerical result, and can be readily applied to more general particle–surface interactions involving short-range hydrodynamics associated with colloidal particles in the near vicinity of a large solid collector surface at intermediate Reynolds number of the supporting flow.     

Slip at the surface of a sphere translating perpendicular to a plane wall in micropolar fluid

The Stokes axisymmetrical flow caused by a sphere translating in a micropolar fluid perpendicular to a plane wall at an arbitrary position from the wall is presented using a combined analytical-numerical method. A linear slip, Basset type, boundary condition on the surface of the sphere has been used. To solve the Stokes equations for the fluid velocity field and the microrotation vector, a general solution is constructed from fundamental solutions in both cylindrical, and spherical coordinate systems. Boundary conditions are satisfied first at the plane wall by the Fourier transforms and then on the sphere surface by the collocation method. The drag acting on the sphere is evaluated with good convergence. Numerical results for the hydrodynamic drag force and wall effect with respect to the micropolarity, slip parameters and the separation distance parameter between the sphere and the wall are presented both in tabular and graphical forms. Comparisons are made between the classical fluid and micropolar fluid.      

Stokes flow past an assemblage of slip eccentric spherical particle‐in‐cell models

The quasisteady axisymmetrical flow of an incompressible viscous fluid past an assemblage of slip eccentric spherical particle‐in‐cell models with Happel and Kuwabara boundary conditions is investigated. A linear slip, Basset type, boundary condition on the surface of the spherical particle is used. Under the Stokesian approximation, a general solution is constructed from the superposition of the basic solutions in the two spherical coordinate systems based on the particle and fictitious spherical envelope. The boundary conditions on the particle's surface and fictitious spherical envelope are satisfied by a collocation technique. Numerical results for the normalized drag force acting on the particle are obtained with good convergence for various values of the volume fraction, the relative distance between the centers of the particle and fictitious envelope and the slip coefficient of the particle. In the limits of the motions of the spherical particle in the concentric position with cell surface and near the cell surface with a small curvature, the numerical values of the normalized drag force are in good agreement with the available values in the literature. Copyright © 2011 John Wiley & Sons, Ltd.     

The contact problem for a two-layer spherical base

Analytical methods for solving problems of the interaction of punches with two-layer bases are described using in the example of the axisymmetric contact problem of the theory of elasticity of the interaction of an absolutely rigid sphere (a punch) with the inner surface of a two-layer spherical base. It is assumed that the outer surface of the spherical base is fixed, that the layers have different elastic constants and are rigidly joined to one anther, and that there are no friction forces in the contact area. Several properties of the integral equation of this problem are investigated, and schemes for solving them using the asymptotic method and the direct collocation method are devised. The asymptotic method can be used to investigate the problem for relatively small layer thicknesses, and the proposed algorithm for solving the problem by the collocation method is applicable for practically any values of the initial parameters. A calculation of the contact stress distribution, the parameters of the contact area, and the relation between the displacement of the punch and the force acting on it is given. The results obtained by these methods are compared, and a comparison with results obtained using Hertz, method is made for the case in which the relative thickness of the layers is large.     

The virtual mass of a sphere in a suspension of spherical particles

The problem of the virtual mass of a sphere, moving in an ideal incompressible fluid when there are other identical spherical particles of arbitrary mass present is considered. A solution is constructed for the velocity potential of the fluid in the form of the superposition of perturbation fields, introduced into the flow by each of the particles. The perturbation fields are obtained in the form of functional series, the coefficients of which are mutually consistent by a defined system of equations. An explicit expression is obtained for the hydrodynamic force acting on the sphere in the form of a function of the coordinates of all the particles. A simple analytical dependence of the mean value of the force and the virtual mass of the sphere on the particle-to-fluid density ratio in a first approximation of the volume fraction of the dispersed phase is obtained for a statistically uniform distribution of the dispersed particles in the suspension, using the procedure of averaging over their different possible configurations in space.     

