THE METHOD OF CONTINUOUS DISTRIBUTION OF SINGULARITIES TO TREAT THE STOKES FLOW OF THE ARBITRARY OBLATE AXISYMWETRICAL BODY

This paper deals with the Stokes flow of the arbitrary oblate axisymmetrical body by means of constant density and quadratic distribution function approximation for the method of continuous distribution of singularities. The Sampson spherical infinite series arc chosen as fundamental singularities. The convergence, accuracy and range of application of both two approximations are examined by the unbounded Stokes flow past the oblate spheroid. It is demonstrated that the drag factor and pressure distribution both conform with the exact solution very well. Besides, the properties, accuracy and the range of application are getting belter with the improving of the approximation of the distribution function. As an example of the arbitrary oblate axisymmetrical bodies, the Stokes flow of the oblate Cassini oval are calculated by these two methods and the results are convergent and consistent. Finally, with the quadratic distribution approximation the red blood cell, which has physiologic meaning, is considered and for the first time the (orresponding drag factor and pressure distribution on the surface of the cell are obtained.     

Stokes flow past a porous spheroid embedded in another porous medium

A study is made with an analysis of an incompressible viscous fluid flow past a slightly deformed porous sphere embedded in another porous medium. The Brinkman equations for the flow inside and outside the deformed porous sphere in their stream function formulations are used. Explicit expressions are investigated for both the inside and outside flow fields to the first order in small parameter characterizing the deformation. The flow through the porous oblate spheroid embedded in another porous medium is considered as the particular example of the deformed porous sphere embedded in another porous medium. The drag experienced by porous oblate spheroid in another porous medium is also evaluated. The dependence of drag coefficient and dimensionless shearing stress on the permeability parameter, viscosity ratio and deformation parameter for the porous oblate spheroid is presented graphically and discussed. Previous well-known results are then also deduced from the present analysis.     

Hydrodynamic action of the fluid slowly flowing around a spheroidal particle covered by a viscous film

A steady problem of a slow axisymmetric flow of a viscous incompressible fluid around an oblate spheroid covered by a viscous film is solved analytically with the use of the Stokes approximation. Surface tension on the interface between the fluids is taken into account. Expressions for velocity components and stream functions are presented. A formula for determining the force action of the incoming flow onto the oblate spheroid is derived.     

Brownian dynamics simulation of multiphase suspension of disc-like particles and polymers

A computational model is proposed for simulating the flow of polymer nanocomposites. This model is based on a multiphase suspension of disc-like particles and polymers. The particles are represented by oblate spheroid particles that interact with each other via the Gay-Berne (GB) potential, and the polymers are modeled by finitely extensible nonlinear elastic (FENE) chains that interact with each other via the repulsive Lennard-Jones potential. The interaction between an oblate spheroid particle and a FENE chain is also considered using a modified GB potential. A Brownian dynamics simulation of the shear flows of this system was conducted to investigate the orientation behavior of disc-like particles and the rheological properties of this system. The orientation of disc-like particles was affected by polymers, and the particles in a suspension were well aligned in flows because of the flow orientation property of polymers. The predicted shear viscosity exhibited shear thinning, and the normal stress differences agree qualitatively with experimental measurements of polymer/clay nanocomposites. The simulation results suggest that the present model has the potential to be used as a computational model for polymer nanocomposites.     

椭球体绕流阻力数值模拟研究

颗粒在无界均匀流场中的阻力是颗粒多相流领域最基本的问题之一。在相关的工程应用与科学研究中,颗粒基本都是非球形。早期基本以相同体积的圆球阻力来近似非球颗粒的实际阻力,近来则提出了多种基于实验数值拟合的非球颗粒阻力经验公式。本文通过高精度的计算方法,系统模拟了从扁平到细长的回转椭球颗粒,在常见的颗粒雷诺数范围内（0.1~1000）,对于各种来流攻角条件下的阻力,数值模拟结果得到了已知理论解的充分验证。在不存在理论解的大多数条件下,将数值结果与多种流行的经验公式结果进行了系统比较。一般而言,当颗粒以最大迎风面积置于来流中时,其所受阻力最大。但是数值模拟发现了特例,在流动进入牛顿阻力区时（雷诺数1000）,扁平的椭球在回转轴与流向一致时（此时具有最大迎风面积）,其阻力反而小于倾斜置于流场中的结果。对流动结构的分析表明,这是因为在非轴对称条件（颗粒倾斜）下,流动更容易失稳,造成紊乱的尾流,使得压差阻力上升。综合分析表明,对于给定的颗粒形状,数值模拟相对于通过各种实验数据拟合得到的适用于一般非球颗粒的经验公式具有更高的精度。累积更大量的高精度数值模拟结果,有望给出具有更高精度、更具适应性的阻力公式。

