Oblate Spheroidal Shell Research Articles

Characteristics of a precessing flow under the influence of a convecting temperature field in a spheroidal shell

We present results from direct numerical simulations of flows in spherical and oblate spheroidal shells, driven both by precession and thermal convection, with Ekman number , non-diffusive Rayleigh numbers from to and unity Prandtl number. The applied precessional forcing spans seven orders of magnitude. Our experiments show a clear transition between a convective state and a precessing flow that can be approximated by a reduced dynamical model. The change in the flow is apparent in visualizations and a decomposition of the velocity into symmetric and antisymmetric components. For the flow dominated by precession, some parameter combinations show two stable solutions that can be realized by a hysteresis or a strong thermal forcing. An increase of the Rayleigh number at a constant precession rate exhibits established scaling properties for the heat transfer, with exponents and .

Oblate Spheroidal Triboelectric Nanogenerator for All‐Weather Blue Energy Harvesting

AbstractTriboelectric nanogenerators (TENGs) provide one of the most promising techniques for large‐scale blue energy harvesting. However, lack of reasonable designs has largely hindered TENG from harvesting energy from both rough and tranquil seas. In this paper, an oblate spheroidal TENG assembled by two novel TENG parts is elaborately designed for both situations. The TENG in the upper part is based on spring steel plates without other substrate materials, which makes it possible to output considerable power in rough seas and occupy small space. The TENG in the lower part consists of two copper‐coated polymer films and a rolling ball which can capture small wave energy from tranquil seas. The working mechanism and output performance are systematically studied. A maximum open‐circuit voltage of 281 V and a short‐circuit current of 76 µA can be achieved by one upper part, enough to charge a commercial capacitor for potential applications. More important, the proposed oblate spheroidal shell not only guarantees high sensitivity of the TENG in the lower part, but also qualifies the TENG with unique self‐stabilization and low consumables for the next generation of TENGs with new structural design toward all‐weather blue energy harvesting.

Coulomb energy of uniformly charged spheroidal shell systems.

We provide exact expressions for the electrostatic energy of uniformly charged prolate and oblate spheroidal shells. We find that uniformly charged prolate spheroids of eccentricity greater than 0.9 have lower Coulomb energy than a sphere of the same area. For the volume-constrained case, we find that a sphere has the highest Coulomb energy among all spheroidal shells. Further, we derive the change in the Coulomb energy of a uniformly charged shell due to small, area-conserving perturbations on the spherical shape. Our perturbation calculations show that buckling-type deformations on a sphere can lower the Coulomb energy. Finally, we consider the possibility of counterion condensation on the spheroidal shell surface. We employ a Manning-Oosawa two-state model approximation to evaluate the renormalized charge and analyze the behavior of the equilibrium free energy as a function of the shell's aspect ratio for both area-constrained and volume-constrained cases. Counterion condensation is seen to favor the formation of spheroidal structures over a sphere of equal area for high values of shell volume fractions.

Light scattering by multilayer axially symmetric particles

An exact solution to the problem of light scattering by multilayer axially symmetric particles is derived and some aspects of its computer-aided implementation are discussed. The main specific features of the solution are (i) separation of the incident, scattered, and internal fields into two parts and special selection of the scalar potentials for each of them; (ii) expansion of the potentials in terms of spherical wave functions; (iii) formulation of the problem in the form of surface integral equations; and (iv) solution of the reduced systems of the linear algebraic equations for the coefficients of the potential expansions. Mathematical justification of the solution is discussed, which is formulated in the recursive and nonrecursive form (for the T-matrix). The developed computer program has shown that the proposed approach makes it possible to consider axially symmetric particles with essentially different internal structures (i.e., with a spherical core, oblate spheroidal shell, or prolate spheroidal intermediate layer). The results of calculations of the optical properties of the multilayer nonspherical particles are presented and discussed.

Geomagnetic effects of the Earth’s ellipticity

We analyse the external field generated by a uniform distribution of magnetic susceptibility contained in an oblate spheroidal shell when it is magnetized by an internal magnetic field of arbitrary complexity. The situation is more relevant to the Earth than that of a spherical shell considered by Runcorn (1975a) (in the context of lunar magnetism), because of the larger flattening of the Earth than that of the Moon. We find that, to first order in the susceptibility, each internal harmonic in a spheroidal harmonic expansion of the magnetic potential generates just one non-vanishing external field coefficient, unlike in the spherical case when all harmonics vanish identically. The field generated is proportional to the susceptibility, thickness of the shell and square of the Earth’s eccentricity, and hence it appears that this field amplification mechanism will be very ineffective for the Earth.

