Showing papers on "Spherical shell published in 1977"

Nonlinear dynamics of boussinesq convection in a deep rotating spherical shell-i

Abstract: Convection in a rotating spherical shell has wide application for understanding the dynamics of the atmospheres and interiors of many celestial bodies. In this paper we review linear results for convection in a shell of finite depth at substantial but not asymptotically large Taylor numbers, present nonlinear multimode calculations for similar conditions, and discuss the model and results in the context of the problem of solar convection and differential rotation. Detailed nonlinear calculations are presented for Taylor number T = 105, Prandtl number P = 1, and Rayleigh number R between 1 |MX 104 and 4 |MX 104 (which is between about 4 and 16 times critical) for a shell of depth 20% of the outer radius. Sixteen longitudinal wave numbers are usually included (all even wave numbers m between 0 and 30) the amplitudes of which are computed on a staggered grid in the meridian plane. The kinetic energy spectrum shows a peak in the wave number range m = 12–18 at R = 104, which straddles the critical wav...

Molecular dynamics and chemical reactivity

Abstract: A computer simulation has been obtained of the atom recombination reaction in which the recombination energy is removed by a third-body. The equations of classical dynamics have been solved for two iodine atoms and six inert gas atoms (He, Ar or Xe) confined to a spherical vessel by a wall potential which is a spatial average of a spherical shell of inert gas atoms. Reactions at a given temperature and concentration are simulated by varying the initial momenta of the atoms and the volume of the sphere. A computer run of 100 trajectories for each physical situation gives the average time for recombination, and from this a macroscopic rate constant has been calculated that agrees well with the experimental result. The model reproduces all the characteristic kinetic mechanisms that have traditionally been used to interpret atom recombination. However, at high inert gas concentrations the steady-state approximation is shown to fail as many of the important intermediate reactions do not reach equilibrium. In t...

A new kind of 3-body problem

Abstract: A new kind of restricted 3-body problem is considered. One body,m1, is a rigid spherical shell filled with an homogeneous incompressible fluid of density ρ1. The second one,m2, is a mass point outside the shell andm3 a small solid sphere of density ρ3 supposed movinginside the shell and subjected to the attraction ofm2 and the buoyancy force due to the fluid ρ1. There exists a solution withm3 at the center of the shell whilem2 describes a Keplerian orbit around it. The linear stability of this configuration is studied assuming the mass ofm3 to beinfinitesimal. Explicitly two cases are considered. In the first case, the orbit ofm2 aroundm1 is circular. In the second case, this orbit is elliptic but the shell is empty (i.e. no fluid inside it) or the densities ρ1 and ρ3 are equal. In each case, the domain of stability is investigated for the whole range of the parameters characterizing the problem.

The closest packing of equal spheres on a spherical surface

Abstract: A table is given of putative solutions to the Fejes problem: to find the maximum value of the smallest angular distance between any two of N points movable on the surface of a sphere. Values of N run without omission up to 27 with six sporadic cases thereafter. Some applications of this system as a model are discussed.

Relativistic dynamics of spherical counter-rotating dust bodies

Abstract: Spherically symmetric dust clusters, of the type first proposed by Einstein, are generalized to nonstationaxy cases. Particular attention is paid to a massive spherical shell of particles orbiting a central body, using a method developed by Israel. Shells provide comparisons with Schwarzschild test particles, and offer as well a simplified approach to continuously distributed clusters.

Convection in a rotating deep compressible spherical shell: application to the sun

Abstract: In this paper we study the interaction of rotation with convection in a deep compressible spherical shell as the Sun's convection zone. We examine how the energy transport and the large scale motions can be affected by rotation. In particular we study how a large scale meridional circulation can give rise to variations of angular velocity with latitude and depth.

Improved design of spherical multimode hydrophone

Abstract: An improved design is made for a spherical multimode hydrophone by dividing each hemisphere into two zones and selecting the position of the line of division between them, so that the resonances associated with higher modes of the spherical shell vibrations, specifically n=3 and n=5 modes in the present case, are eliminated. The elimination of these resonances has led to the provision of a frequency‐independent dipole pattern over a wide range of frequencies. The present theory includes the effect of spherical shell bending, since it is important at higher modes. Accordingly, the resonant frequencies are clearly identified by the associated mode numbers. In both theory and experiment, these resonant effects were eliminated by employing the newly designed hydrophone.

Radiative transfer in spherical shell atmospheres. 2: Asymmetric phase functions

Abstract: Abstract In this paper we investigate the effects of sphericity on the radiation reflected from a planet with a homogeneous, conservative scattering atmosphere of optical thicknesses of 0.25 and 1.0. We considered a Henyey-Greenstein phase function with asymmetry factors of 0.5 and 0.7. Significant differences were found when these results were compared with the plane-parallel calculations. Also, large violations of the reciprocity theorem, which is only true for plane-parallel calculations, were noted. Results are presented for the radiance versus height distributions as a function of planetary phase angle. These results will be useful to researchers in the field of remote sensing and planetary spectroscopy.

