Prolate Spheroidal Coordinates Research Articles

Translational motion of a spheroidal drop in a viscous fluid

The problem of translational motion of a spheroidal drop along its axis of revolution in a viscous incompressible fluid is investigated semi-analytically. The flow fields in the exterior and interior of the drop are governed by the Stokes equations. Stream function formulation is adopted to solve the hydrodynamic equations in both regions. The general solution for the stream function in prolate and oblate spheroidal coordinates is expressed in an infinite-series form of semi-separation of variables. The leading order coefficients in the stream function are obtained using suitable boundary conditions. The hydrodynamic drag force experienced by the spheroidal drop is numerically evaluated with adequate convergence behavior for various values of the internal-to-external viscosity ratio and axial-to-radial aspect ratio of the drop. The numerical values of the drag force for the infinite and infinitesimal viscosity ratios agree with the available corresponding results for the slow translation of a slip spheroidal particle in the limiting conditions of no slip and full slip, respectively. At intermediate values of the viscosity ratio, the hydrodynamic force may not be a monotonic function of the aspect ratio. For a spheroidal drop with a fixed aspect ratio, its drag force increases monotonically with an increase in the viscosity ratio.

Implementation of the Time-Dependent Complete-Active-Space Self-Consistent-Field Method for Diatomic Molecules.

We present a computational approach that implements the time-dependent complete-active-space self-consistent-field method, as introduced in [Phys. Rev. A 88, 023402 (2013)]. Our implementation addresses the challenge of diatomic molecules subjected to an intense laser pulse by considering the full dimensionality of the problem using prolate spheroidal coordinates. The method incorporates the gauge-invariant frozen-core approximation, boosts the evaluation of the electron-electron interaction term using finite-element discrete-variable representation with Neumann expansion, and utilizes an exponential time differencing scheme tailored for the stable propagation of the stiff nonlinear orbital functions. We have successfully applied this methodology to study high-harmonic generation in diatomic molecules such as H2, LiH, and N2, shedding light on the impact of electron correlations in these systems.

Finite difference method in prolate spheroidal coordinates for freely suspended spheroidal particles in linear flows of viscous and viscoelastic fluids

A finite difference scheme is used to develop a numerical method to solve the flow of an unbounded viscoelastic fluid with zero to moderate inertia around a prolate spheroidal particle. The equations are written in prolate spheroidal coordinates, and the shape of the particle is exactly resolved as one of the coordinate surfaces representing the inner boundary of the computational domain. As the prolate spheroidal grid is naturally clustered near the particle surface, good resolution is obtained in the regions where the gradients of relevant flow variables are most significant. This coordinate system also allows large domain sizes with a reasonable number of mesh points to simulate unbounded fluid around a particle. Changing the aspect ratio of the inner computational boundary enables simulations of different particle shapes ranging from a sphere to a slender fiber. Numerical studies of the latter particle shape allow testing of slender body theories. The mass and momentum equations are solved with a Schur complement approach allowing us to solve the zero inertia case necessary to isolate the viscoelastic effects. The singularities associated with the coordinate system are overcome using L'Hopital's rule. A straightforward imposition of conditions representing a time-varying combination of linear flows on the outer boundary allows us to study various flows with the same computational domain geometry. For the special but important case of zero fluid and particle inertia we obtain a novel formulation that satisfies the force- and torque-free constraint in an iteration-free manner. The numerical method is demonstrated for various flows of Newtonian and viscoelastic fluids around spheres and spheroids (including those with large aspect ratio). Good agreement is demonstrated with existing theoretical and numerical results.

Internal and external harmonics in bi-cyclide coordinates

The Laplace equation in three-dimensional Euclidean space is -separable in bi-cyclide coordinates leading to harmonic functions expressed in terms of Lamé–Wangerin functions called internal and external bi-cyclide harmonics. An expansion for a fundamental solution of Laplace’s equation in products of internal and external bi-cyclide harmonics is derived. In limiting cases this expansion reduces to known expansions in bi-spherical and prolate spheroidal coordinates.

The Tomimatsu–Sato Metric Reloaded

In this work, we derive exact analytic formulae for the inner and outer surfaces representing the boundary of the ergoregion appearing in the Tomimatsu–Sato (TS) metric. Exact expressions for the radii of the ergoregion in prolate spheroidal coordinates and in Boyer-Lindquist coordinates are obtained. We also found that in addition to the ring-shaped naked singularity, there is an event horizon placed in the inner region inside the aforementioned curvature singularity. In comparing our results with previous studies, we also uncovered and corrected several errors in the literature. Finally, we provide tables of numerical values for the inner and outer boundaries of the ergoregion for different values of the rotational parameter. We hope this study will be a useful resource for all researchers interested in the Tomimatsu–Sato metric.

