Slow rotation of a sphere with source at its centre in a viscous fluid

In this note, the problem of a sphere carrying a fluid source at its centre and rotating with slow uniform angular velocity about a diameter is studied. The analysis reveals that only the azimuthal component of velocity exists and is seen that the effect of the source is to decrease it. Also, the couple on the sphere is found to decrease on account of the source.     

带有微弱章动的圆球在粗糙水平面上的运动

对于圆球在粗糙水平面上的运动,在文[1]中,作者忽略了章动,得到了近似解析解。本文在此基础上给出了有章动情况下的控制方程。通过求解这些方程,证明文[1]关于接触点速度的结论在有章动时仍然正确。还得到其它一些有趣的结果,例如:球心和接触点的速度与球的自转角速度和章动角速度有一定联系;球心和接触点的速度的方向具有不变性。在进一步假设微弱章动的情况下,文中得到近似解析解,从而证明文[1]结果的正确性。     

On motion of a symmetric gyrostat in a Newtonian force field

A spherical cavity in a sphere-shaped gyrostat contains a spherical rotor, which is rotating at a constant angular velocity relative to the outer sphere. The centres of the outer sphere, the cavity and the rotor coincide. Attached to the outer sphere are identical point masses, placed at the vertices of an octahedron. A study is presented of the influence of the rotation of the rotor on the existence and stability of steady motions of the gyrostat about its mass centre in the Newtonian field of a fixed attracting centre. Interest centres on motions in which the radius vector of the gyrostat centre and the gyrostatic moment are collinear. It is shown that the existence of a gyrostatic moment may essentially modify the stability properties of the steady motions discovered.     

Point source excitation in direct and inverse scattering: The soft and the hard small sphere

A spherical wave emanating from a point source is scatteredby either a soft or a hard body. The incident spherical wavehas a wavelength which is much larger than the characteristicdimension of the scatterer and it is modified in such a wayas to recover the plane wave incidence when the source pointrecedes to infinity. Using low frequency expansions the scatteringproblem is transformed to a sequence of exterior potential problemsin the presence of a monopole singularity located at the sourceof the incident wave field. Complete expansions for the scatteringamplitude are provided. The method is applied to the cases ofa soft and a hard sphere and the first three approximationsfor the near, as well as the far, field are evaluated. It isobserved that every one, after the first, low frequency approximationof the far field, involves one spherical multipole more thanthe corresponding approximation for the case of an incidentplane wave. As the point singularity tends to infinity, therelative results recover all the known expressions for planeincidence. It is shown that for point excitation the Rayleighapproximation of the scattering amplitude for a hard sphereis of the second order, in contrast to the case of plane excitationwhich is of the third order. Simple algorithms that specifythe radius and the position of a soft and a hard sphere areproposed, which are based on the additional dependence of thescattering amplitude represented by the distance from the pointsource to the centre of the scatterer. The inversion algorithmis shown to be stable whenever the source point is not too faraway from the target sphere. A simple way to decide whetherthe sphere is a soft or a hard body is also provided.     

Hypersonic flow past a sphere

The effects of dissociation or ionization of air on the analytical solution for hypersonic flow past a sphere are considered here, under certain assumptions. It has been assumed that the shock wave is in the shape of a sphere, that the density ratio across the shock is constant, that the flow behind the shock is at constant density and that dissociation or ionization only occurs behind the shock wave. Thus the effects of the compressibility of the air, variation of density ratio along the shock, and the department of the shock shape from being circular are not taken into account. Here the velocity, pressure, temperature, pressure coefficient and vorticity, etc., at any point between the shock and the surface of the sphere in the presence of dissociation or ionization are obtained. In addition, shock detachment distance, drag coefficient, stagnation point velocity gradient and sonic points on the shock and the surface have also been obtained. The results have been compared with the corresponding results obtained in the case when dissociation or ionization does not occur behind the shock.     

