Oblate Spheroidal Geometry Research Articles

Rapidly rotating self-gravitating Boussinesq fluid. IV. Onset of multimodal thermal convection influenced by oblate spheroidal geometry

Rapidly rotating self-gravitating Boussinesq fluid. IV. Onset of multimodal thermal convection influenced by oblate spheroidal geometry

Relativistic Mechanics in Gravitational Field within Oblate Spheroidal Coordinates Based upon Riemannian Geometry for Rotating Homogeneous Mass Distribution

The emergence of the geometrical theory of gravitation (general relativity) by Albert Einstein in his quest to unite special relativity and the Newtonian law of universal gravitation has led to several Mathematical approaches for the exact and analytical solution for all gravitational fields in nature. The first and the most famous analytical solution was the Schwarzchild’s which can be constructed by finding a mapping where the metric tensor takes a simple form i.e. the vanishing of the non-diagonal elements. In this paper, we construct exact solution of the Einstein geometrical gravitational field equation using Riemannian metric tensor called the golden metric tensor that was first developed by Howusu, in the year 2009, for the rotating homogeneous mass distribution within oblate Spheroidal Coordinates. The equations of motion for test particles in the Oblate Spheroidal Geometry were derived using the coefficient of affine connection. Then the law of conservation of momentum and energy are equivalently formulated using the generalized Lagrangian as compared to the analytical solution of the Schwarzchild’s gravitational field. We also derived the planetary equation of motion in the equatorial plane of the Oblate Spheroidal body for this gravitational field.

The Radiation Problem of a Submerged Oblate Spheroid in Finite Water Depth Using the Method of the Image Singularities System

This study examines the hydrodynamic parameters of a unique geometry that could be used effectively for wave energy extraction applications. In particular, we are concerned with the oblate spheroidal geometry that provides the advantage of a wider impact area on waves, closer to the free surface where the hydrodynamic pressure is higher. In addition, the problem is formulated and solved analytically via a method that is robust and most importantly very fast. In particular, we develop an analytical formulation for the radiation problem of a fully submerged oblate spheroid in a liquid field of finite water depth. The axisymmetric configuration of the spheroid is considered, i.e., the axis of symmetry is perpendicular to the undisturbed free surface. In order to solve the problem, the method of the image singularities system is employed. This method allows for the expansion of the velocity potential in a series of oblate spheroidal harmonics and the derivation of analytical expressions for the hydrodynamic coefficients for the translational degrees of freedom of the body. Numerical simulations and validations are presented taking into account the slenderness ratio of the spheroid, the immersion below the free surface and the water depth. The validations ensure the correctness and the accuracy of the proposed method. Utilizing the same approach, the whole process is implemented for a disc as well, given that a disc is the limiting case of an oblate spheroid since its semi-minor axis approaches zero.

Non-Equilibrium Thermodynamics-Based Convective Drying Model Applied to Oblate Spheroidal Porous Bodies: A Finite-Volume Analysis

Commonly based on the liquid diffusion theory, drying theoretical studies in porous materials has been directed to plate, cylinder, and sphere, and few works are applied to non-conventional geometries. In this sense, this work aims to study, theoretically, the drying of solids with oblate spheroidal geometry based on the thermodynamics of irreversible processes. Mathematical modeling is proposed to describe, simultaneously, the heat and mass transfer (liquid and vapor) during the drying process, considering the variability of the transport coefficients and the convective boundary conditions on the solid surface, with particular reference to convective drying of lentil grains at low temperature and moderate air relative humidity. All the governing equations were written in the oblate spheroidal coordinates system and solved numerically using the finite-volume technique and the iterative Gauss–Seidel method. Numerical results of moisture content, temperature, liquid, vapor, and heat fluxes during the drying process were obtained, analyzed, and compared with experimental data, with a suitable agreement. It was observed that the areas near the focal point of the lentil grain dry and heat up faster; consequently, these areas are more susceptible to the appearance of cracks that can compromise the quality of the product. In addition, it was found that the vapor flux was predominant during the drying process when compared to the liquid flux.