     

The three-dimensional fundamental solution to Stokes flow in the oblate spheroidal coordinates with applications to multiples spheroid problems

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Indirect boundary element method for shear flow over prolate and oblate hemispheroidal protuberances on plane walls

The indirect boundary element method was used to study the hydrodynamics of longitudinal shear flow and cross flow with a longitudinal rate of shear over prolate and oblate hemispheroidal protuberances projecting from a plane wall. Analytic techniques such as Fourier analysis, spheroidal co-ordinates, and the method of images were used to make the numerical methods more efficient. A novel method for computing the hydrodynamic torque was used—instead of directly calculating the torque from the weightings of the Green's functions (a method that is only valid when the weightings have physical significance) the hydrodynamic torque was computed indirectly using a Green's function for torque that derived here. As a test of this method, the present scheme was applied to determine the hydrodynamic torque of full spheroids, where exact solutions are known, and excellent results were obtained. Our results for hemispheroids projecting from plane walls were, except for extremely wide oblate hemispheroids, within a factor of two of those of full spheroids. Our results also agreed with those of previous study of oblate hemispheroidal protuberances. © 1998 John Wiley & Sons, Ltd.     

Simulation of the rheological properties of suspensions of oblate spheroidal particles in a Newtonian fluid

A simulation algorithm was developed to predict the rheological properties of oblate spheroidal suspensions. The motion of each particle is described by Jeffery’s solution, which is then modified by the interactions between the particles. The interactions are considered to be short range and are described by results from lubrication theory and by approximating locally the spheroid surface by an equivalent spherical surface. The simulation is first tested on a sphere suspension, results are compared with known experimental and numerical data, and good agreement is found. Results are then presented for suspensions of oblate spheroids of two mean aspect ratios of 0.3 and 0.2. Results for the relative viscosity η _r, normal stress differences N ₁ and N ₂ are reported and compared with the few available results on oblate particle suspensions in a hydrodynamic regime. Evolution of the orientation of the particles is also observed, and a clear alignment with the flow is found to occur after a transient period. A change of sign of N ₁ from negative to positive as the particle concentration is increased is observed. This phenomenon is more significant as the particle aspect ratio increases. It is believed to arise from a change in the suspension microstructure as the particle alignment increases.     

Creeping flow past and within a permeable spheroid

Flow past and within an isolated permeable spheroid directed along its axis of symmetry is studied. The flow velocity field is solved using the Stokes creeping flow equations governing the fluid motion outside the spheroid, and the Darcy equation within the spheroid. Expressions for the hydrodynamic resistance experienced by oblate and prolate spheroids are derived and analyzed. The limiting cases of permeable circular disks and elongated rods are examined. It is shown that the spheroid’s resistance varies significantly with its aspect ratio and permeability, expressed via the Brinkman parameter.     

The orientation dynamics of small prolate and oblate spheroids in linear shear flows

The present work deals with the stable orientation of oblate and prolate spheroids in general steady linear flows and with the mode of convergence to these stable orientations. The orientation dynamics is governed by the Jeffery equation. The stable orientations are either fixed points or limit cycles in the orientation space. The type of stable orientation depends on whether the eigenvalues of the linear part of Jeffery equation are real or complex. We define prolate and oblate spheroids to be equivalent if the aspect ratio of one is the reciprocal of the other. We show that, in a given flow, equivalent oblate and prolate spheroids possess the same number of fixed points and limit cycles of which only one is stable. If they possess only fixed points, then their corresponding stable fixed points are orthogonal. If they possess one fixed point and one limit cycle each, then the stable fixed point of one is orthogonal to the plane of the limit cycle of the other. The rate of convergence to these attractors is important to consideration of the orientations in time-space varying flow fields. We show that non-normal growth (NNG) of the distance to these attractors may delay the convergence by several characteristic shear time scales. We derive conditions for occurrence of NNG and explicit expressions for the maximal duration of the growth. We consider a specific case of which the vorticity is a stable orientation of prolate spheroids. We analyze the conditions that imply monotonic or, conversely, non-monotonic convergence to this orientation due to NNG. We thereby find the corresponding conditions for convergence of the equivalent oblate spheroids to their attractors, normal to the vorticity. We show that the convergence is monotonic if the vorticity is parallel to the strain tensor’s largest eigenvector, but that NNG occurs if the vorticity is parallel to the strain tensor’s intermediate eigenvector. The NNG duration decreases with increasing vorticity-strain ratio and with the strain intermediate eigenvalue approaching the largest eigenvalue.     