稀相气固悬浮体越过沿平面匀速运动激波后的流动结构*

本文给出气固悬浮体中激波感生边界层的渐近数值分析,其中计及了作用于固体粒子的Saf-fman升力.研究结果表明粒子横越边界层的迁移导致了粒子轨道的交叉,因此对目前通用的含灰气体模型应做相应的修正.本文利用匹配渐近展开方法得到了匀速运动激波后方的两相侧壁边界层方程,详细描述了在Lagrange坐标下计算颗粒相流动参数的方法,并给出了粒子浓度很低情况下的数值结果.     

The lift on a small sphere in a linear shear flow near the interface of two immiscible fluids

Analytical expressions have been derived which predict, to lowest order, the inertial lift and the lateral migration velocity of a rigid sphere translating and rotating in a linear shear flow field near the flat interface of two immiscible fluids. This asymptotic analysis is primarily based on the assumption that the two Reynolds numbers defined by the gap width between the interface and the sphere, the shear rate and the translational slip velocity with which the spherical particle moves parallel to the interface are small. Furthermore, the radius of the sphere is assumed to be small compared to the gap width. To leading order in this creeping flow regime, the linear Stokes equations are obtained and a symmetry argument can be used to show that the Stokes solution does not predict any lift force. The transverse force experienced by the sphere and its migration velocity are due to the small but finite inertial terms in the Navier-Stokes equations, which can be studied by perturbation techniques. By applying a Green's function approach and matched asymptotic methods, which also incorporate the effects of the outer Oseen-like flow regime, the three components comprising the lift velocity have been calculated in closed form: the one induced by the shear rate only, the purely slip induced one and the one due to the interaction of the slip velocity with the shear flow field. The thus obtained expressions for the case of two immiscible fluids with arbitrary density and viscosity ratios extend the results that already exist in the literature for other flow configurations, such as an unbounded shear flow field [1] or a wall-bounded one, where the wall lies either within the leading order Stokes region [2] or in the outer Oseen region [3]. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Wall effects on the slow steady motion of a particle in a viscous incompressible fluid

In this paper, asymptotic expansions with respect to small Reynolds numbers are proved for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid past a single plane wall. The flow problem is modelled by a certain boundary value problem for the stationary, nonlinear Navier-Stokes equations. The coefficients of these expansions are the solutions of various, linear Stokes problems which can be constructed by single layer potentials and corresponding boundary integral equations on the boundary surface of the particle. Furthermore, some asymptotic estimates at small Reynolds numbers are presented for the slow steady motion of an arbitrary particle in a viscous, incompressible fluid between two parallel, plane walls and in an infinitely long, rectilinear cylinder of arbitrary cross section. In the case of the flow problem with a single plane wall, the paper is based on a thesis, which the author has written under the guidance of Professor Dr. Wolfgang L. Wendland.     

Mechanical Modeling of Particle-Droplet Interaction Motivated by the Study of Self-Cleaning Mechanisms

Motivated by the study of self-cleaning surfaces, the interaction between a spherical solid particle and a water droplet is studied on the microsale level, in terms of the balance forces and the surface properties. To define the forces acting on the solid-liquid interface, the meniscus depression is computed from the Young-Laplace equation, using a non-linear finite element formulation. The equilibrium force is derived in terms of the contact angle, particle size, and the penetration. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic expansion of the solution to the convective diffusion problem in the wake of a spherical particle

The exterior boundary value problem of steady-state diffusion around a spherical particle placed in a Stokes flow is considered at high Peclet numbers. A complete asymptotic expansion of the solution in the wake of the particle is constructed by the method of matched asymptotic expansions.     

Stresses in elastic conical tubes of transversely isotropic materials with spherical anisotropies under temperature and force loadings

Analytic solutions are proposed for a number of new problems on determining the state of stress of a transversely-isotropic hollow cone with spherical anisotropy. An exact solution of the problem of the axisymmetric deformation of a long conical tube (or continuous cone) from an elastic transversely-isotropic material with spherical anisotropy subjected to an axial force is obtained in a spherical coordinate system R, , θ, the material axis of symmetry is directed along the spherical radius R. A rigorous solution is given of the problem of the uniform heating of a conical tube of transversely-isotropic material with spherical anisotropy for particular values of Poisson's ratios; the material axis of symmetry is directed along the θ-axis. For arbitrary Poisson's ratios an asymptotic solution is found for the temperature problem for a tube with small conicity.     