Plates and shells with cracks : a collection of stress intensity factor solutions for cracks in plates and shells

Abstract: 1 Interaction of arbitrary array of cracks in wide plates under classical bending.- 1.1 Introduction.- 1.2 Basic relations.- 1.3 Complex potentials for traction free cracks.- 1.4 Arbitrary array of cracks in wide plate.- 1.5 Numerical results.- 1.6 Discussions.- References.- 2 Improved approximate theories of the bending and extension of flat plates.- 2.1 Introduction.- 2.2 Approximate theories by variational methods.- 2.3 Applications to crack problems.- 2.4 Guidelines for practical applications.- References.- 3 Through cracks in multilayered plates.- 3.1 Introduction.- 3.2 Minimum complementary energy applied to a layered plate.- 3.3 An approximate three-dimensional theory of multi-layered plates.- 3.4 Through crack in a layered plate.- 3.5 Stress distribution across the plate thickness.- 3.6 Discussion of numerical results.- 3.7 Appendix: Definition of constants.- References.- 4 Asymptotic approximations to crack problems in shells.- 4.1 Introduction.- 4.2 General theory - classical.- 4.3 The stress field in a cracked spherical shell.- 4.4 The stress field in a cracked plate.- 4.5 The stress field in a cracked cylindrical shell.- 4.6 Approximate stress intensity factors for other shell geometries.- 4.7 Plates on elastic foundations.- 4.8 Particular solutions.- 4.9 Discussion.- References.- 5 Crack problems in cylindrical and spherical shells.- 5.1 Introduction.- 5.2 Formulation of the specially orthotropic cylindrical shell problem.- 5.3 The skew-symmetric problem.- 5.4 The symmetric problem.- 5.5 Results for a specially orthotropic cylindrical shell.- 5.6 The effect of Poisson's ratio.- 5.7 Interaction of two cracks.- 5.8 Further results for isotropic shells.- References.- 6 On cracks in shells with shear deformation.- 6.1 Introduction.- 6.2 Shell theory with shear deformation.- 6.3 Symmetric loading.- Appendix: Integrand and Kernel functions.- References.- 7 Dynamic analysis of cracked plates in bending and extension.- 7.1 Introduction.- 7.2 Classical plate bending theory.- 7.3 Mindlin's theory of plate bending.- 7.4 Kane-Mindlin's equation in plate extension.- 7.5 Plates subjected to sudden loading.- References.- 8 A specialized finite element approach for three-dimensional crack problems.- 8.1 Introduction.- 8.2 Three-dimensional elastic calculations.- 8.3 Finite element method - background.- 8.4 Specialized elements for the crack edge.- 8.5 Applications to crack problems.- 8.6 Details of the analysis.- 8.7 Results of the finite element analysis.- 8.8 Summary.- References.- Author's Index.

Ion beam inertial confinement target

Abstract: A target for implosion by ion beams composed of a spherical shell of frozen DT surrounded by a low-density, low-Z pusher shell seeded with high-Z material, and a high-density tamper shell. The target has various applications in the inertial confinement technology. For certain applications, if desired, a low-density absorber shell may be positioned intermediate the pusher and tamper shells.

Dynamic expansion of a compressible hyperelastic spherical shell

Abstract: This paper is concerned with the finite spherically symmetric motion of a compressible hyperelastic spherical shell, subjected to a spatially uniform step funtion application of pressure at its inner surface. A method, given in a previous paper [1], for the determination of the field of characteristics, for expansion of a spherical cavity in an unbounded solid is adapted to consider the spherical shell problem. Results are presented graphically for a particular strain energy function and are compared with results obtained for an incompressible material and from linear elasticity theory.

Convection in rotating stars

Abstract: It is shown that many features of convection in rotating spheres and spherical shells can be understood on the basis of plane layer models. The phenomenon of differential rotation generated by convection is emphasized. The potential applications and limitations of analytical and numerical models for problems of astrophysical interest are briefly discussed.

Transient displacement and stress in non-homogeneous elastic shells

Abstract: Transient displacement and stress in thick inhomogeneous cylindrical and spherical shells is investigated for situations when the surfaces are subjected to dynamic loads consistent with the production of radial vibrations. In the first instance, transformation of the governing equations is sought to a form amenable to solution via a finite Hankel transform technique. Such reduction is available for a broad class of radial variations in the material density and elastic parameters. However, the solutions generated by this method are rather involved and two other approaches are indicated. Thus, reduction of the system to the conventional wave equation is obtained under certain restrictions on the nature of the inhomogeneities and a simple correspondence principle is presented for the imposed boundary stress boundary value problem. Finally an asymptotic wave front analysis is presented which has a wide generality of application.

