Prolate Spheroidal Coordinate System Research Articles

A Fourier-Chebyshev spectral collocation method for simulating flow past spheres and spheroids

An accurate Fourier-Chebyshev spectral collocation method has been developed for simulating flow past prolate spheroids. The incompressible Navier-Stokes equations are transformed to the prolate spheroidal co-ordinate system and discretized on an orthogonal body fitted mesh. The infinite flow domain is truncated to a finite extent and a Chebyshev discretization is used in the wall-normal direction. The azimuthal direction is periodic and a conventional Fourier expansion is used in this direction. The other wall-tangential direction requires special treatment and a restricted Fourier expansion that satisfies the parity conditions across the poles is used. Issues including spatial and temporal discretization, efficient inversion of the pressure Poisson equation, outflow boundary condition and stability restriction at the pole are discussed. The solver has been validated primarily by simulating steady and unsteady flow past a sphere at various Reynolds numbers

Moisture diffusion modelling of drying in parboiled paddy components. Part II: Bran and Husk

Multi-component prolate spheroid geometry was used in developing a non-dimensional isothermal liquid diffusion drying model for studying the moisture movement in the composite solids using a prolate spheroidal co-ordinate system. The developed models were solved using a finite difference method and were applied to simulate the drying process of multi-component parboiled brown rice and parboiled paddy. The observed drying characteristics of the input materials prepared by suitable hydrothermal treatment were obtained using a fluidised bed dryer at drying air temperatures ranging from 50 ∘C to 100 ∘C. In the multi-component parboiled brown rice drying model, the diffusion coefficients of bran were determined from the diffusion coefficients of the starchy endosperm, and for the parboiled paddy model the diffusion coefficients of husk were determined from the values of starchy endosperm and bran successively. By minimising the sum of squared differences (SSD) between experimentally observed and model predicted characteristics the liquid diffusion coefficients of bran and husk were determined at the various air temperatures studied. The developed models showed good agreement with the observed drying characteristics of the input materials. The temperature dependence of the diffusion coefficients of the parboiled paddy components was expressed by Arrhenius type of equations Bran: D vb =3.985948×10 −6 exp (−4096.003100/T a ), ( r=0.9923), Husk: D vh =1.655885×10 −5 exp (−4230.121506/T a ), ( r=0.9859). The diffusion coefficients of the parboiled paddy components were comparable to the recent findings on raw paddy and the differences encountered may be attributed to the properties of the component material.

Moisture diffusion modelling of drying in parboiled paddy components. Part I: starchy endosperm

A non-dimensionalised isothermal liquid diffusion drying model in prolate spheroid geometry using prolate spheroidal co-ordinate system was developed, and applied to drying of parboiled polished rice. Using appropriate hydrothermal treatment the parboiled polished rice was prepared, and its drying characteristics were determined using fluidised bed drying at air temperatures varying from 50 to 100 ∘C. A mathematical model simulating the drying was solved using the finite difference methodology. The liquid diffusion coefficients of starchy endosperm of parboiled polished rice while drying with various air temperatures were determined by minimising the sum of squared differences between experimentally observed and model predicted characteristics. The developed models showed good agreement with the experimentally observed data. Temperature dependence of liquid diffusion coefficients was expressed by an Arrhenius type of equation, which had a high correlation coefficient: D vs =1.079706×10 −7 exp (−2036.252438/T a ) [ r=0.9887]

Infinite elements in the time domain using a prolate spheroidal multipole expansion

Many phenomena in acoustically loaded structural vibrations are better understood in the time domain, particularly transient radiation, shock, and problems involving non-linearities, cavitation, and bulk structural motion. In addition, the geometric complexity of structures of interest drives the analyst toward domain-discretized solution methods, such as finite elements or finite differences, and large numbers of degrees of freedom. In such methods, efficient numerical enforcement of the Sommerfeld radiation condition in the time domain becomes difficult. Although a great many methodologies for doing so have been demonstrated, there seems to exist no consensus on the optimal numerical implementation of this boundary condition in the time domain. Here, we present theoretical development of several new boundary operators for conventional finite element codes. Each proceeds from successful domain-discretized, projected field-type harmonic solutions, in contrast to boundary integral equation operators or those derived from analyses of outgoing waves. We exploit the separable prolate-spheroidal co-ordinate system, which is sufficiently general for a large variety of problems of naval interest, to obtain finite element-like operators (matrices) for the boundary points. Use of this co-ordinate system results in element matrices that can be analytically inverse transformed from the frequency to the time domain, without imposing continuity requirements on the solution above those imposed by the underlying partial differential equation. In addition, use of element-like boundary operators does not alter the banded structure of assembled system matrices. Results presented here include theoretical derivation of the infinite elements, resolution of the Fourier inversion issues, and element matrices for the boundary operators which introduce no new continuity requirements on the fluid field variable. The simplest infinite elements are verified in a coupled three-dimensional context against DAA2 and Helmholtz integral equation results. © 1998 John Wiley & Sons, Ltd.