Extension of SPH to simulate non-isothermal free surface flows during the injection molding process

This article presents an extension of smoothed particle hydrodynamics (SPH) to non-isothermal free surface flows during the injection molding process. Specifically, we use the method presented by Xu and Yu, Appl. Math. Model. 48 (2017) pp. 384–409, in which the corrected kernel gradient is implemented to increase the computational accuracy and the Rusanov flux is introduced into the continuity equation to alleviate large and random pressure oscillations. To model non-isothermal free surface flows, a working SPH discretization of the temperature equation is derived. An enhanced treatment of the wall boundary is further developed, which can model arbitrary-shaped mold walls. The proposed SPH method is first validated by solving non-isothermal Couette flow and non-isothermal injection molding of a circular disc with a core and comparing the SPH results with those obtained by other numerical methods or experiments. We then extend the numerical method to non-isothermal injection molding of F-shaped and N-shaped cavities. The convergence of the method is examined with several different particle sizes. The effects of the operating conditions (e.g., injection temperature, temperature of the mold wall, and injection velocity) on the flow behavior are analyzed. All the results illustrate that the present SPH method is a powerful computational tool for simulations of non-isothermal free surface flows during the injection molding process.     

Stokes flow between eccentric rotating spheres with slip regime

The steady axisymmetric flow problem of a viscous fluid contained between two eccentric spheres that rotate about an axis joining their centers with different angular velocities is considered. A linear slip of Basset-type boundary condition at both surfaces of the spherical particle and the container is used. Under the Stokesian assumption, a general solution is constructed from the superposition of basic solutions in the spherical coordinate systems based on the inner solid particle and the spherical container. The boundary conditions on the particle’s surface and spherical container are satisfied by a collocation technique. Numerical results for the coupling coefficient acting on the particle are obtained with good convergence for various values of the ratio of particle-to-container radii, the relative distance between the centers of the particle and container, the slip coefficients and the relative angular velocity. In the limiting cases, the numerical values of the coupling coefficient for the solid sphere in concentric position with the container and when the particle is near the inner surface of the container are obtained, and the results are in good agreement with the available values in the literature. The variation of the coupling coefficient with respect the parameters considered are tabulated and displayed graphically.     

Linear stationary surface waves over a rough bottom

We consider the linear formulation of the 3-dimensional problem of a stationary flow with a free surface over a rough bottom of an irrotational capillary heavy ideal incompressible fluid. For fast near-critical flows we derive an approximate equation for the free surface elevation. We express a fundamental solution to the problem in terms of contour integrals and establish its asymptotic behavior at large distances from the origin.     

Effects of Suction on the Nonstationary Lower Branch Modes of a Compressible Boundary Layer Flow Due to a Rotating Disk

In this paper, suction and injection effects are investigated theoretically on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the compressible boundary layer flow due to a rotating disk. In a recent study [ 1 ], it was demonstrated that the short-wavelength stationary/nonstationary compressible crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies can be described by an asymptotic expansion procedure as set up in [ 2 ] for the incompressible stationary modes, which rigorously takes into account the nonparallel effects. Employing this rational asymptotic technique, it is shown here that the wavenumber and the orientation of the compressible lower branch modes are governed by an eigenrelation that is under the strong influence of a suction/injection parameter     , which, when set to zero, the relation turns out to be the one obtained previously by Turkyilmazoglu [ 1 ] for zero-suction compressible modes.
The boundary layer growth contributes in the way of destabilizing all the modes, in particular for the compressible modes, though the wall cooling in the case of suction and the wall insulation and heating in the case of injection are found to persist to the destabilization for the modes in the vicinity of the stationary mode. From a linear stability analysis point of view, suction is found to be stabilizing, whereas injection enhances the instability as compared to the no suction through the surface of the disk. In both cases, positive frequency waves are found to be highly destabilized as compared to the waves having negative frequencies. The findings of the work are also fully supported after a comparison between the numerical results obtained from directly solving the linearized compressible system with a usual parallel flow approximation and the asymptotic compressible data obtained at a high Reynolds number.     