Drying of Oblate Spheroidal Solids via Model Based on the Non-Equilibrium Thermodynamics

Food drying is one of the most used methods of preservation. To accurately describe moisture migration within biological products (grains, fruits, vegetables, etc.) during drying and explain the effects of this process on the quality of the material, have been proposed several mathematical models, but few incorporate the phenomena of simultaneous heat and mass transport applied to complex geometry. In this sense, this paper aims to present a mathematical model, based on the thermodynamics of irreversible processes to describe the heat and mass transfer (liquid and vapor) during the drying of bodies with oblate spheroidal geometry. This model was applied to describe drying of lentil, considering the variables transport coefficients and equilibrium conditions at the surface of the solid. Results of the average moisture content, average temperature, liquid flux, vapor flux, and moisture content and temperature distributions inside a lentil kernel during drying process, at different temperatures (40 and 60 oC) were presented and analyzed.

Analytical Solution to the Vinti Problem in Oblate Spheroidal Equinoctial Orbital Elements

Equinoctial orbital elements were recently generalized from spherical geometry to the oblate spheroidal geometry of Vinti theory. For the symmetric Vinti potential, which accounts exactly for oblateness, these nonsingular elements are defined for all nondegenerate orbital regimes and resolve the usual problems found in the classical elements associated with angle ambiguities. In the present work, the generalized equinoctial elements are used to solve Vinti’s initial value problem, leading to a fully nonsingular analytical solution for bounded orbits. The result is akin to deriving the equinoctial form of Kepler’s equation for the two-body problem and then solving it, formally completing the introduction of the new equinoctial element set. Derivations of the equinoctial integrals of the motion are included, as well as techniques to eliminate all polar orbit singularities and solve a generalized Kepler’s equation. Multiple examples are presented. Code for predicting the Vinti orbit in oblate spheroidal equinoctial elements is provided online as supplementary material.

Equinoctial elements for Vinti theory: Generalizations to an oblate spheroidal geometry

Vinti theory constructs orbits on an oblate spheroidal geometry, naturally encoding the gravitational potential of an oblate spheroid in the coordinates. Classical techniques use spherical geometry. Recent work applied Vinti theory to the relative motion problem by way of a linear dynamical model, which is nonsingular in the oblate spheroidal element space. But as with classical spherical elements, the linear mapping between classical spheroidal elements and inertial rectangular coordinates becomes singular for small eccentricities and/or inclinations in the sense of linear dependence of columns in the Jacobian. To mitigate these practical issues, the standard (spherical) equinoctial elements are chosen to inform in a natural way their generalization to a new nonsingular element set: the oblate spheroidal equinoctial orbital elements. The spherical equinoctial elements can be considered a special case of the spheroidal equinoctial elements in the same way that spherical coordinates can be considered a special case of oblate spheroidal coordinates. The new element set is defined and algorithms for converting between spheroidal equinoctial elements and inertial coordinates are derived. Similarities and differences between spheroidal and spherical equinoctial elements are emphasized for clarity, both in terms of the form of equations and geometrical interpretation. The transformations are valid away from the nearly rectilinear orbit regime and are exact except near the poles. When near the poles, the transformations match the accuracy of the approximate analytical solution, which has been developed to the third order in J2 in the literature. As a result, the singularity on the poles is completely eliminated for the first time.

Radiation damping of, and scattering from, an arbitrarily shaped bubble.

The work by Strasberg on small volume oscillations of bubbles with arbitrary shape [J. Acoust. Soc. Am. 25, 536-537 (1953)] is here extended to include the effects of radiation damping. An expression for the far-field scattering amplitude from an arbitrarily shaped bubble is given in terms of a quantity that is mathematically equivalent to the electrostatic capacitance of the bubble shape. This general approach is then applied to prolate and oblate spheroidal geometries, for which simple analytic expressions are available for the electrostatic capacitance, and the resulting far-field scattering amplitudes are compared to previous work in the literature.