Thermal stability in a rotating spherical fluid shell with non-uniform heating

Abstract: The theory is developed for the convective stability of a rotating spherical shell of fluid upon which is initially imposed a stable thermally induced shear flow. The fluid shell contains heating sources which are distributed proportional to the sine of the polar angle squared. Thus the analysis has a number of similarities to some geophysical flow situations. It is found that the properties of the solution are strongly dependent on the initial conditions. Thus to obtain further insight concerning the stability of the system numerical solutions are obtained at two shell thicknesses. The critical values of the Taylor number ($T$) and the Rayleigh number ($C$) are generally similar to those found in previous studies of rotating fluid shells. However the effect of the initial shear flow is to reduce the critical value of $C$ for a given $T$, below that found for uniform heating and an initially quiescent state. The flows obtained at the onset of instability are toroidal cells which vary in number dependent on $T$ and $C$. A maximum of six cells are found at large values of $T$. A significant effect of the initial shear flow is the occurrence of a rapid change in stability when the number of toroidal cells changes.

The asymptotic distribution of torsional eigenfrequencies of a spherical shell. II.

Abstract: In the two previous papers of this series (SATO and LAPWOOD, 1977a, b) we examined approximate methods for calculating eigenfrequencies of radial overtones of torsional oscillations of spherically symmetrical shells. For shells composed of uniform layers we were able to obtain an exact frequency equation, in terms of spherical Bessel functions, for which roots could be computed with any desired precision. They thus supplied a standard for the measurement of the accuracy of approximate methods.In applications to shells of two and three uniform layers, which were simple representations of an Earth with inner surfaces of discontinuity, we noted the presence of the solotone effect, which is the existence of recurring patterns of eigenfrequencies owing to internal reflection.In this paper we take up the analysis of the solotone effect, showing how it may be predicted from knowledge of the Shell structure, and how it may be interpreted in terms of ray theory. Applications to the same Earth-models as used before show that for them the theory of the solotone gives an excellent fit to the precisely computed eigenfrequencies. The pattern of eigenfrequencies proves to be very sensitive to changes in layer thickness, and thus offers the possibility of future use in determining the positions of surfaces of discontinuity within the mantle of the Earth.

Radial deformation of a nonhomogeneous spherically isotropic elastic sphere with a concentric spherical inclusion

Abstract: The elasticity of a spherically isotropic medium bounded by two concentric spherical surfaces subjected to normal pressures is discussed. The material of the structure is spherically isotropic and, in addition, is continuously inhomogeneous with mechanical properties varying exponentially as the square of the radius. An exact solution of the problem in terms of Whittaker functions is presented. The St. Venant’s solution in the case of homogeneous material and Lame’s solution in the case of homogeneous isotropic material are derived from the general solution. The problem of a solid sphere of the same medium under the external pressure is also solved as a particular case of the above problem. Finally, the displacements and stresses of a composite sphere consisting of a solid spherical body made of homogeneous material and a nonhomogeneous concentric spherical shell covering the inclusion, both of them being spherically isotropic, are obtained when the sphere is under uniform compression.

Two New Charged-Particle Diagnostics for Laser-Fusion Experiments

Abstract: Illuminating a gas-filled spherical shell pellet by a high power laser is now a conventional laser-fusion experiment. During laser energy deposition, the target material can be conveniently divided into three regions — the outer corona, consisting of a hot plasma; the tamper, which is the relatively cool remainder of the shell; and the fuel gas at the interior. The fusion burn reaction is most efficient if the fuel can be compressed to a very high density before appreciable heating has occurred. A number of experiments have demonstrated that compression and heating do occur, and that neutrons are generated by the fusion reaction1, but many questions remain about the exact nature of the energy transfer process from the laser to the fuel.

Hhshell water heater

Abstract: PURPOSE:Facilitation of production and assembly,and improvement of reliabilty which are accomplished by forming a water heater in a hair-pin multi-tube heat exchanger out of two drums joined by a spherical shell.

Low‐frequency spherical‐shell projectors

Abstract: The spherical‐shell projector comprises a piezoelectric ceramic ring driving two concave spherical shell segments in flexure. One model resonant at 600 Hz in water weighs 15 kg and delivers a souce level of 201.5 dB re 1 μPa at 1 m. Its mechanical Q is 5 and efficiency 80% at resonance. An axially‐symmetric finite element model is being used to study the performance of spherical‐shell projectors and generally good agreement is achieved between the calculated and experimental results.

The gravimetric density formula for a spherical shell

Abstract: The formula for the gravimetric density (ρ) of material, between two stations a vertical distance ΔZ apart in a borehole, is (Hammer, 1950): ρ=14πkF-ΔgΔZ,where k is the gravitational constant, F the free air gradient (the derivative of gravity with respect to depth in the absence of matter), and Δg the change in gravity. This formula is generally derived by integrating the vertical component of gravity caused, at a measuring point, by an infinite horizontal plate of thickness t, and then doubling it to account for one measuring point above and below the plate. This results in (Heiskanen and Vening‐Meinesz, 1958), Δg=4πkρt.In fact, however, the earth is not made up of infinite horizontal plates; it can perhaps more reasonably be regarded as a series of spherical shells. It is therefore instructive to derive the expression for the change in gravity associated with passing through one of these spherical shells.