Mixed convection boundary layer flow over a permeable vertical surface with prescribed wall heat flux

The effects of suction/injection on the laminar mixed convection boundary-layer flow on a vertical wall with a prescribed heat flux are considered. The conditions which allow the equations to be reduced to similarity form are derived and numerical solutions of the resulting equations are obtained for a range of values of the suction/injection and buoyancy parameters. Two specific cases, corresponding to a stagnation point flow and uniform wall heat flux, are treated in detail. Results are presented in terms of the skin friction and wall temperature with a selection of velocity and temperature profiles also being given. Dual solutions are found to exist for assisting flow, these are an addition to what has been reported previously for opposing flows. Solutions for some limiting values of the parameters are also derived.     

存在滑移时两圆球间的幂律流体挤压流动

基于Reynolds润滑理论分析了壁面滑移对任意圆球颗粒间幂律流体的挤压流动的影响。研究表明有壁面滑移时挤压流动的粘性力可通过引进本文定义的滑移修正系数分离出无滑移解。推导出的挤压力滑移修正系数是一积分表达式，依赖于滑移参数、幂律指数、球间隙和积分上限。一般地壁面滑移导致粘性力减小，粘性力的减小量随幂律指数的增大而增大，表明壁面滑移对剪切增稠流变材料有更大的影响；粘性力的减小量还随着滑移参数的增大而增加，而这恰与假设相符合；粘性力的减小量又随球间隙减小或积分上限的增大（从液桥情况到完全浸渍）而减小直到趋于常数，这一特性在离散元模拟时可以有效地减少计算量。     

Asymptotic analysis of the Stokes flow in a thin cylindrical elastic tube

The purpose of this article is to perform an asymptotic analysis for an interaction problem between a viscous fluid and an elastic structure when the flow domain is a three-dimensional cylindrical tube. We consider a periodic, non-steady, axisymmetric, creeping flow of a viscous incompressible fluid through a long and narrow cylindrical elastic tube. The creeping flow is described by the Stokes equations and for the wall displacement we consider the Koiter's equation. The well posedness of the problem is proved by means of its variational formulation. We construct an asymptotic approximation of the problem for two different cases. In the first case, the stress term in Koiter's equation contains a great parameter as a coefficient and dominates with respect to the inertial term while in the second case both the terms are of the same order and contain the great parameter. An asymptotic analysis is developed with respect to two small parameters. Analysing the leading terms obtained in the second case, we note that the wave phenomena takes place. The small error between the exact solution and the asymptotic one justifies the below constructed asymptotic expansions.     

On Transonic Marginal Separation

Laminar boundary‐layer separation near the leading edge of a thin airfoil is one of the principal factors that limits the lift force acting on the airfoil. Marginal separation, in particular, denotes the onset of separation which is accompanied by the formation of a short separation bubble. Using asymptotic analysis this effect is studied in the limit of high Reynolds number and for transonic external flow conditions. It is assumed that the fluid under consideration is a perfect gas and the airfoil surface is taken to be thermally insulated. Results to be presented include the analytical investigation of the emerging three layer structure, the associated transonic far field and the calculation of representative wall shear stress distributions. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

First stages of drop impact on a dry surface: asymptotic model

After impact of a viscous liquid drop on a dry wall surrounded by a gas, the drop surface is highly deformed, leading to the formation of an axisymmetrical lateral lamella along the wall. A local asymptotic model for the potential flow and unsteady boundary layer flow is developed to describe the lamella dynamics at early stages after impact. The second-order potential flow displaced by the unsteady boundary layer is taken into account. The lamella shape, its velocity and pressure are calculated with this model in parametrical forms. The three model parameters are evaluated here by fitting with recent experimental findings.