三维斯托克斯流动在扁球坐标系中的基本解及其应用

通过把Lamb基本解中的调和函数转换为扁球坐标系下的表达式,这项研究成功地得到了一个新的Stokes流动三维基本解.此基本解可用于解决任意多个扁椭球处于任意位置和方向时的流动问题.应用最小二乘法,三维流动问题中常遇到的收敛性差的困难在此得以完全克服.结果表明该方法具有准确度高,收敛性好和计算量小的特点.由于扁球可用于模拟从圆盘到圆球的多种物体形状,此基本解被用于系统地分析了各种几何因素对两个扁球所受力和力矩的影响.为了显示此方法的通用性,该基本解还用于研究了两例三个扁球的问题.     

Stokes flow with slip and Kuwabara boundary conditions

The forces experienced by randomly and homogeneously distributed parallel circular cylinder or spheres in uniform viscous flow are investigated with slip boundary condition under Stokes approximation using particle-in-cell model technique and the result compared with the no-slip case. The corresponding problem of streaming flow past spheroidal particles departing but little in shape from a sphere is also investigated. The explicit expression for the stream function is obtained to the first order in the small parameter characterizing the deformation. As a particular case of this we considered an oblate spheroid and evaluate the drag on it.     

Drawdown in prolate spheroidal–spherical coordinates obtained via Green’s function and perturbation methods

When investigating aquifer behaviour it is important to note that there exists a close relationship between the geometrical properties of the aquifer and the behaviour of the solution. In this paper our concern is to solve the flow equation described by prolate spheroidal coordinates by means of perturbation and the Green’s function method, where the spheroid is considered to be a perturbation of a sphere. We transformed the spheroidal coordinates to spherical polar coordinates in the limit, as the shape factor tends to zero. The new groundwater flow equation is solved via an asymptotic parameter expansion and the Green’s function method. The approximate solution of the new equation is compared with experimental data from real world. To take into account the error committed while approximating, we estimate the error in the asymptotic expansion. The error functions obtained suggest that the error would be very small for the shape factor tending to zero if the first two terms of the expansion are taken as an approximation.     

Analytic solution of an exterior Neumann problem in a non‐convex domain

A new method, based on the Kelvin transformation and the Fokas integral method, is employed for solving analytically a potential problem in a non‐convex unbounded domain of ?², assuming the Neumann boundary condition. Taking advantage of the property of the Kelvin transformation to preserve harmonicity, we apply it to the present problem. In this way, the exterior potential problem is transformed to an equivalent one in the interior domain which is the Kelvin image of the original exterior one. An integral representation of the solution of the interior problem is obtained by employing the Kelvin inversion in ?² for the Neumann data and the ‘Neumann to Dirichlet’ map for the Dirichlet data. Applying next the ‘reverse’ Kelvin transformation, we finally obtain an integral representation of the solution of the original exterior Neumann problem. Copyright © 2010 John Wiley & Sons, Ltd.     

The stressed state of a transversally isotropic medium with a spheroidal inclusion under nonideal mechanical contact

We study the problem of the stressed state of a transversally isotropic medium containing a foreign inclusion in the form of a prolate spheroid under an arbitrary homogeneous stress field at infinity. On the “medium-inclusion” interface there is slipping without flaking. The stressed state is constructed in the medium and in the inclusion using the exterior and interior problems for a prolate spheroid on the basis of potential functions. The solution of the problem is reduced to studying infinite systems of linear algebraic equations. The results of numerical studies are shown as graphs that describe the stress distribution in both the transversally isotropic medium and in the inclusion under various boundary conditions. Four figures. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 15–26.     

Modeling the Sedimentation of Red Blood Cells in Flow under Strong External Magnetic Body Force Using a Lattice Boltzmann Fictitious Domain Method

Experimental observations show that a strong magnetic field has a dramatic influence on the sedimentation of RBCs, which motivates us to model the sedimentation of red blood cell (RBC) under strong external magnetic body force. To model the sedimentation of a RBC in a square duct and a circular pipe, a recently developed technique derived from the lattice Boltzmann and the distributed Lagrange multiplier/fictitious domain methods (LBM-DLM/FD) is extended to employ the mesoscopic network model for simulations of the sedimentation of a RBC in flow. The flow is simulated by the LBM with a strong magnetic body force, while the network model is used for modeling RBC deformation. The fluid-RBC interactions are enforced by the Lagrange multiplier. The sedimentation of RBC in a square duct and a circular pipe is simulated, which demonstrates the developed method's capability to model the sedimentation of RBCs in various flows. Numerical results illustrate that the terminal settling velocity increases incrementally with the exerted body force. The deformation of RBC has a significant effect on the terminal settling velocity due to the change in the frontal area. The larger the exerted force, the smaller the frontal area and the larger the RBC deformation become. Additionally, the wall effect on the motion and deformation of RBC is also investigated.     