Time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energy dissipation

In this paper, an attempt is made to estimate the influence due to a time-harmonic normal point load or thermal source in a homogeneous isotropic magneto-thermo-viscoelastic half-space. The system of fundamental equations is solved by using Hankel transform. The inverse transform integrals using Romberg integration with adaptive stepwise after using the results from successive refinements of the extended trapezoidal rule followed by extrapolation of the results to the limit when the step-size tends to zero. The two special cases: (i) normal point load acting on the surface and (ii) thermal point source acting on the surface, are studied in the cases of with and without energy dissipation of magneto-thermo-viscoelasticity. The displacements, temperature and stress components have been obtained in analytical form. Finally, the results obtained are displayed numerically and presented graphically for copper material.     

Identification of objects in an acoustic wave guide inversion II: Robin–Dirichlet conditions

In this paper we investigate the unknown body problem in a wave guide where one boundary has a pressure release condition and the other an impedance condition. The method used in the paper for solving the unknown body inverse problem is the intersection canonical body approximation (ICBA). The ICBA is based on the Rayleigh conjecture, which states that every point on an illuminated body radiates sound from that point as if the point lies on its tangent sphere. The ICBA method requires that an analytical solution be known exterior to a canonical body in the wave guide. We use the sphere of arbitrary centre and radius in the wave guide as our canonical body. We are lead then to analytically computing the exterior solution for a sphere between two parallel plates. We use the ICBA to construct solutions at points ranging over the suspected surface of the unknown object to reconstruct the unknown object using a least‐squares matching of computed, acoustic field against the measured, scattered field. Copyright © 2005 John Wiley & Sons, Ltd.     

The motion of a body with a plane of symmetry over a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation

The problem of the motion of a rigid body possessing a plane of symmetry over the surface of a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation forces is considered. Approaches to introducing spherical analogues of the concepts of centre of mass and centre of gravity are discussed. The spherical analogue of “satellite approach” in the problem of the motion of a rigid body in a central field, which arises on the assumption that the dimensions of the body are small compared with the distance to the gravitating centre, is studied. Within the framework of satellite approach, assuming plane motion of the body, the question of the existence and stability of steady motions is investigated. A spherical analogue of the equation of the plane oscillations of a body in an elliptic orbit is derived.     

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 motion of chaplygin''s sphere on a horizontal plane

The motion of a heavy sphere on a fixed horizontal plane is considered. It is assumed that the centre of mass of the sphere is at its geometric centre, while the principal central moments are different (Chaplygin's sphere). Using the method of averaging, the motion of the sphere is investigated under slip conditions when there is low viscous and also low dry friction. It is shown that when the sphere moves with viscous friction it tends, for the majority of initial data, to rotate about the longest of the axes of the principal central moments of inertia. The motion of the sphere centre tends to become uniform so that the slip velocity approaches zero exponentially. A system of averaged equations, which is fully integrable, is obtained in the case of almost equal moments of inertia, when the friction is dry. The solutions are analyzed.     

Distribution of a distance function

Summary The distributions of the square of the distance between a random point and a fixed point on ann-dimensional unit sphere when (i) the two points lie on the surface of whole sphere and (ii) the two points lie on the surface in the positive quadrant, have been derived and their moments obtained. Some test statistics are also proposed.     

Automatic integration over a sphere

We present an algorithm for automatic integration over an N-dimensional sphere. The quadrature formula is obtained by using a trapezoidal rule after a non-linear transformation, and allows to deal with integrand singularities on the surface or in the centre of the sphere.At the basis of the theoretical development lie the construction and the selection of suitable transformations.The algorithm is cast into an automatic integration program coded as a Standard Fortran sub-routine.     

四面体的十二点二次曲面

垂心四面体中四条高的垂足,四个面的重心及从各顶点与四面体的垂心连线的三等分点,共十二个点共球.试图把垂心改为四面体内的任意点,相应地把四条高线改换为过该点与每个顶点连线的共点直线组时,则将把垂心四面体的十二点球有趣地推广为四面体的十二点二次曲面.     

