Showing papers on "Spherical shell published in 1992"

Spiralling columnar convection in rapidly rotating spherical fluid shells

Abstract: It is shown that the fundamental features of both thermal instabilities and the corresponding nonlinear convection in rapidly rotating spherical systems (in the range of the Taylor number 109 < T < 1012) are determined by the fluid properties characterized by the size of the Prandtl number. Coefficients of the asymptotic power law for the onset of convection at large Taylor number are estimated in the range of the Prandtl number 0.1 ≤ Pr ≤ 100. For fluids of moderately small Prandtl number, a new type of convective instability in the form of prograde spiralling drifting columnar rolls is discovered. The linear columnar rolls extend spirally from near latitude 60° to the equatorial region, and each spans azimuthally approximately five wavelengths with the inclination angle between a spirally elongated roll and the radial direction exceeding 45°. As a consequence, the radial lengthscale of the linear roll becomes comparable with the azimuthal lengthscale. A particularly significant finding is the connection between the new instability and the predominantly axisymmetric convection. Though non-axisymmetric motions are preferred at the onset of convection, the nonlinear convection (at the Rayleigh number of the order of (R—Rc)/Rc = O(0.1)) bifurcating supercritically from the spiralling mode is primarily dominated by the component of the axisymmetric zonal flow, which contains nearly 90% of the total kinetic energy. For fluids of moderately large Prandtl numbers, thermal instabilities at the onset of convection are concentrated in a cylindrical annulus coaxial with the axis of rotation; the position of the convection cylinder is strongly dependent on the size of the Prandtl number. The associated nonlinear convection consists of predominantly non-axisymmetric columnar rolls together with a superimposed weak mean flow that contains less than 10% of the total kinetic energy at (R—Rc)/Rc = O(0.1). A double-layer structure of the temperature field (with respect to the basic state) forms as a result of strong nonlinear interactions between the nonlinear flow and the temperature field. It is also demonstrated that the aspect ratio of the spherical shell does not substantially influence the fundamental properties of convection.

Three-dimensional convection of an infinite-Prandtl-number compressible fluid in a basally heated spherical shell

Abstract: A numerical investigation is made of the effects of compressibility on threedimensional thermal convection in a basally heated, highly viscous fluid spherical shell with an inner to outer radius ratio of approximately 0.55, characteristic of the Earth’s whole mantle. Compressibility is implemented with the anelastic approximation and a hydrostatic adiabatic reference state whose bulk modulus is a linear function of pressure. The compressibilities studied range from Boussinesq cases to compressibilities typical of the Earth’s whole mantle. Compressibility has little effect on the spatial structure of steady convection when the superadiabatic temperature drop across the shell AT,, is comparable to a characteristic adiabatic temperature. When AT,& is approximately an order of magnitude smaller than the adiabatic temperature, compressibility is significant. For all the non-Boussinesq cases, the regular polyhedral convective patterns that exist at large AZ, break down at small AT, into highly irregular patterns ; as AZ, decreases convection becomes penetrative in the upper portion of the shell and is strongly time dependent at Rayleigh numbers only ten times the critical Rayleigh number, (Ra),,. Viscous heating in the compressible solutions is concentrated around the upwelling plumes and is greatest near the top and bottom of the shell. Solutions with regular patterns (and large AEJ remain steady up to fairly high Rayleigh numbers (100(Ra),,), while solutions with irregular convective patterns are time dependent at similar Rayleigh numbers. Compressibility affects the pattern evolution of the irregular solutions, producing fewer upwelling plumes with increasing compressibility.

Galactic accretion and angular momentum re-orientation

Abstract: The generation of galactic angular momentum by tidal forces is re-examined The tidal torques are calculated in linear theory on a spherical shell centred either on a random point or on a peak in the smoothed density field It is shown that when the torque is decomposed into contributions from multipoles, the dipole term is more important than the quadrupole, and higher multipoles are unimportant When the dipole contribution is taken into account, the torques of thin shells of radii r and r' are anticorrelated for r/r' ≥ 2 Significant contributions to the torque on any shell come from shells that are more than five times as big The magnitude of the torque on a shell does not depend greatly on whether the shell is centred on a peak in the smoothed density field rather than on a random point The dimensionless angular momentum parameter λ of virialized objects remains fairly constanty at λ ⇒ 005 as the object grows as a result of cosmic infall

Bellows sealed ball joint

Abstract: A bellows sealed ball joint comprising an inner spherical shell, a bellows and an outer spherical shell. The inner spherical shell comprises a plurality of elongated non-spherical areas or indentations which receive a plurality of locking surfaces on the outer spherical shell. The inner and outer spherical shells can then be rotated with respect to each other a predetermined distance until the spherical locking surfaces on the outer spherical shell operatively engage concentric spherical locking surfaces on the inner spherical shell. Once the locking surfaces of the inner and outer spherical shells are in operative engagement, the bellows ball joint can pivot freely without becoming disconnected. A final joining seal is then made at each end of the bellows to prevent the spherical sections from unlocking and to provide a hermetic seal. In an alternate embodiment, the inner spherical shell is shown as having elongated slots in place of the non-spherical areas. A support ring may also be used to facilitate supporting the locking surfaces on the outer spherical shell against the locking surfaces of the inner spherical shell.