Numerical Simulation of Red Blood Cell Suspensions Behind a Moving Interface in a Capillary

Computational modeling and simulation are presented on the motion of red blood cells behind a moving interface in a capillary. The methodology is based on an immersed boundary method and the skeleton structure of the red blood cell (RBC) membrane is modeled as a spring network. As by the nature of the problem, the computational domain is moving with either a designated RBC or an interface in an infinitely long two-dimensional channel with an undisturbed flow field in front of the computational domain. The tanking-treading and the inclination angle of a cell in a simple shear flow are briefly discussed for the validation purpose. We then present and discuss the results of the motion of red blood cells behind a moving interface in a capillary, which show that the RBCs with higher velocity than the interface speed form a concentrated slug behind the moving interface.     

Visco-elastic flow past a spheroid

Summary The creeping flow of an Oldroyd visco-elastic liquid past a spheroid whose shape varies slightly from that of a sphere is examined. An explicit expression is obtained for the stream function characterizing the flow. As a particular case the flow past an oblate spheroidal particle is investigated. Two special known cases are then deduced.     

A fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array

In this paper, we propose a fundamental solution method for three-dimensional viscous flow problems with obstacles in a periodic array. Our problem is mathematically a boundary value problem of the Stokes equation with periodic boundary conditions, to which it is difficult to give a good approximation by the ordinary fundamental solution method. Our method gives an approximate solution by a linear combination of the periodic fundamental solutions. In addition, we can compute the drag forces on the obstacles by using the data obtained in our method. Numerical examples for the problems of flows past spheres show the effectiveness of our method.     

A finite transform involving generalized prolate spheroidal wave functions and its applications

The author has extended his previous results pertaining to spheroidal functions by introducing a new finite transform involving generalized prolate spheroidal functions. The inversion has also been found. In the end its application has also been given in solving certain boundary value problems.     

解任意形状扁轴对称体Stokes流动的连续奇点分布法

本文以Sampson球形无穷级数作为基本奇点,采用分段等强度和分段二次抛物分布两种体内连续分布法解任意形状扁轴对称体的Stokes流动.通过扁球的无界绕流问题,对这两种方法的收敛性,精度和适用范围做了检验和比较.结果表明,在一定的范围内,无论是阻力系数或压力分布,它们的计算结果都和精确解符合得很好,而且,随着分布函数逼近程度的提高,其收敛性得到改善,适用范围也随之扩大.作为一般算例,分别用这两种方法解决了卡西尼扁卵形体的绕流问题,得到了一致的结果.最后,用分段二次连续分布法计算了具有一定生理意义的红细胞体的Stokes流动,首次得到了它的阻力系数和表面压力分布.     

A DLM/FD/IB Method for Simulating Compound Cell Interacting with Red Blood Cells in a Microchannel

In this article, a computational model and related methodologies have been tested for simulating the motion of a malaria infected red blood cell (iRBC for short) in Poiseuille flow at low Reynolds numbers. Besides the deformability of the red blood cell membrane, the migration of a neutrally buoyant particle (used to model the malaria parasite inside the membrane) is another factor to determine the iRBC motion. Typically an iRBC oscillates in a Poiseuille flow due to the competition between these two factors. The interaction of an iRBC and several RBCs in a narrow channel shows that, at lower flow speed, the iRBC can be easily pushed toward the wall and stay there to block the channel. But, at higher flow speed, RBCs and iRBC stay in the central region of the channel since their migrations are dominated by the motion of the RBC membrane.     

通过可渗透近球体的轴对称流动

提出了可渗透近球体轴对称流动的分析方法.用修正边界条件的办法反映可渗透性.用正规摄动法求解了Stokes方程,达到ε的2阶修正.ε是描述不变形球体半径偏差的小参数.计算了阻力和流量,并从几何方面和表面渗透性方面考查了计算结果.还尝试将此理论应用于过滤供水问题.小型的生态学上重要的水生生物体的过滤器,被模型化为轴对称可渗透物体,用扁球体或近球体建立了该问题的初级模型.     