Numerical Simulation of Plastic Deformation of Pressurized Oblate Spheroid-Inscribed Single-Curvature Shells

The integral hydro-bulge forming (IHBF) technology is used for manufacturing oblate spheroidal shells. A mild steel oblate spheroidal shell and a stainless steel shell were manufactured by using this new technology. The IHBF processes of the experimental shells are simulated by using an explicit finite element code, the numerical results are discussed and compared with the experimental results. Based on the numerical simulations and experimental results, a few suggestions are made for improving the new technology. [S1087-1357(00)71201-9]

Spherical and spheroidal shells as models in magnetic detection

The expressions for the scalar potentials of prolate and oblate spheroidal shells immersed in a dc uniform magnetic field are obtained. The expressions for the induced dipole moment of the shells are also evaluated. The problem is solved by finding solutions for the Laplace equation that satisfy boundary conditions at the shell surfaces. The shell thickness effect on the induced dipole moment and on its orientation are evaluated, The results appear to be useful for the analysis and for the prediction of magnetic signatures of hidden ferromagnetic objects belonging to a relatively large family.

Research on the integral hydrobulge forming technology of spheroidal shells

In the present paper an altemative novel process of manufacturing oblate spheroidal vessels is proposed: the integral hydro‐bulge forming (IHBF) technology of oblate spheroidal shells. A few mild steel and stainless steel oblate spheroidal shells for industrial use were manufactured using this new technology with one of the major diameters as high as 3 m. There is a critical value of the ratio between the vertical diameter and the horizontal diameter of the shell before forming, which can determine whether the shell will be wrinkled during hydro‐bulging. The processes were analysed afterwards, using the explicit finite element code LS‐DYNA3D. The numerical results are discussed and compared with the practical processes. Based on the numerical results a few proposals for the improvement of the IHBF technology of the spheroidal shells are presented.

AN APPROXIMATION TO THE VIBRATIONS OF OBLATE SPHEROIDAL SHELLS

This paper presents the results of investigating the axi-symmetric free vibrations of an isotropic thin oblate spheroidal shell. An oblate spheroid is considered as a continuous system constructed from two spherical shell caps by matching the continuous boundary conditions. This approximation technique is used to express the results in a continuous function analytical formation. The radii and opening angles of the spherical elements are chosen according to the eccentricity of the oblate spheroid.It is shown that for an eccentricity: smaller than 0·6 the Rayleigh method reasonably estimates natural frequencies; larger than 0·95 both of the shallow and non-shallow spherical shell theories predict natural frequencies in close agreement; and equal to zero an exact thin sphere solution emerges.The method presented herein is clearly an engineering approximation for special purpose shells and does not require the computer storage of numerical methods. Also, it may be quite useful when utilised judiciously in many applications, such as in force response analysis. Comparison is made between the present technique, the Rayleigh method, and experimental data for two laboratory models. The formulation can be extended to many types of oblate spheroids with different support conditions and still retain the functional representation.

Theoretical analysis of a class V flextensional fiber-optic interferometric hydrophone

When an oblate spheroidal shell having an aspect ratio a/b>(2 − v), where v = Poisson's ratio, is subject to hydrostatic pression, the major (a) and minor (b) axes experience strain of opposite sign [R. A. Clark and E. Reissner, J. Mech. Phys. Solids 6, 63 (1957)]. If two optical fibers, which comprise the arms of an interferometer, are wound around the equatorial and meridional circumferences of the spheroid, pressure changes induce a differential optical phase shift in the interferometer. Calculations of circumferential strain and of the optical pressure sensitivity and buckling pressure of such a sensor will be presented for thin shells (t/b≪b3/a3, where t is the shell thickness). Sample designs based on these calculations will be compared to other fiber-optic sensors of similar dimensions and materials.

Stress amplification and plastic flow for spheroidal shells: Applications to myopia

Amplification of effective stress and plastic strain rates after yielding are derived for the anisotropic stress fields of the prolate and oblate spheroidal shells as models of the myopic and hyperopic eyes. Dimensionless closed-form results are presented for arbitrary axis ratio with both constant shell thickness and variable shell thickness, using the constant scleral mass assumption. The results show that the myopic and hyperopic eyes are susceptible to failure by plastic yielding at the equator and pole, respectively, for high intraocular pressures. Experimental data from the equatorial zone of rabbit sclera shows scleral yielding and plastic flow for intraocular pressures greater than 32 mm Hg. Practical applications include glaucoma and pathological myopia.

Free vibration of an inflated oblate spheroidal shell

An approximate analytical solution of the subject problem is developed. The vibration of the oblate spheroid is assumed to be governed by the membrane theory of shells. It is also assumed that the square of eccentricity of the oblate spheroid is small and constant during inflation and that the spheroid is composed of a neo-Hookean material. The first step in the solution process concerns an exact solution of the free vibration problem of an inflated spherical shell. The free vibration of the oblate spheroid is then obtained by Galerkin's method with the modal solutions of the inflated spherical shell being used. The frequencies and mode shapes of both types of shell are given and compared to similar linear solutions. Comparisons of the behavior of oblate spheroidal and spherical shells are given. An interesting instability phenomenon of the vibration of inflated spherical shells is discussed and the constant eccentricity assumption is justified by comparisons with test results.