Hydrodynamics of non-axisymmetric oblate spheroids below a free surface

We develop an analytical methodology for solving the hydrodynamic diffraction and the radiation problems by immersed oblate spheroids in a liquid field of infinite extend bounded from above by a free-surface. In turn, free-surface effects are included in the formulation of the hydrodynamic problem allowing the investigation of the hydrodynamic characteristics of the multi-purpose oblate spheroidal geometry that runs from a sphere to a circular disk depending on its slenderness parameter. The case considered here is that of a “non-axisymmetric” placement of a fully wetted spheroid, namely when its semi-minor axis is parallel to the free-surface, extending the existing efforts which have considered only the “symmetric” case (i.e. semi-minor axis perpendicular to the free-surface). The analytic solution is sought by constructing and using the multipole potentials of the problem that express the fundamental Green’s function in terms of oblate spheroidal harmonics. The adopted methodology is based on a Fredholm integral equation (of the second kind) approach that provides the source distribution over the surface of the spheroid. The source distribution is accordingly obtained by considering the potentials in the interior and the exterior of the spheroid and expressing them in spheroidal harmonics. The derivation of the multipole potentials, enable us to obtain analytic expressions for the diffraction and the radiation velocity potentials in terms of the involved harmonics. Several numerical tests have been performed to calculate the exciting forces and moments as well as the hydrodynamic parameters of the spheroid. Special attention has been paid to the limiting case of a circular disk subjected to head seas, for which the hydrodynamic effects are most pronounced. The other limiting case, i.e. that of a sphere, is well documented and was used for validation purposes.

Moisture profiles during intermittent soaking of an oblate spheroid

Oblate spheroidal geometry and Fickian diffusion with constant diffusivity are used to model moisture absorption in a sorghum kernel with an aspect ratio of 1.48 during intermittent soaking. During the soaking phases the surface is assumed to be saturated while during the drain phases convective transport between the surroundings and the surface is assumed. An explicit finite difference scheme is used to solve the dimensionless form of the diffusion equation. For a three-cycle scenario with a 30-min soak, 2-h drain, 30-min soak, 5-h drain, and a 64-h soak, it is shown that the moisture distribution in the kernel is more uniform (as measured by the standard deviation of the moisture profile) than during a constant soaking scenario regardless of the assumed values of the Biot modulus and the equilibrium surface moisture concentration during the drain phases.

On the transversally isotropic pressure effect on avascular tumor growth

A typical mathematical formulation for describing avascular tumor growth is considered in the present work, in the frame of oblate spheroidal geometry focusing on the kind of exterior conditions under which such a geometrical structure is attainable. It turns out that given a transversally isotropic pressure field, an avascular tumor can exhibit an oblate spheroidal growth, only if the nutrient supply is provided in a specific form related to the pressure and following the tumor evolution. A geometrical reduction to the prolate geometry and recovering of existing results for the sphere are included. Copyright © 2013 John Wiley & Sons, Ltd.

The Dielectric Behavior of Nonspherical Biological Cell Suspensions: An Analytic Approach

The influence of the cell shape on the dielectric and conductometric properties of biological cell suspensions has been investigated from a theoretical point of view presenting an analytical solution of the electrostatic problem in the case of prolate and oblate spheroidal geometries. The model, which extends to spheroidal geometries the approach developed by other researchers in the case of a spherical geometry, takes explicitly into account the charge distributions at the cell membrane interfaces. The presence of these charge distributions, which govern the trans-membrane potential Δ V, produces composite dielectric spectra with two contiguous relaxation processes, known as the α-dispersion and the β-dispersion. By using this approach, we present a series of dielectric spectra for different values of the different electrical parameters (the permittivity ɛ and the electrical conductivity σ, together with the surface conductivity γ due to the surface charge distribution) that define the whole behavior of the system. In particular, we analyze the interplay between the parameters governing the α-dispersion and those influencing the β-dispersion. Even if these relaxation processes generally occur in well-separated frequency ranges, it is worth noting that, for certain values of the membrane conductivity, the high-frequency dispersion attributed to the Maxwell-Wagner effect is influenced not only by the bulk electrical parameters of the different adjacent media, but also by the surface conductivity at the two membrane interfaces.

SUNYAEV-ZEL'DOVICH EFFECT OBSERVATIONS OF THE BULLET CLUSTER (1E 0657–56) WITH APEX-SZ

We present observations of the Sunyaev-Zel'dovich effect (SZE) in the Bullet cluster (1E 0657–56) using the APEX-SZ instrument at 150 GHz with a resolution of 1'. The main results are maps of the SZE in this massive, merging galaxy cluster. The cluster is detected with 23σ significance within the central 1' radius of the source position. The SZE map has a broadly similar morphology to that in existing X-ray maps of this system, and we find no evidence for significant contamination of the SZE emission by radio or IR sources. In order to make simple quantitative comparisons with cluster gas models derived from X-ray observations, we fit our data to an isothermal elliptical β model, despite the inadequacy of such a model for this complex merging system. With an X-ray-derived prior on the power-law index, β = 1.04+0.16 –0.10, we find a core radius r c = 142'' ± 18'', an axial ratio of 0.889 ± 0.072, and a central temperature decrement of –771 ± 71 μKCMB, including a ±5.5% flux calibration uncertainty. Combining the APEX-SZ map with a map of projected electron surface density from Chandra X-ray observations, we determine the mass-weighted temperature of the cluster gas to be T mg = 10.8 ± 0.9 keV, significantly lower than some previously reported X-ray spectroscopic temperatures. Under the assumption of an isothermal cluster gas distribution in hydrostatic equilibrium, we compute the gas mass fraction for prolate and oblate spheroidal geometries and find it to be consistent with previous results from X-ray and weak-lensing observations. This work is the first result from the APEX-SZ experiment, and represents the first reported scientific result from observations with a large array of multiplexed superconducting transition-edge sensor bolometers.