Laplace方程有孔椭圆区域边值问题的解析解

本文利用相似流动替换方法 ,解决了中心有圆孔的椭园形区域上 Laplace方程第一类边值问题 ;采用分区域解法 ,给出了中心有椭园孔的椭园形区域上 Laplace方程第一类边值问题的解析通解 .这一结果在许多工程领域有重要应用 ,本文给出了油藏工程实例     

Rolling of a non-homogeneous ball over a sphere without slipping and twisting

Consider the problem of rolling a dynamically asymmetric balanced ball (the Chaplygin ball) over a sphere. Suppose that the contact point has zero velocity and the projection of the angular velocity to the normal vector of the sphere equals zero. This model of rolling differs from the classical one. It can be realized, in some approximation, if the ball is rubber coated and the sphere is absolutely rough. Recently, J. Koiller and K. Ehlers pointed out the measure and the Hamiltonian structure for this problem. Using this structure we construct an isomorphism between this problem and the problem of the motion of a point on a sphere in some potential field. The integrable cases are found.      

Slow rotation of a sphere in a non-Newtonian fluid with or without suction or injection

The flow generated by the rotation of a sphere in an infinitely extending fluid has recently been studied by Goldshtik. The corresponding problem for non-Newtonian Reiner-Rivlin fluids has been studied by Datta. Bhatnagar and Rajeswari have studied the secondary flow between two concentric spheres rotating about an axis in the non-Newtonian fluids. This last investigation was further generalised by Rajeswari to include the effects of small radial suction or injection. In Part A of the present investigation, we have studied the secondary flow generated by the slow rotation of a single sphere in non-Newtonian fluid obeying the Rivlin-Ericksen constitutive equation. In Part B, the effects of small suction or injection have been studied which is applied in an arbitrary direction at the surface of the sphere. In the absence of suction or injection, the secondary flow for small values of the visco-elastic parameter is similar to that of Newtonian fluids with inclusion of inertia terms in the Oseen approximation. If this parameter exceeds K_c = 18R/219, whereR is the Reynolds number, the breaking of the flow field takes place into two domains, in one of which the stream lines form closed loops. For still higher values of this parameter, the complete reversal of the sense of the flow takes place. When suction or injection is included, the breaking of the flow persists under certain condition investigated in this paper. When this condition is broken, the breaking of the flow is obliterated.     

Slow motion of a micropolar fluid through a porous sphere bounded by a solid sphere

The paper examines the slow motion of a micropolar fluid produced by the relative motion of a solid sphere to an inside porous sphere. The result extends the Cunningham’s problem to micropolar fluid when the inner sphere is porous with prescribed radial suction/injection velocity at the surface of the sphere. The result can also be taken as an extension of the work of Ramkissoon and Majumdar when the fluid is bounded at a radiusr=b (b>a) but the solid sphere is replaced by a porous sphere. The force experienced by the inner sphere has been calculated and particular cases of interest have been deduced.     

Free vibration analysis of a rigidly fixed viscothermoelastic hollow sphere

In this paper an exact analysis of homogenous rigidly fixed vibrations of viscothermoelastic hollow sphere is presented. The basic governing partial differential equations have been reduced to ordinary differential equations by using Helmholtz decomposition equations. The uncoupled equation is taken for first class vibrations and remains independent of temperature variations, while coupled system of equations are taken for second class vibrations. Matrix Fröbenious method of extended power series has been applied in the coupled system of differential equations to get displacements and temperature. Numerical results have been presented, giving lowest frequency, dissipation factor, displacements and temperature change.     

The motion of a particle in the non-stationary field of a logarithmic potential

The plane motion of a material point, driven by a force inversely proportional to the distance from a fixed centre of variable mass, is studied. Consideration is given to the case in which the motion may be integrated by using a specially obtained first integral.     

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