Asymptotically exact Coulomb spheroidal wavefunctions

The one-Coulomb-centre problem is considered in the prolate spheroidal coordinate system when the distances between the foci of the spheroidal system are large. The asymptotically exact Coulomb spheroidal quasiradial and quasiangular wavefunctions are derived in the whole space of electron coordinates. The constructed wavefunctions are compared with the exact Coulomb spheroidal wavefunctions which are also derived.

Quasi-translational addition expressions of prolate scalar wave functions for thin prolate spheroids

The conventional, translational addition theorems for both spherical and prolate spheroidal wave functions applied to incoming or outgoing waves possess a well-known convergence sphere near which the addition series fail to converge rapidly. To overcome this deficiency the quasi-translational addition expressions for prolate wave functions are developed by using a physical-geometrical transformation which allows the separation in the azimuthal direction in the translated prolate spheroidal coordinate system. These quasi-translational addition expressions are valid everywhere and, in particular, exhibit a fast convergence characteristics for thin spheroids which are commonly used in physical and engineering applications.

The one-Coulomb-centre problem in the spheroidal coordinate system: asymptotic solutions

The one-Coulomb-centre problem is considered in the prolate spheroidal coordinate system. The asymptotic expansions for the separation constant and the Coulomb spheroidal quasiradial and quasiangular wavefunctions are derived when the distances between the foci of the spheroidal system R are large. The constructed wavefunctions are in excellent agreement with the exact wavefunctions in intra-atomic space, when the condition is fulfilled. The formulae obtained in the present paper can easily be generalized for the case of the two-Coulomb-centre problem.

Methods for calculating electrostatic quantities due to a free charge in a nanoscale three-dimensional tip/base junction

The exact solutions for fully three-dimensional electrostatic quantities due to a free charge in a nanoscale tip/base junction, such as the scanning-tunneling microscope, are given for the problem as modeled in the prolate-spheroidal coordinate system. These exact solutions consist of summation series of integrals of conical functions, and methods for calculating these exact solutions with high accuracy are described in detail in this paper. Calculated results for the surface-charge distribution on the tip and the base due to a free electron in a nanoscale tip/base junction are also presented. This analysis has important implications for the ultimate objective of quantifying electron tunneling, from first principles, in nonplanar geometries such as sharp vacuum field emitters.

Three-dimensional electrostatic potential, and potential-energy barrier, near a tip-base junction

The geometry of an atomically sharp or nearly atomically sharp tip in proximity to a planar anode may be closely approximated in the prolate-spheroidal coordinate system. An exact three-dimensional electrostatic-potential solution for a free charge in such a tip/base junction is given in this letter, including calculations for both the symmetrical on-axis case and the asymmetric off-axis case. An exact solution for the potential-energy barrier is also given; this solution has immediate applications in three-dimensional tunneling studies and in calculations of electron trajectories in micron- and submicron-sized field-emitter arrays.

Calculations of particle decay widths of hyper-deformed nuclei using prolate spheroidal coordinates

We address the problem of calculating the particle decay widths from hyper-deformed nuclei. To simplify the calculation we treat the deformed nucleus as a prolate spheroid, and the particle-nucleus interaction as sharp-edged. This enables us to describe the deformed nuclear shape as an equipotential surface in a prolate spheroidal coordinate system. We model the particle-nucleus interaction by means of a spheroidal “square well” potential. Next we make the adiabatic approximation. This enables us to consider the deformed nucleus as being fixed in space. We then solve a system of coupled Schrödinger equations which describe the scattering of the particle from the deformed nucleus. These partial differential equations are separable in our chosen coordinate system. We generate prolate spheroidal wave functions to describe the particle both inside and outside the nuclear volume using a matrix diagonalization technique. The scattering problem is then solved by matching appropriate linear combinations of internal and external wave functions, and their first derivatives, at the nuclear boundary and searching for resonances as the Q-value for particle emission is varied. Using this formalism we calculate the alpha-decay widths of the recently discovered α-chain state in 24Mg and of the proposed 16O+ 16O+ 16O configuration in 48Cr.