Low-frequency scattering from perfectly conducting spheroidal bodies in a conductive medium with magnetic dipole excitation

Inductive electromagnetic means that are currently employed in the exploration of the Earth’s subsurface and embedded voluminous bodies often call for an intensive use, primary at the modeling stage and later on at the inversion stage, of analytically demanding tools of field calculation. Under the aim of modeling implementation, this contribution is concerned with some interesting aspects of the low-frequency interaction of arbitrarily orientated (i.e. three-dimensional) time-harmonic magnetic dipoles, with 3-D perfectly conducting spheroidal bodies embedded in an otherwise homogeneous conductive medium. For many practical applications involving buried obstacles such as Earth’s subsurface electromagnetic probing at low-frequency or any other physical cases (e.g. geoelectromagnetics), non-axisymmetric spheroidal geometry approximates sufficiently such kind of metallic shapes. On the other hand, our analytical approach deals with prolate spheroids, since the corresponding results for the oblate spheroidal geometry can be readily obtained through a simple transformation. The particular physical model concerns a solid impenetrable (metallic) body under a magnetic dipole excitation, where the scattering boundary value problem is attacked via rigorous low-frequency expansions for the incident, scattered and total electric and magnetic fields in terms of positive integral powers of ( ik), that is ( ik) n for n ⩾ 0, where k stands for the complex wavenumber of the exterior medium. The purpose of the modeling is to contribute to a simple yet versatile tool to infer information on an unknown body from measurements of the three-component electric and magnetic fields nearby. Our goal is to obtain the most important terms of the low-frequency expansions of the electromagnetic fields, that is the static (for n = 0) and the dynamic ( n = 1, 2, 3) terms. In particular, for n = 1 there are no incident fields and thus no scattered ones, while for n = 0 the Rayleigh electromagnetic expression is easily obtained in terms of infinite series. Emphasis is given on the calculation of the next two non-trivial terms (at n = 2 and at n = 3) of the aforementioned fields. Consequently, those are found in closed form from exact solutions of coupled (at n = 2, to the one at n = 0) or uncoupled (at n = 3) Laplace equations and they are given in compact fashion, as infinite series expansions for n = 2 or finite forms for n = 3. Nevertheless, the difficulty of the Poisson’s equation that has to be solved for n = 2 is presented, whereas our analytical approach demands the use of the well-known cut-off method in order to obtain an analytical closed solution. Finally, this research adds useful reference results to the already ample library of scattering by simple shapes using analytical methods.

Prestack wave-equation depth migration in elliptical coordinates

We extend Riemannian wavefield extrapolation (RWE) to prestack migration using 2D elliptical-coordinate systems. The corresponding 2D elliptical extrapolation wavenumber introduces only an isotropic slowness model stretch to the single-square-root operator. This enables the use of existing Cartesian finite-difference extrapolators for propagating wavefields on elliptical meshes. A poststack migration example illustrates advantages of elliptical coordinates for imaging turning waves. A 2D imaging test using a velocity-benchmark data set demonstrates that the RWE prestack migration algorithm generates high-quality prestack migration images that are more accurate than those generated by Cartesian operators of the equivalent accuracy. Even in situations in which RWE geometries are used, a high-order implementation of the one-way extrapolator operator is required for accurate propagation and imaging. Elliptical-cylindrical and oblate-spheroidal geometries are potential extensions of the analytical approach to 3D RWE-coordinate systems.