Stokes flow in a cylindrical tube containing a line of spheroidal particles

Viscous flow in a circular cylindrical tube containing an infinite line of rigid spheroidal particles equally spaced along the axis of the tube is considered for (a) uniform axial translation of the spheroids (b) flow past a line of stationary spheriods and (c) flow of the suspending fluid and spheroids under an imposed pressure gradient. The fluid is assumed to be incompressible and Newtonian. The Reynolds number is assumed to be small and the equations of creeping flow are used. Two types of solutions are developed: (i) an exact solution in the form of an infinite series which is valid for ratios of the spheroid diameter to the tube diameter up to 0.80, (ii) an approximate solution using lubrication theory which is valid for spheroids which nearly fill the tube. The drag on each spheroid and the pressure drop are computed for all cases. Both prolate and oblate spheroids are considered. The results show that the drag and pressure drop depend on the spheroidal diameter perpendicular to the axis of tube primarily and the effects of the spheroidal thickness and spacing are secondary. The results are of interest in connection with mechanics of capillary blood flow, sedimentation, fluidized beds, and fluid-solid transport.     

Diffusion flux to a distorted gas bubble at large Reynolds numbers

The diffusion flux to a distorted gas bubble situated in a uniform viscous incompressible fluid flow is determined for large Reynolds and Péclet numbers and finite Weber numbers. The bubble has the shape of an ellipsoid of revolution, oblate in the flow direction, making it possible to use the flow field derived by Moore [1] in the form of a two-term expansion with respect to the flow parameter =R^–1/2 (R is the Reynolds number; the zeroth term of the expansion corresponds to potential flow). The dependence of the diffusion flux onto the bubble surface on the Weber and Reynolds numbers is determined. The results of Winnikow [2] and Sy and Lightfoot [3] are thus generalized to the case of finite Weber numbers and a broader range of Reynolds numbers.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 70–76, July–August, 1976.     

Dynamics of disk-like particles in turbulent vertical channel flow

The dynamical behavior of inertial disk-like particles in turbulent vertical channel flow is investigated by an Eulerian–Lagrangian point-particle approach. Gravity effects on distribution, translation, rotation and orientation statistics of non-spherical particles modeled as oblate spheroids have been studied both in an upward and a downward flow and compared with results obtained in the absence of gravity. Altogether 12 different particle classes have been studied, with inertia and shape parameterized by means of Stokes number St and aspect ratio λ ≤ 1. The St = 5 disk-like particles distribute more evenly across the channel in upward than in downward flow. The gravity effect on the particle concentration diminishes with large inertia and the spheroid shape has only a modest influence. Although the gravity significantly affects the streamwise and wall-normal mean slip velocities with increasing inertia, the particle shape rarely has any impact on the translational motion, except for the mean wall-normal velocity. The fluctuations of the velocity of disk-like particles are mainly ascribed to inertia, whereas the gravity and shape only have marginal effects. The presence of gravity is moreover found to have a negligible effect on the particles’ orientation and rotation, in spite of the striking effect of λ on the orientation and rotation seen in the near-wall region. The tendency of the disks to align their symmetry axis orthogonal to the fluid vorticity in the channel center is stronger for particles with modest inertia. In the near-wall region, however, oblate spheroids preferentially align with the fluid vorticity for St >> 1. The observed behavior is believed to be caused by the influence of the gravity force on the turbophoresis; i.e. that inertial particles move towards low-turbulence regions.     

Numerical investigation on motion of an ellipsoidal particle inside confined microcavity flow

The behaviors of a neutrally buoyant ellipsoidal particle in vortical flow confined by a microcavity are numerically studied using the Lattice-Boltzmann method. For specific initial position, an isolated ellipsoid may develop a stable limit cycle orbit inside microcavity due to the interaction between particle and the carrier flow. It is observed that ellipsoidal particles of different shapes exhibit two different stable rotational modes depending on the initial orientation and lateral position. A prolate spheroid tends to enter a tumbling mode whereas an oblate spheroid is apt to achieve a rolling mode. The evolution of rotational velocities along the stable orbit is also analyzed for particles of different shapes.     