Numerical models of mantle convection

Abstract: An overview of numerical methods describing the structure and dynamics of the mantle is presented with attention given to novel 3D modeling techniques. The paper reviews 3D spherical and Cartesian models for constant viscosity emphasizing the assumptions regarding style of convection, time dependence, and implications for the mantle. Similarly treated are 3D Cartesian models with temperature-dependent viscosities, and briefly examined are models that are based on compressibility, nonlinear viscosity, or plates. Extensive illustrations are presented detailing: (1) temperature variations from models of 3D thermal convection in spherical shells; (2) thermal anomalies in equatorial cross sections; and (3) temperature variations in a spherical shell heated from within. The discussion relates the numerical results of the models with real mantle-convection events, and the simulations are shown to yield increasingly realistic representations of material behavior.

Elastic Helmholtz resonators

Abstract: The influence of wall elasticity on the response of a Helmholtz resonator is examined by analyzing the canonical case of a thin elastic spherical shell with a circular aperture subject to plane wave excitation. By neglecting the thickness of the wall and representing the elasticity via a ‘‘thin shell’’ theory the problem is reduced to one of solving an integral equation over the aperture for the polarization velocity, which is related to, but distinct from, the radial particle velocity of the fluid. The integral equation can be solved by asymptotic methods for small apertures, yielding closed‐form expressions for the major resonator parameters. In general, wall compliance reduces the resonance frequency in comparison with an identically shaped rigid cavity. The Q value of the resonance is increased and the scattering strength of the cavity at resonance is enhanced by wall compliance. The asymptotic results are supported and supplemented by numerical calculations for thin steel shells in water.

Ab Initio Phase Determination for Viruses with High Symmetry: a Feasibility Study

Abstract: Conditions that would permit the complete structure determination of spherical viruses that have high internal symmetry were examined starting only from an initial spherical shell model. Problems were considered that might arise due to the following. 1. Creation of centric phases due to the simple shell model and its position in the unit cell. The centric symmetry can generally be broken on averaging an initial electron density map based on observed structure amplitudes, provided that the internal molecular symmetry is sufficiently non-parallel to the crystallographic symmetry. 2. Choice of the average model shell radius. Some incorrect radii led to the Babinet opposite solution (electron density is negative instead of positive). Phases derived from other models with incorrect radii failed to converge to the correct solution. 3. Error in structure amplitude measurements. 4. Lack of a complete data set. 5. Error in positioning the initial spherical-shell model within the crystal unit cell. It was found that an error of 1.6 A caused noticeable phasing error at a resolution greater than 20 A.

Ab initio phase determination for spherical viruses: parameter determination for spherical-shell models.

Abstract: The structure determination of canine parvovirus depended on the extension of phases calculated initially from a spherical-shell model [Tsao, Chapman, Wu, Agbandje, Keller & Rossmann (1992). Acta Cryst. B48, 75-88]. Such ab initio phasing holds the promise of obviating initial experimental phasing by isomorphous or molecular replacement, thereby expediting the structure determinations of spherical virus capsids. In this paper, it is shown how parameters such as radii, DNA density and particle positions may be determined and refined from diffraction data with sufficient precision to start phase extension from 20 A resolution for a virus of approximately 122 A radius.

Sound scattering from an internally loaded cylindrical shell

Abstract: Sound scattering from a cylindrical shell with a mass‐spring system diametrically attached inside it is examined analytically. Explicit solutions for both the shell motions and the scattered acoustic field are derived. It is shown that the attachment of the simple mass‐spring loading causes significant changes in the acoustic behavior of the shell. The scattering from the internally loaded shell is shown to be dominated by the interactions between structural waves in the shell and the attachment, which have an overall resonant effect, manifested by a series of scalloplike variations in the scattering form function in the low‐ and intermediate‐frequency domain. In the low‐frequency region, it is shown that in contrast to the case of an empty shell where scattering is essentially controlled by the total mass of the scatterer, the acoustic behavior of the internally loaded shell is basically determined by waves in the shell, and, hence, is elasticity controlled. It is found that the shell deformation is domi...