A Monte Carlo study of polarization structures in the Thomson-scattered line radiation

Thomson scattering is often invoked to explain broad wing features that are seen in various objects including active galactic nuclei and symbiotic stars. Despite the wavelength-independent scattering cross-section of Thomson scattering, the line flux may exhibit wavelength-dependent linear degree of polarization, because various parts of emission wings are contributed by photons with different scattering numbers. Specifically, more scattered and hence more weakly polarized photons tend to fill the farther wing parts from the line centre, while the neighbourhood of the line centre is dominated by less-scattered photons with higher degree of polarization. Using a Monte Carlo technique, we investigate the polarization structure of Thomson-scattered line radiation. A detailed analysis of polarization structure formation is conducted by investigating the dependence of the polarization and profile width on the scattering number for various finite electron scattering slabs. Significantly varying degree of polarization is obtained when the scattering medium has Thomson optical depth τTh≥ 1. We present our high-resolution spectrum of the symbiotic star V1016 Cyg obtained with the Bohyunsan Optical Echelle Spectrograph (BOES) in order to fit the broad profile around Hα by electron scattering wings adopting an oblate spheroidal geometry with Thomson optical depth τTh= 0.5 and electron temperature Te= 6.2 × 104 K. Local maxima in the linear degree of polarization of Thomson-scattered line radiation are expected to appear in the spectral regions characterized by the average scattering number ≃1.

Inverse Source Problem in an Oblate Spheroidal Geometry

The canonical inverse source problem of reconstructing an unknown source whose region of support is describable as a spheroidal (oblate or prolate) volume from knowledge of the far-field radiation pattern it generates is formulated and solved within the framework of the inhomogeneous scalar Helmholtz equation via a linear inversion framework in Hilbert spaces. Particular attention is paid to the analysis and computer illustration of flat, aperture-like sources whose support is approximated by an oblate spheroidal volume

Optical phonons and resonant Raman scattering in II–VI spheroidal quantum dots

AbstractFirst order and multiphonon resonant Raman scattering processes from confined and interface polar optical phonons in II–VI nanocrystallites are analyzed. Fröhlich interaction between excitons and optical phonons has been considered and general selection rules for the exciton–phonon matrix elements of the scattering processes in the case of spheroidal quantum dots are given. A comparison is made between the Raman selection rules and non‐linear hyper‐Raman selection rules for spherical dots. It is shown that the relative intensities between phonon overtones in II–VI spherical quantum dots can be correctly described in the framework of excitonic model and the optical polar vibrational continuum model satisfying both the mechanical and electrostatic matching conditions at the interface. Detailed discussions of the surface optical phonons are provided in the case of a spherical Quantum‐Dot/Quantum‐Well (QD/QW) heterostructure. Their frequency dependence on the geometry, material parameters, as well as the electron–phonon interaction Hamiltonian are also reported. In addition, the interface optical phonons and the corresponding electron–phonon interaction in nanostructures with prolate and oblate spheroidal geometries and their contribution to the first order scattering process have been studied. This surface‐optical phonon modes are strongly dependent on the nanocrystal geometry and new Raman selection rules are obtained given a plausible explanation for the observed low frequency shoulders present in the spectrum of several II–VI semiconductor nanostructures. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Interface optical phonons in spheroidal quantum dots

Interface optical phonons are studied in the case of a quantumdot (QD) with prolate and oblate spheroidal geometries within thedielectric continuum approach. We considered CdSe or CdS QDsimbedded in a host material which is modelled as an infinitemedium. The surface optical phonon modes, the correspondingfrequencies, and the electron-phonon interaction Hamiltonian arereported. Comparison is made with previous works which onlyconsidered strictly spherical dots. We conclude that deviationsfrom the perfect spherical shape could be responsible forobservable physical effects in Raman spectra.

TOA Lightning Location Retrieval on Spherical and Oblate Spheroidal Earth Geometries

A simple linear algebraic solution is introduced for retrieving the location and time-of-occurrence of lightning ground strikes on a spherical Earth from a network of four or more time-of-arrival (TOA) sensors. Since the solution accounts for Earth curvature, it represents an extension to earlier planar model results described by Koshak et al. A test of the retrieval method is provided using computer-simulated data sets. The method is easy to comprehend and completely avoids reference to the mathematics of spherical hyperbolas such as discussed by Lewis. A quasi-analytic extension to the spherical Earth solution is provided for an oblate spheroidal Earth geometry, and the importance/relevance of oblate effects are discussed. Future application of these methods in support of the North American National Lightning Detection Network (NALDN) described by Cummins et al. is desirable, but additional theoretical investigations are required to incorporate magnetic bearing information into the present solution process.