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.     

On grid generation for numerical models of geophysical fluid dynamics

A simple geometric condition that defines the class of classical (stereographic, conic and cylindrical) conformal mappings from a sphere onto a plane is derived. The problem of optimization of computational grid for spherical domains is solved in an entire class of conformal mappings on spherical (geodesic) disk. The characteristics of computational grids of classical mappings are compared for different spherical radii of geodesic disk. For a rectangular computational domain, the optimization problem is solved in the class of classical mappings and respective area of the spherical domain is evaluated.     

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.     

On spherical-wave scattering by a spherical scatterer and related near-field inverse problems

A spherical acoustic wave is scattered by a bounded obstacle.A generalization of the ‘optical theorem’ (whichrelates the scattering cross-section to the far-field patternin the forward direction for an incident plane wave) is proved.For a spherical scatterer, low-frequency results are obtainedby approximating the known exact solution (separation of variables).In particular, a closed-form approximation for the scatteredwavefield at the source of the incident spherical wave is obtained.This leads to the explicit solution of some simple near-fieldinverse problems, where both the source and coincident receiverare located at several points in the vicinity of a small sphere.     

Existence and Stability for Spherical Crystals Growing in a Supersaturated Solution

We consider the growth of a spherical crystal in a supersaturatedsolution. In the first part, existence and uniqueness resultsfor radially symmetric growth are obtained, provided that thesupersaturation is not too large; conversely, when the far-fieldsupersaturation exceeds a critical value, it is shown that theradially symmetric solution ceases to exist in finite time.In the second part, we examine the linear stability of a radiallysymmetric similarity solution (in which the radius grows ast?) to shape perturbations. The results are compared with previousquasi-static analyses, and, in particular, the critical radiusat which the crystal becomes unstable is found to be largerfor small supersaturations, but smaller for large supersaturations,than those predicted by the quasi-static analysis     

Chaplygin ball over a fixed sphere: an explicit integration

We consider a nonholonomic system describing the rolling of a dynamically nonsymmetric sphere over a fixed sphere without slipping. The system generalizes the classical nonholonomic Chaplygin sphere problem and it is shown to be integrable for one special ratio of radii of the spheres. After a time reparameterization the system becomes a Hamiltonian one and admits a separation of variables and reduction to Abel-Jacobi quadratures. The separating variables that we found appear to be a non-trivial generalization of ellipsoidal (spheroconic) coordinates on the Poisson sphere, which can be useful in other integrable problems. Using the quadratures we also perform an explicit integration of the problem in theta-functions of the new time.      

The resistive coated sphere in the presence of a point generated wave field

We consider the low‐frequency scattering problem of a point source generated incident field by a small penetrable sphere. The sphere, which is also lossy, contains in its interior a co‐ecentric spherical core on the boundary of which an impedance boundary condition is satisfied. An appropriate modification of the incident wave field allows for the reduction of the solution to the corresponding scattering problem of plane wave incidence, by moving the point source to infinity. For the near field, we obtain the low‐frequency coefficients of the zeroth and the first order. This was done with the help of the corresponding solution for the hard core problem and an appropriate use of linearity with respect to the Robin parameter. In the far field, we derive the leading non‐vanishing terms for the normalized scattering amplitude and the scattering cross‐section, which are both of the second order, as well as for the absorption cross‐section, which is of the zeroth order. The special cases of a lossy or a lossless penetrable sphere, of a resistive sphere, and of a hard sphere are recovered by an appropriate choice of the physical or the geometrical parameters. Copyright © 1999 John Wiley & Sons, Ltd.     

On the sphericity testing of single source digraphs

A digraph D=(V,A) is called spherical, if it has an upward embedding on the round sphere which is an embedding of D on the round sphere so that all edges are monotonic arcs and all point to a fixed direction, say to the north pole. It is easy to see that [S.M. Hashemi, Digraph embedding, Discrete Math. 233 (2001) 321-328] for upward embedding, plane and sphere are not equivalent, which is in contrast with the fact that they are equivalent for undirected graphs. On the other hand, it has been proved that sphericity testing for digraphs is an NP-complete problem [S.M. Hashemi, A. Kisielewicz, I. Rival, The complexity of upward drawings on spheres, Order 14 (1998) 327-363]. In this work we study sphericity testing of the single source digraphs. In particular, we shall present a polynomial time algorithm for sphericity testing of three connected single source digraphs.     

