scholarly journals Gravity field due to a homogeneous oblate spheroid: Simple solution form and numerical calculations

2011 ◽  
Vol 41 (4) ◽  
pp. 307-327 ◽  
Author(s):  
Milan Hvoždara ◽  
Igor Kohút
Keyword(s):  
Gravity Field ◽  
Separation Of Variables ◽  
Oblate Spheroid ◽  
Numerical Calculations ◽  
Simple Solution ◽  
Simple Derivation ◽  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Solution Form ◽  
Oblate Spheroidal Harmonics

Gravity field due to a homogeneous oblate spheroid: Simple solution form and numerical calculations We present a simple derivation of the interior and exterior gravitational potentials due to oblate spheroid and also its gravity field components by using the fundamental solution of the Laplace equation in oblate spheroidal coordinates. Application of the method of separation of variables provides an expression for the potential in terms of oblate spheroidal harmonics of degree n = 0, 2. This solution is more concise and suitable for the numerical calculations in comparison with infinite series in spherical harmonics. Also presented are the computations in the form of potential isolines inside and outside the spheroid, as well as for the gravity field components. These reveal some interesting properties of the gravity field of this fundamental geophysical body useful for the applied gravimetry.

scholarly journals Ultimate Image Singularities in Oblate Spheroidal Coordinates with Applications in Hydrodynamics

2020 ◽  
Vol 8 (1) ◽  
pp. 32 ◽  
Author(s):  
Ioannis K. Chatjigeorgiou ◽  
Eva Loukogeorgaki ◽  
Eirini Anastasiou ◽  
Nikos Mantadakis
Keyword(s):  
Free Surface ◽  
Coordinate System ◽  
Oblate Spheroid ◽  
Fundamental Solutions ◽  
Special Focus ◽  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Oblate Spheroidal Harmonics ◽  
Spheroidal Coordinates ◽  
Accurate Computations

This study exploits the Touvia Miloh oblate spheroid theorem with a special focus on hydrodynamical applications. The theorem provides explicit relations that express the oblate spheroidal harmonics, given in terms of the fundamental solutions of the Laplace equation. Here, the theorem is employed to transform the underlying Green’s function into the relevant coordinate system and, consequently, to formulate the diffraction potential. The case considered refers to the axisymmetric placement of the spheroid, namely, symmetrical axis perpendicular to the free surface. The mathematical formulations have been implemented numerically providing exceptionally accurate computations, which manifests the consistency and robustness of the relevant formulas.

scholarly journals II.—Oblate spheroidal harmonics and their applications

1924 ◽  
Vol 224 (616-625) ◽  
pp. 49-93 ◽  
Keyword(s):  
Harmonic Functions ◽  
Integral Method ◽  
Large Body ◽  
Oblate Spheroid ◽  
Analytical Form ◽  
Legendre Functions ◽  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Physical Problems ◽  
Oblate Spheroidal Harmonics

The harmonic functions appropriate to the oblate spheroid, which are of the form P n (ζ), q n (ζ), or P n (ιζ), Q n (ιζ), when the large letters denote the usual Legendre functions, have received but little attention. Yet they provide, as we shall show in this memoir, a very elegant analysis of a variety of physical problems. We propose to exhibit a series of illustrations of their use, together with a large body of analysis whose applications extend very far, and lead to elegant solutions, in an analytical form, of problems which are in many cases new. In other cases—for example, the classical problems of electrified circular discs under influence—geometrical methods which lead to serious limitations have alone been effective hitherto. The analysis by spheroidal harmonics is shown to be intimately associated with that by other methods, such as the Fourier-Bessel integral method, and important theorems of analysis are involved. We may begin with a brief summary of the more important expressions already known for these functions. If a potential function ϕ satisfies ∇ 2 ϕ = 0 and a transformation to cylindrical coordinates ( z, ρ, ω ) is made, ∂ 2 ϕ / ∂ ρ 2 + 1/ ρ ∂ ϕ / ∂ ρ ∂ 2 ϕ / ∂ z 2 + 1/ ρ 2 ∂ 2 ϕ / ∂ ω 2 = 0, where ρ is distance from the axis.