The Scalar Scattering of a Plane Wave by an Ellipsoid

The problem of scalar Dirichlet scattering by a general ellipsoidis discussed. An exact solution of the wave equation is determinedvia the method of separation of the variables leading to expressionsfor the total field and the far field amplitude in terms ofellipsoidal wave function products. Particular attention ispaid to the case when the ellipsoid is almost a prolate spheroid.Finally methods of numerical solution are discussed and twonew results in ellipsoidal wave function theory are obtained.     

Singularities on Free Surfaces of Fluid Flows

Isolated singularities on free surfaces of two-dimensional and axially symmetric three-dimensional steady potential flows with gravity are considered. A systematic study is presented, where known solutions are recovered and new ones found. In two dimensions, the singularities found include those described by the Stokes solution with a 120° angle, Craya's flow with a cusp on the free surface, Gurevich's flow with a free surface meeting a rigid plane at 120° angle, and Dagan and Tulin's flow with a horizontal free surface meeting a rigid wall at an angle less than 120°. In three dimensions, the singularities found include those in Garabedian's axially symmetric flow about a conical surface with an approximately 130° angle, flows with axially symmetric cusps, and flows with a horizontal free surface and conical stream surfaces. The Stokes, Gurevich, and Garabedian flows are exact solutions. These are used to generate local solutions, including perturbations of the Stokes solution by Grant and Longuet-Higgins and Fox, perturbations of Gurevich's flow by Vanden-Broeck and Tuck, asymmetric perturbations of Stokes flow and nonaxisymmetric perturbations of Garabedian's flow. A generalization of the Stokes solution to three fluids meeting at a point is also found.     

A variational approach to estimate incompressible fluid flows

A variational approach is used to recover fluid motion governed by Stokes and Navier–Stokes equations. Unlike previous approaches where optical flow method is used to track rigid body motion, this new framework aims at investigating incompressible flows using optical flow techniques. We formulate a minimization problem and determine conditions under which unique solution exists. Numerical results using finite element method not only support theoretical results but also show that Stokes flow forced by a potential are recovered almost exactly.     

Solution to the Neumann problem exterior to a prolate spheroid by radial basis functions

We consider the exterior Neumann problem of the Laplacian with boundary condition on a prolate spheroid. We propose to use spherical radial basis functions in the solution of the boundary integral equation arising from the Dirichlet–to–Neumann map. Our approach is particularly suitable for handling of scattered data, e.g. satellite data. We also propose a preconditioning technique based on domain decomposition method to deal with ill-conditioned matrices arising from the approximation problem.     

The Limits of Porous Materials in the Topology Optimization of Stokes Flows

We consider a problem concerning the distribution of a solid material in a given bounded control volume with the goal to minimize the potential power of the Stokes flow with given velocities at the boundary through the material-free part of the domain.We also study the relaxed problem of the optimal distribution of the porous material with a spatially varying Darcy permeability tensor, where the governing equations are known as the Darcy–Stokes, or Brinkman, equations. We show that the introduction of the requirement of zero power dissipation due to the flow through the porous material into the relaxed problem results in it becoming a well-posed mathematical problem, which admits optimal solutions that have extreme permeability properties (i.e., assume only zero or infinite permeability); thus, they are also optimal in the original (non-relaxed) problem. Two numerical techniques are presented for the solution of the constrained problem. One is based on a sequence of optimal Brinkman flows with increasing viscosities, from the mathematical point of view nothing but the exterior penalty approach applied to the problem. Another technique is more special, and is based on the “sizing” approximation of the problem using a mix of two different porous materials with high and low permeabilities, respectively. This paper thus complements the study of Borrvall and Petersson (Internat. J. Numer. Methods Fluids, vol. 41, no. 1, pp. 77–107, 2003), where only sizing optimization problems are treated.     

A univalent spiral-vortex model for separated flow past a polygonal obstacle

Summary A free-streamline flow model for flow past a polygonal obstacle with a near-wake terminating in Tulin's double spiral vortices is constructed. The flows are univalent for a large class of geometries. In addition a criterion is given for determining the underpressure as function of the Reynolds number using the Stokes solution for diffusion of a vortex sheet, and an extension of Tulin and Hsu's matching theory to transitional flows.

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Sunto Si costruisce un modello di flusso con scia e vortici a doppia spirale alla Tulin per un ostacolo poligonale arbitrario. Il flusso risulta univalente per un' ampia classe di geometrie. Inoltre viene proposto un criterio per correlare il parametro del modello al numero di Reynolds del corrispondente flusso viscoso, combinando la soluzione di Stokes per la diffusione di uno strato di vortici con la teoria di Tulin e Hsu (1980).