Poisson比为1/2材料中球形空穴突变和球形空穴萌生的分岔问题研究

研究Ｐｏｉｓｓｏｎ比为１／２的Ｈｏｏｋｅ材料中,空穴的突变和萌生现象·求解一个球对称几何非线性弹性力学的移动边界（ｍｏｖｉｎｇｂｏｕｎｄａｒｙ）问题,空穴为球形,远离空穴处为三向均匀拉伸应力状态,在当前构形上列控制方程;在当前构形边界上列边界条件·找到了这个自由边界问题的封闭解并得到空穴半径趋于零时的叉型分岔解·计算结果显示,在位移＿载荷曲线上存在一个切分岔型分岔点（或鞍结点型分岔点、极值型分岔点）,这个分岔点说明在外力作用下空穴会发生突变,即突然“长大”;当球腔半径趋于零时,这个切分岔转化为叉型分岔（或分枝型分岔）,这个叉型分岔可以解释实心球中的空穴萌生现象     

Mathematical Analysis of an Approximation Model for a Spherical Cloud of Cavitation Bubbles

In the paper we introduce a new approximation scheme for modelling a spherical cloud of cavitation bubbles based upon a model developed in (Wang and Brennen in J. Fluids Eng. 121(4):872–880, 1999) which consists of fully nonlinear continuum mixture equations coupled with the Rayleigh-Plesset equation for dynamics of the bubbles. We prove existence of a unique, local-in-time solution to the equations using the Banach fixed-point theorem which also provides us with the convergent scheme for a numerical simulation of the solution. We further demonstrate acquired numerical results.     

On catastrophe and cavitation for spherical cavity

This work deals with catastrophe of a spherical cavity and cavitation of a spherical cavity for Hooke material with 1/2 Poisson's ratio. A nonlinear problem, which is the Cauchy traction problem, is solved analytically. The governing equations are written on the deformed region or on the present configuration. And the conditions are described on moving boundary. A closed form solution is found. Furthermore, a bifurcation solution in closed form is given from the trivial homogeneous solution of a solid sphere. The results indicate that there is a tangent bifurcation on the displacement-load curve for a sphere with a cavity. On the tangent bifurcation point, the cavity grows up suddenly, which is a kind of catastrophe. And there is a pitchfork bifurcation on the displacement-load curve for a solid sphere. On the pitchfork bifurcation point, there is a cavitation in the solid sphere.     

On the effective viscosity of a dilute emulsion of gas bubbles

The problem of the motion of a rigid spherical body in a homogeneous emulsion of gas bubbles is considered in the Stokes approximation, using the self-consistent field method. An expression is obtained for the correction factor in the Stokes formula for the drag of the body in the first approximation with respect to the volume concentration of the dispersed phase. An analytical relation between the correction factor and the ratio of the sizes of the bubbles and the body is found. It is shown that, in the limit when this ratio tends to zero, the correction factor obtained is identical to Taylor's result for the effective viscosity of an emulsion of gas bubbles. In the case of non-point bubbles, the coefficient on the volume concentration in the expression for the effective viscosity of the emulsion can be considerably different from Taylor's result. A similar conclusion was also obtained in the case of the problem of the motion of a spherical bubble of arbitrary size in an emulsion of gas bubbles.     

Some problems of conformal mappings of spherical domains

In this study the problem of finding the conformal mapping from a sphere onto a plane with a given scale function independent of longitude is solved for an arbitrary spherical domain. The obtained results are compared with the well-known projections used in cartography and geophysical fluid dynamics. The problem of minimization of the distortion under conformal mappings is solved for domains in the form of the spherical disk. The distortions of some extensively used conformal mappings are compared with the distortions of orthogonal mappings.     

Some problems of conformal mappings of spherical domains

In this study the problem of finding the conformal mapping from a sphere onto a plane with a given scale function independent of longitude is solved for an arbitrary spherical domain. The obtained results are compared with the well-known projections used in cartography and geophysical fluid dynamics. The problem of minimization of the distortion under conformal mappings is solved for domains in the form of the spherical disk. The distortions of some extensively used conformal mappings are compared with the distortions of orthogonal mappings.     