Axisymmetric flow through a permeable near-sphere

An analytical approach is described for the axisymmetric flow through a permeable near-sphere with a modification to boundary conditions in order to account permeability. The Stokes equation was solved by a regular perturbation technique up to the second order correction in epsilon representing the deviation from the radius of nondeformed sphere. The drag and the flow rate were calculated and the results were evaluated from the point of geometry and the permeability of the surface. An attempt also was made to apply the theory to the filter feeding problem. The filter appendages of small ecologically important aquatic organisms were modeled as axisymmetric permeable bodies, therefore a rough model for this problem was considered here as an oblate spheroid or near-sphere.     

Flow around λ wings with a bend in the leading edge

In recent years a considerable number of studies have been published on flow around wings at high supersonic velocities. The researches have been conducted in two directions: there are studies of hypersonic flow around wings of traditional shape and a search is carried out for new types of lay-out which possess optimal aerodynamic characteristics. The second direction relates to the numerous studies of flow around wings with shaped transverse cross sections [1–7]. The calculation of the aerodynamic quality of a shaped delta wing composed of plane surfaces on the basis of the relationships on an oblique shock [1, 2], from the results of experiments on the pressure distribution and from weight tests [3, 4], showed that the shaped wing has a higher quality than the plane delta wing.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 1, pp. 171–175, January–February, 1985.     

Numerical solution of 2D and 3D turbulent internal flow problems

The paper describes a method for solving numerically two-dimensional or axisymmetric, and three-dimensional turbulent internal flow problems. The method is based on an implicit upwinding relaxation scheme with an arbitrarily shaped conservative control volume. The compressible Reynolds-averaged Navier-Stokes equations are solved with a two-equation turbulence model. All these equations are expressed by using a non-orthogonal curvilinear co-ordinate system. The method is applied to study the compressible internal flow in modern power installations. It has been observed that predictions for two-dimensional and three-dimensional channels show very good agreement with experimental results.     

Void-shape effects on strength properties of nanoporous materials

In this paper, strength properties of nanoporous materials with spheroidal nanocavities are investigated via a Molecular Dynamics approach applied to a nanovoided aluminium single crystal, in the case of a fixed porosity level, and for prolate, oblate and spherical void shapes. Estimates of the effective strength domain are provided, by considering several mechanical loadings including axisymmetric and shear strain-rate states. Void-shape effects are quantified for different values of the void aspect ratio, mainly resulting in an overall weakening of the sample as the spheroidal nanovoid assumes either an oblate or a prolate shape, in comparison to the case of a spherical void. Finally, it is observed that the computed strength profiles exhibit the following specific features: (i) a strong dependence on the hydrostatic, second-order and third-order deviatoric stress invariants, (ii) more significant void-shape effects for triaxial-expansion stress states with a small hydrostatic component, and (iii) a more pronounced influence of the spheroid shape, as the aspect ratio is varied, in the presence of an oblate nanovoid rather than of a prolate one.     

3D phase field modeling of electrohydrodynamic multiphase flows

A 3D phase field model is developed to investigate the electrohydrodynamic (EHD) two phase flows. The explicit finite difference method, enhanced by parallel computing, is employed to solve the coupled nonlinear governing equations for the electric field, the fluid flow field and free surface deformation. Numerical tests indicate that an appropriate interpolation of densities within the interface is critical in ensuring numerical stability for highly stratified flows. The 3D phase field model compares well with the Taylor theory for the deformation of a single dielectric droplet in an electric field. Computed results show that the deformation of a leaky dielectric droplet in an electric field undergoes various stages before it reaches the final oblate shape. This is caused by the free charge relaxation near the fluid–fluid interface. The coalescence of four droplets in an electric field illustrates a truly 3D deformation behavior and a complex evolving fluid flow field associated with the participating droplets. The coalescence is a result of combined actions produced by the global electric force, the circulatory flows generated by the local electrohydrodynamic stress and the electrically-induced deformation. The 3D phase field model is also applied in modeling of an electrohydrodynamic patterning process for manufacturing nanoscaled structures, in which complex 3D flow structures develop as the electrically-induced deformation evolves.     

屈服后表面裂纹与穿透裂纹的等效裂纹长度换算

在较小范围屈服条件下，将D-M模型和弹塑性线弹簧模型相结合，建立了屈服后一般扁壳表面裂纹的计算模型，导出了其控制方程和相应的COD计算公式。在此基础上，提出了屈服后一般扁壳表面裂纹与平板穿透裂纹的等COD换算方法。通过各种影响因素的分析并去除次要因素，最终给出了该等效换算关系的简单拟合公式。计算结果表明，该分析方法简便且有足够的工程精度，适应于在役压力容器等结构的现场快速断裂评定。