A mathematical model of heat conduction in a prolate spheroidal coordinate system with applications to the theory of welding

The steady-state heat conduction equation is solved in a prolate spheroidal coordinate system. Some simple solutions are derived in the limiting cases of both low and high Peclet numbers. The analysis was carried out in such a way as to avoid the physical details of conditions inside the weld pool, so that the solutions are restricted to the solid region outside the weld pool. This procedure was specifically adopted because these conditions are difficult to gain access to experimentally, as is the precise detailed shape of the pool; the solutions obtained can be verified experimentally. The high-Peclet-number approximation is likely to be particularly useful in the case of laser welding where large translation speeds of the weld piece are of interest. The solution of the problem is given in the form of a series as well as in an asymptotic form. The asymptotic method of solution presented can be adapted to any smooth shape of weld pool with only minor alterations, since the method involves integration in the tangent plane to the weld pool and the results of such an integration are independent of the global form of the weld pool. The asymptotic result is compared with the exact solution in a number of special geometric configurations. These are prolate spheroidal weld pool geometries with various aspect ratios and cylindrical weld pool geometries with elliptical or circular cross sections. The results of these comparisons were found to be satisfactory.

Electrode heating in a wire-to-plane arc

A steady wire-to-plane electric discharge has been modeled in a prolate spheroidal coordinate system with the wire shape taken as a hyperboloid of revolution. A set of continuum conservation equations for the charged particle densities and temperatures together with Poisson’s equation for the self-consistent electric potential describe the steady electric discharge process. These equations have been solved numerically to obtain ion and electron densities, temperature distribution, and electrode heat fluxes. Particle densities show the main body of the arc is quasineutral bounded by space charge sheaths at both electrodes. The temperature is greatest in a region around the discharge axis about one-third of the distance from the wire to the plane. Strong electric fields are concentrated in the electrode sheaths. The heat flux to the wire is sharply peaked near the tip but on the plane it decays slowly away from the discharge axis. The knowledge of heat transfer from the arc to the electrodes is useful in determining arc parameters that govern the ball formation process used in wire bonding of microelectronic semiconductor chips as well as welding processes.

Breakdown of a wire-to-plane discharge: Transient effects

A wire-to-plane discharge during the early phases of breakdown has been studied. The discharge has been modeled in a prolate spheroidal coordinate system with the wire shape taken as a hyperboloid of revolution. Four simultaneous coupled, time-dependent, nonlinear partial differential equations describe the electrical discharge. These are the conservation equations for ion and electron densities, the energy equation for electron temperature, and Poisson’s equation for the self-consistent electric field. By solving this formulation subject to appropriate initial and boundary conditions, charged particle densities and temperature variations have been obtained as the ionization progresses in the discharge. The results show that both the electron temperature and the charged particle densities increase with the progress of ionization. The effect of different wire polarities is also examined. With a positive wire polarity, the increases in electron temperature and charged particle densities are confined to regions of the discharge in the vicinity of the wire tip. With a negative wire polarity, the breakdown occurs in the entire gap at a faster rate than with a positive wire polarity. The wire polarity affects the magnitude of energy transfer between the particles.

Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation

A new family of exact solutions of the scalar Helmholtz equation is presented. The 0, 0 order of this family represents a new mathematical model for the fundamental mode of a propagating Gaussian beam. The family consists of nonseparable functions in the oblate spheroidal coordinate system and can easily by transformed into a different set of solutions in the prolate spheroidal coordinate system, where the 0, 0 order is a spherical wave. This transformation consists of two substitutions in the coordinate system parameters and represents a more general method of obtaining a Gaussian beam from a spherical wave than assuming a complex point source on axis. Finally, each higher-order member of the family of solutions possesses an amplitude consisting of a finite number of higher-order terms with a zero-order term that is Gaussian.