Point Sets on the Sphere \mathbb{S}^{2} with Small Spherical Cap Discrepancy

In this paper we study the geometric discrepancy of explicit constructions of uniformly distributed points on the two-dimensional unit sphere. We show that the spherical cap discrepancy of random point sets, of spherical digital nets and of spherical Fibonacci lattices converges with order?N ^?1/2. Such point sets are therefore useful for numerical integration and other computational simulations. The proof uses an area-preserving Lambert map. A?detailed analysis of the level curves and sets of the pre-images of spherical caps under this map is given.     

Electroencephalography in ellipsoidal geometry

The human brain is shaped in the form of an ellipsoid with average semiaxes equal to 6, 6.5 and 9 cm. This is a genuine 3-D shape that reflects the anisotropic characteristics of the brain as a conductive body. The direct electroencephalography problem in such anisotropic geometry is studied in the present work. The results, which are obtained through successively solving an interior and an exterior boundary value problem, are expressed in terms of elliptic integrals and ellipsoidal harmonics, both in Jacobian as well as in Cartesian form. Reduction of our results to spheroidal as well as to spherical geometry is included. In contrast to the spherical case where the boundary does not appear in the solution, the boundary of the realistic conductive brain enters explicitly in the relative expressions for the electric field. Moreover, the results in all three geometrical models reveal that to some extend the strength of the electric source is more important than its location.     

Classification of Harmonic Functions in the Exterior of the Unit Ball

We solve the Laplace equation in an exterior infinite spherical domain with nonlinear (quadratic) boundary conditions on the spherical boundary. We linearize the problem and, under the additional assumption that the distinguishing function is spherically symmetric, write the solution by using the formal power series method with recursion of the series coefficients. Applying the Poincaré--Perron theorem, we describe the space of convergent formal power series and calculate its dimension. Estimating the roots of the fourth-degree characteristic polynomial corresponding to the given problem, we also calculate the dimension of the space of functions whose gradient at each point of the sphere is orthogonal to the linear combination of an axially symmetric dipole and a quadrupole. In conclusion, we state several unsolved problems arising in geophysical applications.     

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.     

Latitudinally deforming rotating sphere

In this study, we consider a sphere with a surface that is fully covered by a stretchable elastic material. The radius of the sphere is fixed and it is also rotating about its radial axis. We investigate how the axisymmetric motion of a triggered fluid flow around the sphere is affected by the presence of both sphere rotation and latitudinal stretching. Considering that the deformation over the sphere commences at the pole, the problem is formulated such that the fluid flow near the pole is similar to the induced flow due to a linearly stretchable rotating disk, which has been described well in previous studies. When the rotation is omitted, the flow develops two-dimensionally under the action of pure stretching; otherwise, a three-dimensional axisymmetric fluid flow occurs, which is computed at each latitudinal angle both numerically and using a perturbation approach. The solution with wall deformation is different from the traditional character of the solution due to a solely rotating sphere. This solution is then used to compute the surface shears due to the physical drag and torque acting over the sphere. The contribution of wall stretching reduces the drag, whereas high rotation suppresses the effects of stretching to enhance the drag. More torque is required to rotate the sphere when both stretching and rotation mechanisms are in action.     

Time dependent random fields on spherical non-homogeneous surfaces

We introduce a class of isotropic time dependent random fields on the non-homogeneous sphere which is represented by a time-changed spherical Brownian motion of order ν∈(0,1]$\nu \in (0,1]$. We can capture some anisotropies in Cosmology with this model. This process is a time-changed rotational diffusion (TRD) or the stochastic solution to the equation involving the spherical Laplace operator and a time-fractional derivative of order ν$\nu$. TRD is a diffusion on the non-homogeneous sphere and therefore, the spherical coordinates given by TRD represent the coordinates of a non-homogeneous sphere by means of which an isotropic random field is indexed. The time dependent random fields we present in this work are therefore realized through composition and can be viewed as isotropic random field on randomly varying sphere.