Blind Robust 3-D Mesh Watermarking Based on Oblate Spheroidal Harmonics

2009 ◽  
Vol 11 (1) ◽  
pp. 23-38 ◽  
Author(s):  
J.M. Konstantinides ◽  
A. Mademlis ◽  
P. Daras ◽  
P.A. Mitkas ◽  
M.G. Strintzis
Keyword(s):  
Spheroidal Harmonics ◽  
Oblate Spheroidal ◽  
Oblate Spheroidal Harmonics

Torsional oscillation of a homogeneous elastic spheroid

1962 ◽  
Vol 52 (3) ◽  
pp. 469-484 ◽  
Author(s):  
Tatsuo Usami ◽  
Yasuo Satô
Keyword(s):  
Fundamental Mode ◽  
Free Oscillation ◽  
Torsional Oscillation ◽  
Numerical Calculations ◽  
Higher Harmonics ◽  
Oblate Spheroidal ◽  
The Earth ◽  
Spectral Peaks ◽  
The Difference ◽  
Spheroidal Coordinates

abstract There are several causes for the observations of splitting of the spectral peaks determined from the free oscillation of the earth. In this paper, the splitting due to the ellipticity is studied assuming a homogeneous earth described by oblate spheroidal coordinates. Ellipticity causes the iTn mode to split into (n + 1) modes, while the earth's rotation causes it to split into (2n + 1) modes. 1/297.0 is adopted as the ellipticity of the earth. Numerical calculations are carried out for the fundamental mode (n = 2, 3, 4) and for the first higher harmonics (n = 1). The difference between the extreme frequencies for each value of n is 0.7% (n = 2), 0.5% (n = 3), and 0.4% (n = 4).

Thermal Diffusivity Determination of High-Temperature Levitated Oblate Spheroidal Specimen by a Flash Method

1998 ◽  
Vol 120 (3) ◽  
pp. 777-781 ◽  
Author(s):  
F. Shen ◽  
J. M. Khodadadi
Keyword(s):  
Thermal Diffusivity ◽  
Transfer Problem ◽  
Oblate Spheroid ◽  
Single Step ◽  
Flash Method ◽  
Conduction Heat Transfer ◽  
Oblate Spheroidal ◽  
Flash Technique ◽  
Transient Conduction

In extending the range of applicability of a recently developed method, a single-step containerless flash technique for determining the thermal diffusivity of levitated oblate spheroidal oblate spheroidal samples is proposed. The flash method is modeled as an axisymmetric transient conduction heat transfer problem within the oblate spheroid. It is shown that by knowing the sample geometric parameters and recording the temperature rise history at least at two different points on the surface simultaneously, the thermal diffusivity can be determined without knowing needed for determining the thermal diffusivity of oblate spheroidal samples are provided.

Viscous drop in compressional Stokes flow

2013 ◽  
Vol 720 ◽  
pp. 169-191 ◽  
Author(s):  
Michael Zabarankin ◽  
Irina Smagin ◽  
Olga M. Lavrenteva ◽  
Avinoam Nir
Keyword(s):  
Stokes Flow ◽  
Entire Range ◽  
Extensional Flow ◽  
Oblate Spheroid ◽  
Chebyshev Series ◽  
Drop Shape ◽  
Spheroidal Harmonics ◽  
Viscous Drop ◽  
Two Parameter ◽  
Flow Case

AbstractThe dynamics of the deformation of a drop in axisymmetric compressional viscous flow is addressed through analytical and numerical analyses for a variety of capillary numbers, $\mathit{Ca}$, and viscosity ratios, $\lambda $. For low $Ca$, the drop is approximated by an oblate spheroid, and an analytical solution is obtained in terms of spheroidal harmonics; whereas, for the case of equal viscosities ($\lambda = 1$), the velocity field within and outside a drop of a given shape admits an integral representation, and steady shapes are found in the form of Chebyshev series. For arbitrary $Ca$ and $\lambda $, exact steady shapes are evaluated numerically via an integral equation. The critical $\mathit{Ca}$, below which a steady drop shape exists, is established for various $\lambda $. Remarkably, in contrast to the extensional flow case, critical steady shapes, being flat discs with rounded rims, have similar degrees of deformation ($D\sim 0. 75$) for all $\lambda $ studied. It is also shown that for almost the entire range of $\mathit{Ca}$ and $\lambda $, the steady shapes have accurate two-parameter approximations. The validity and implications of spheroidal and two-parameter shape approximations are examined in comparison to the exact steady shapes.

Sign in / Sign up

Export Citation Format

Share Document