Eigenstates of Wannier excitons near a semiconductor surface

The exact energies and wave functions, in the effective-mass approximation, of an electron moving in the potential of a massive hole near a single rigid surface of a semiconductor are presented. The electron-hole interaction potential is taken to be the Coulombic $\frac{1}{\ensuremath{\epsilon}r}$ inside the semiconductor and $+\ensuremath{\infty}$ outside. The Schr\"odinger equation for this electron-hole interaction potential is separable in the prolate spheroidal coordinate system, and thus can be solved numerically to any desired accuracy rather easily. The ground state of the exciton is seen to change continously from a hydrogenic $1s$ state, when the hole is well inside the bulk, to a hydrogenic $2{p}_{z}$ state, when the hole is located right on the surface in agreement with the known results in these two limits. The variation of the energies and wave functions of the ground and the excited states as a function of the distance of the hole from the surface are discussed and compared with some previous calculations.

Two-electron excited states and adiabatic quantum defects ofH2: Analysis of elastic scattering of electrons fromH2+

The spheroidal ssigma-, psigma-, p..pi..-, and dsigma-wave phase shifts for the scattering of electrons from H/sub 2/ /sup +/ are calculated at various internuclear distances by the Kohn variational method in the prolate-spheroidal coordinate system. The positions and widths of the two-electron excited resonance states are obtained for the symmetries /sup 1/..sigma../sub g/, /sup 1,3/..sigma../sub u/, and /sup 1,3/Pi/sub u/. Adiabatic quantum defects are also obtained by extrapolating the phase shifts to the low-energy limit. Importance of these quantities is discussed briefly in connection with the study of various dynamic processes of highly excited states of molecules.

Borehole resistivity curves near spheroidal masses

Modeling of in situ energy extraction processes (e.g., coal gasification and oil shale retorting) has been examined by assuming the affected zones may be described by oblate or prolate spheroids of lowered electrical resistivity. The purpose of the model study is to determine if such subsurface bodies may be monitored and defined by electrical resistivity measurements made in boreholes away from the reaction zone. Solutions of the Laplace equation in (1) oblate and (2) prolate spheroidal coordinate systems enable theoretical determination of the electrical potential distribution as would be measured in a borehole near an anomalous body of oblate or prolate shape. The body is assumed to be enclosed within an infinite, homogeneous mass. Normal (pole‐pole) type curves of (1) electrical resistivity for various electrode spacings (sounding curves) and (2) electrical resistivity for various positions in the borehole (vertical profiling) are developed for both oblate and prolate spheroidal models using spheroids of increasing size as a sensitivity parameter. Modeling results verify that an oblate body of the same cross‐sectional shape as a prolate body produces the larger anomaly. For the various size and resistivity parameters specified in this study, deviations from the homogeneous case (i.e., no spheroid) range from 3 percent for the smallest oblate and prolate bodies up to 60 percent for the largest modeled oblate body. If a ±5 percent uncertainty because of instrumentation, field technique, or random noise may be assumed, this implies a lower limit to the size of a detectable body. On this basis, the smaller spheroids of this study would not be detectable. However, both sounding and profiling type measurements offer promise as effective monitoring tools, and, if repeated over the period of process time, may enable process growth estimates. Further, because the oblate case is independent of azimuth while the prolate is not, measurements made in a series of borings surrounding the process center may describe the shape tendency (oblate versus prolate) of the process and determine directivity (azimuth of prolate major axis) of the process if existent.

Elastic scattering of electrons from H2+: phaseshifts, quantum defects and two-electron excited states

The spheroidal s- and p-wave phaseshifts for the scattering of electrons from H2+ are calculated by the Kohn variational method in the prolate spheroidal coordinate system. Two-centre Coulomb wavefunctions are employed as the long-range basis functions. Adiabatic quantum defects are obtained by extrapolation of the phaseshifts to the low-energy limit. Two-electron excited resonances found in the p-wave scattering are analysed.

Elastic scattering of electrons from H2+: two-centre variational calculation of s-wave phaseshift

The spheroidal s-wave phaseshift for the scattering of electrons from H2+ is calculated by the Kohn variational method in the energy range 0.2 approximately 1.0 Ryd. The prolate spheroidal coordinate system and the two-centre Coulomb wavefunctions are employed. The results are compared with those of the single-centre calculation made by Temkin et al.

The use of two-centre expansions in the scattering of electrons by N2molecules

The use of a two-centre prolate spheroidal coordinate system in ab initio calculations of scattering of slow electrons by N2 molecules is studied. The speedier convergence of the resulting multipole expansion of the static potential near the nuclei, compared with that of its one-centre form, is demonstrated. Eigenphase sums calculated using two-centre coordinates also converge more rapidly.