On the Analytical Solution of the Kuwabara-Type Particle-in-Cell Model for the Non-Axisymmetric Spheroidal Stokes Flow via the Papkovich–Neuber Representation

Modern engineering technology often involves the physical application of heat and mass transfer. These processes are associated with the creeping motion of a relatively homogeneous swarm of small particles, where the spheroidal geometry represents the shape of the embedded particles within such aggregates. Here, the steady Stokes flow of an incompressible, viscous fluid through an assemblage of particles, at low Reynolds numbers, is studied by employing a particle-in-cell model. The mathematical formulation adopts the Kuwabara-type assumption, according to which each spheroidal particle is stationary and it is surrounded by a confocal spheroid that creates a fluid envelope, in which the Newtonian fluid moves with a constant velocity of arbitrary orientation. The boundary value problem in the fluid envelope is solved by imposing non-slip conditions on the surface of the spheroid, which is also considered as non-penetrable, while zero vorticity is assumed on the fictitious spheroidal boundary along with a uniform approaching velocity. The three-dimensional flow fields are calculated analytically for the first time, in the spheroidal geometry, by virtue of the Papkovich–Neuber representation. Through this, the velocity and the total pressure fields are provided in terms of a vector and the scalar spheroidal harmonic potentials, which enables the thorough study of the relevant physical characteristics of the flow fields. The newly obtained analytical expressions generalize to any direction with the existing results holding for the asymmetrical case, which were obtained with the aid of a stream function. These can be employed for the calculation of quantities of physical or engineering interest. Numerical implementation reveals the flow behavior within the fluid envelope for different geometrical cell characteristics and for the arbitrarily-assumed velocity field, thus reflecting the different flow/porous media situations. Sample calculations show the excellent agreement of the obtained results with those available for special geometrical cases. All of these findings demonstrate the usefulness of the proposed method and the powerfulness of the obtained analytical expansions.

New relationships between spherical and spheroidal harmonics and applications

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

New relationships between spherical and spheroidal harmonics and applications

<p>This thesis deals with solutions to Laplace's equation in 3D, finding new relationships between solutions, manipulating these to find new approaches to physical problems, and proposing a new class of solutions. We mainly consider spherical and prolate spheroidal geometry and their corresponding solutions - spherical and spheroidal solid harmonics. We first present new relationships between these, expressing for example spherical harmonics as a series of spheroidal harmonics. Similar relationships are known but we work with the spherical and spheroidal coordinate systems being offset from each other. We also propose a new class of solutions which we call logopoles which have many links with spherical and spheroidal harmonics, and are related to the potential created by simple finite line charge distributions. Through the logopoles we find another relationship between the spheroidal harmonics and the often discarded alternate spherical harmonics. Then we apply one of the new spherical-spheroidal harmonic relationships to problems involving a point charge/dipole outside a dielectric sphere. We find new solutions where the potential is expanded as a series of spheroidal harmonics instead of the standard spherical ones, and we show that the convergence is much faster. We also solve these problems with logopoles and the solutions converge even faster, although they are more complicated as they involve a combination of logopoles and spherical harmonics.</p>

Dark matter halo with charge in pseudo-spheroidal geometry

The concept of dark matter (DM) hypothesis comes out as a result from the input of the observed flat rotational velocity. With the assumption that the galactic halo is pseudo-spheroidal and filled with charged perfect fluid, we have obtained a solution which has inkling to a (nearly) flat universe, compatible with the modern day cosmological observations. Various other important aspects of the solution such as attractive gravity in the halo region and the stability of the circular orbit are also explored. Also, the matter in the halo region satisfies the known equation of state which indicates its non-exotic nature.

SPHEROIDAL SPLINE INTERPOLATION AND ITS APPLICATION IN GEODESY

The aim of this paper is to study the spline interpolation problem in spheroidal geometry. We follow the minimization of the norm of the iterated Beltrami-Laplace and consecutive iterated Helmholtz operators for all functions belonging to an appropriate Hilbert space defined on the spheroid. By exploiting surface Green’s functions, reproducing kernels for discrete Dirichlet and Neumann conditions are constructed in the spheroidal geometry. According to a complete system of surface spheroidal harmonics, generalized Green’s functions are also defined. Based on the minimization problem and corresponding reproducing kernel, spline interpolant which minimizes the desired norm and satisfies the given discrete conditions is defined on the spheroidal surface. The application of the results in Geodesy is explained in the gravity data interpolation over the globe.

Results for Initial Trials Comparing Conformal with Non-Conformal Hypocenter Scanning Algorithms

This present paper sets out to perform a limited comparison between that method of P2P (point to point) scanning for Hypocenters, which uses the structure, imparted by a conformal mapping, to accelerate its ray tracing activity, and a method of scanning which incorporates the Earth spheroidal geometry directly within its ray tracing technique and which does not currently use a conformal mapping technique. The “conformal” method uses a mean-radius spherical Earth or can take a mean radius over the region covered by Epicenter and the sensor-field.In this comparison an attempt is made to compare and contrast a set of some possible functions (“indicator functions”) which serve to define the position of a Hypocenter within their response curve which is formed during the scan. Depending on the function, the expected indication may be either a minimum or a maximum, occurring within the range of the response.

A General Method for Estimating Bulk 2D Projections of Ice Particle Shape: Theory and Applications

The orientation of falling ice particles directly influences estimates of microphysical and radiative bulk quantities as well as in situ retrievals of size, shape, and mass. However, retrieval efforts and bulk calculations often incorporate very basic orientations or ignore these effects altogether. To address this deficiency, this study develops a general method for projecting bulk distributions of particle shape for arbitrary orientations. The Amoroso distribution provides the most general bulk aspect ratio distribution for gamma-distributed particle axis lengths. The parameters that govern the behavior of this aspect ratio distribution depend on the assumed relationship between mass, maximum dimension, and aspect ratio. Individual spheroidal geometry allows for eccentricity quantities to linearly map onto ellipse analogs, whereas aspect ratio quantities map nonlinearly. For particles viewed from their side, this analytic distinction leads to substantially larger errors in projected aspect ratio than for projected eccentricity. Distribution transformations using these mapping equations and numerical integration of projection kernels show that both truncation of size distributions and changes in Gaussian dispersion can alter the modality and shape of projection distributions. As a result, the projection process can more than triple the relative entropy between the spheroidal and projection distributions for commonly assumed model and orientation parameters. This shape uncertainty is maximized for distributions of highly eccentric particles and for particles like aggregates that are thought to fall with large canting-angle deviations. As a result, the methods used to report projected aspect ratios and the corresponding values should be questioned.