Free Vibrations of Thin Orthotropic Oblate Spheroidal Shells

The problem of the free vibrations of a thin orthotropic, oblate spheroidal shell has been solved by making the assumptions of orthotropic membrane theory and harmonic axisymmetric motion. The theory has reduced the differential equations of motion to a single ordinary second-order differential equation with variable coefficients. The resulting eigenvalue problem is solved by Galerkin's method. The principal directions of the elastic compliances are assumed to be along parallels of latititude and along meridians. Both the oblate spheroidal and spherical shells were studied with various orthotropic constants. The existence of finite bounds for the corresponding mode shapes has been shown. Special consideration is given to the isotropic shell as a limiting case of the orthotropic problem. The present theory in the isotropic case yields the author's previously published results for the isotropic oblate spheroidal shell, as well as the known exact solution of the isotropic spherical shell.

Time-Dependent Green's Function for Electromagnetic Waves in Moving Conducting Media

This paper treats the problem of radiation from sources of arbitrary time dependence in a moving conducting medium. The medium is assumed to be homogeneous and isotropic, with permittivity ε, permeability μ, and conductivity σ, and to move with constant velocity v with respect to a given inertial reference frame. v may have any value up to the speed of light. It is shown how the Maxwell-Minkowski equations for the electromagnetic fields in the moving medium can be integrated by means of a pair of vector and scalar potential functions analogous to those commonly used with stationary media. The wave equation associated with these potential functions is derived, and an associated scalar Green's function is defined. The solution for the Green's function is obtained in closed form, by means of a technique making use of the relation between the fundamental solution of a radiation problem and that of a corresponding Cauchy initial-value problem. The resulting Green's function is found to consist of an oblate spheroidal shell, similar to that which occurs in a lossless medium, plus a residue which persists after the shell. In addition, the Green's function is exponentially damped in both space and time, an effect not present in a lossless medium.

Free Vibrations of Thin Orthotropic Oblate Spheroidal Shells

The problem of the free vibrations of a thin orthotropic, oblate spheroidal shell has been solved by the assumptions of orthotropic membrane theory and harmonic axisymmetric motion. The theory has reduced the differential equations of motion to a single ordinary second-order differential equation with variable coefficients. The resulting eigenvalue problem is solved by Galerkin's method. The principal directions of the elastic compliances are assumed to be along parallels of latitude and along meridians. Both the oblate spheroidal and spherical shells were studied by varying the orthotropic constants. The existence of finite bounds for the corresponding mode shapes has been shown. Special consideration is given to the isotropic shell as a limiting case of the orthotropic problem. The present theory in the isotropic case yields to the author's previously published results for the isotropic oblate spheroidal shell as well as the well-known exact solution of the isotropic spherical shell.

Free Vibrations of Thin Isotropic Oblate-Spheroidal Shells

The problem of the free vibration of a thin isotropic, oblate spheroidal shell has been solved by Galerkin's method. Membrane theory and harmonic axisymmetric motion have been assumed in order to derive the differential equations of motion. This derivation leads to two ordinary differential equations with variable coefficients that can be reduced to one ordinary second-order differential equation with variable coefficients. The resulting eigenvalue problem is solved by Galerkin's method. It is shown that Galerkin's solution for the oblate spheroid yields the exact solution for the sphere as the eccentricity of the oblate spheroid goes to zero. It is also shown that two sets of frequencies exist for the oblate spheroidal shell. After the frequencies are determined, the tangential displacements are obtained as solutions of a homogeneous system of equations.

Free Vibrations of Thin Isotropic Oblate Spheroidal Shells

The free-vibration problem of a thin, isotropic, oblate spheroidal shell was solved by Galerkin's method. Membrane theory and harmonic axisymmetric motion were assumed to derive the differential equations of motion. The equations of motion lead to two ordinary differential equations with variable coefficients. The two differential equations can be reduced to one ordinary second-order differential equation with variable coefficients. This eigenvalue problem is solved by Galerkin's method. It was shown that Galerkin's solution for the oblate spheroid yields the exact solution for the sphere as the eccentricity of the oblate spheroid goes to zero. It is shown that two sets of frequencies exist for the oblate spheroidal shell. Galerkin's method converges rapidly for those of the lower set. After the frequencies are determined, the tangential displacements are obtained as solutions of a homogeneous system of equations. For numerical illustration, an oblate spheroidal shell with eccentricity equal to 0.688 was considered.