Resonance and interference scattering near the coincidence frequency of a thin spherical shell: An approximate ray synthesis

Abstract: An enhancement of the steady‐state backscattering by an evacuated thin spherical shell is synthesized using surface‐guided rays. This midfrequency enhancement has been predicted to occur near the critical coincidence frequency of thin shells in water. On the high‐frequency side of the enhancement, the nearly periodic scattering variations primarily arise in the ray picture from the interference of a contribution of a subsonic guided wave with the specular reflection. Distinct resonances become important at lower frequencies. The pole locus of the complex partial‐wave index is shown for the subsonic guided wave in the region where the coupling of the wave with the incident and scattered acoustic fields increases. The sphere studied is 2.5% thick stainless steel.

Weight adjustable dumbbell

Abstract: A weight adjustable dumbbell including two spherical shells joined by a short bar, each spherical shell comprising two equal parts connected by a respective lock bolt, with circular cushion plates retained therebetween to hold any of a variety of counter weights, wherein the circular cushion plates have a center hole, through which the respective lock screw bolt is inserted, a plurality of projections spaced around a peripheral edge thereof at equal intervals extending out of the respective spherical shell for protecting the spherical shells against impact, and two annular grooves on two opposite sides around the border into which the peripheral edge of either semi-spherical shell of the respective spherical shell fits respectively.

Thermal convection in the Earth's mantle: Mode coupling induced by temperature‐dependent viscosity in a three‐dimensional spherical shell

Abstract: Temperature-dependent viscosity places constraints on the spherical harmonic spectrum of thermal convection in the earth's mantle. Viscosity, being a strong function of temperature, gives rise to quantized feed-back loops among low-order modes in the governing equations. This nonlinear resonance is proposed as the cause of the dominance of a low-order convective pattern in the earth's mantle.

Magnetic field sources for producing high-intensity variable fields

Abstract: A permanent magnet having a shell of magnetic material and a hollow cavity. The shell is permanently magnetized to produce an axially tapered magnetic field in the cavity. In one embodiment, a segmented spherical magnetic shell has a concentric spherical cavity. An access port in the form of an axial hole passes through polar segments of the shell along its polar axis P. The shell ("magic sphere") is magnetized such that it is capable of producing a tapered magnetic field in the cavity. Both the magnitude and direction of the remanence BR of the shell material vary from segment to segment. Another embodiment shows a segmented cylindrical shell in the form of a ("magic ring"). The shell material has a magnetic remanence BR that varies in magnitude and direction from segment to segment. The shell produces a axially tapered field in the cavity.

Galerkin Finite Element Displacement Formulation of Coupled Thermoelasticity Spherical Problems

Abstract: When the time rate of change of the thermomechanical forces in a continuum is high enough to produce stress wave, the solution must be sought through the simultaneous consideration of the first law of thermodynamics and the equations of thermoelasticity. The general finite element formulation of coupled thermoelasticity along with a general discussion for inclusion of mechanical and thermal boundary conditions is presented. The case is then considered for a spherical symmetry and the governing coupled thermoelastic equations are reduced for a thick sphere. Based on the Galerkin method, a Kantrovich approximation is applied to the displacement and temperature field and the finite element formulation of the problem is obtained.

Some new closed-form solutions describing spherical inflation in compressible finite elasticity

Abstract: The qualitative features of the inflation of thin shells of these special materials are studied. The related problems of spherical eversion and cylindrical inflation are also briefly considered

Axisymmetric non-linear analysis of laminated orthotropic annular spherical shells

Abstract: The problem of axisymmetric non-linear behaviour of laminated orthotropic spherical shells is formulated using the principle of virtual work. The efficacy and applicability of the method of global interior collocation for the solution of the title problem are investigated. The zeroes of a Chebyshev polynomial are used as the collocation points. Parametric studies are conducted to bring out the effects of factors like orthotropic ratio, R / h ratio, shear deformation and opening angle on the non-linear static response of two-layered annular cross-ply shells. The results using the analytical solution presented herein should serve as bench-mark numerical results to test the accuracy of non-linear composite shell elements employed in finite element analysis.

Nonlinear planetary dynamos in a rotating spherical shell. II: The post-Taylor equilibration for α2-dynamos

Abstract: The magnetic field of a rapidly rotating α2 dynamo is computed in a spherical shell as a function of Reynolds and Ekman numbers. It is found that at high enough Reynolds number a Taylor state is reached, after which the solution becomes inviscid. This result confirms the recent work of Hollerbach and Ierley (1991) in a full sphere.

Exact controllability of a shallow shell model

Abstract: A coupled reversible system for the vibrations of a shallow spherical shell is given. The exact controllability is studied using HUM method of J.L. Lions and uniqueness results are obtained for small values of the coupling parameter. The efficiency of a spectral numerical approximation method is illustred by the numerical experiments.