Continuum solutions to the two–center Coulomb problem in prolate spheroidal coordinates

We present a computational and theoretical framework for solving the Schrödinger equation (SE) for the two–center Coulomb problem in prolate spheroidal coordinates when the energy of the SE is positive. A general and robust computer code has been produced that calculates the separation constants, spheroidal harmonic expansion coefficients, regular quasi–radial two–center Coulomb wave functions, and two–center Coulomb phase shifts. These quantities can be calculated over a range of internuclear separations, angular momentum projections, and continuum electron momenta. A representative set of results are presented and compared with previous calculations, excellent agreement is found in many cases while significant disagreements are found in others. Program summaryProgram title:spheroidal-cwCPC Library link to program files:https://doi.org/10.17632/nf5gw7vjnh.1Code Ocean capsule:https://codeocean.com/capsule/7638744Licensing provisions: MITProgramming language: Fortran 90Nature of problem: A robust and accurate computer code for calculating Coulomb wave functions in prolate spheroidal coordinate is necessary for the calculation of cross sections for photonionization, and electron and positron scattering on charged diatomic targets.Solution method: The angular solution is expanded in a series of spherical harmonics leading to the diagonalization of a pentadiagonal matrix. The eigenvectors contain the expansion coefficients and the eigenvalues are the separation constants which, unlike the SE in spherical coordinates, are also functions of the energy. The radial two–center Coulomb wave functions are started using a power series solution and then propagated using a linear multistep method. An asymptotic expansion is used to calculate the two–center Coulomb phase shift, and to normalize the radial wave function.Additional comments including restrictions and unusual features: Two–center Coulomb radial wave functions and phase shifts can be accurately calculated for internuclear separations from 0.001 to 60.0 a0, angular momentum projection |m| from 0 to 44, momentum k from 0.01 to 100.0 a.u., and a combined nuclear charge Z+ from 1 to 12.

JRAF: A Julia package for computation of relativistic molecular auxiliary functions

The evaluation of relativistic molecular integrals over exponential−type spinor orbitals requires the use of relativistic auxiliary functions in prolate spheroidal coordinates, and has been recently achieved (Bağcı and Hoggan (2015) [14]). This process is used in the solution of the molecular Dirac equation for electrons moving in a Coulomb potential. A series of papers on a method for fully analytical evaluation of relativistic auxiliary functions has been published [2, 3, 4] From the perspective of computational physics, these studies demonstrate how to deal with the integrals of the product of power functions with non−integer exponents and incomplete gamma functions. The computer program package used to calculate these auxiliary functions with high accuracy is presented. It is designed using the Julia programming language and yields highly accurate results for molecular integrals over a wide range of orbital parameters and quantum numbers. Additionally, the program package facilitates the efficient calculation of the angular momentum coefficients that arise from the product of two normalized Legendre functions centered at different atomic positions, and the determination of the rotation angular functions used for both complex and real spherical harmonics. Sample calculations are performed for two−center one−electron integrals over non−integer Slater−type orbitals, and the results prove the robustness of the package. Program summaryProgram Title: JRAFCPC Library link to program files:https://doi.org/10.17632/942xsbvfdf.1Developer's repository link:https://github.com/abagciphys/JRAF.jlLicensing provisions: MITProgramming language:Julia programming languageSupplementary material: An experimental version of the computer program package written in Mathematica programming language [5].External routines/libraries:Nemo computer algebra package for the Julia programming language [6], Cuba multidimensional numerical integration using different algorithms in Julia [7].Nature of problem: Relativistic molecular auxiliary function integrals result from the expression of a two−center two−electron Coulomb energy associated with a charge density. The Coulomb energy is transformed into kinetic energy integrals using Poisson's equation and the single−center potential, considering that the Laplace expansion for the Coulomb interactions is expressed in terms of normalized non−integer Slater−type orbitals [1]. Using the resulting expression for the two−center two−electron integrals, relativistic auxiliary function integrals are derived in prolate ellipsoidal coordinates. These auxiliary functions are generalized to the entire set of physical potential operators for the Coulomb potential case.The integral of the relativistic auxiliary functions have no closed−form solutions except that their parameters are integers. As such, the analytical evaluation of these functions is challenging. They are used in the solution of the matrix form representation of the molecular Dirac−Fock self−consistent field (SCF) equation.Solution method: A criterion that considers the symmetry properties of two−center two−electron molecular integrals is initially proposed [2]. This obviates the need for the computation of incomplete and complementary incomplete gamma functions, and utilizes their sum (P+Q=1). The resulting form of the integral of the relativistic molecular auxiliary functions is expressed in terms of the convergent series representation of incomplete beta functions. Recurrence relationships are then derived for each of these sub−functions [3]. The algorithm for computation of the auxiliary functions is based on the vectorization procedure defined in [4].

Spheroidal magnetic stars rotating in vacuum

Context. Gravity shapes stars to become almost spherical because of the isotropic nature of gravitational attraction in Newton’s theory. However, several mechanisms break this isotropy, such as their rotation generating a centrifugal force, magnetic pressure, or anisotropic equations of state. The stellar surface therefore slightly or significantly deviates from a sphere depending on the strength of these anisotropic perturbations. Aims. In this paper, we compute analytical and numerical solutions of the electromagnetic field produced by a rotating spheroidal star of oblate or prolate nature. This study is particularly relevant for millisecond pulsars for which strong deformations are produced by rotation or a strong magnetic field, leading to indirect observational signatures of the polar cap thermal X-ray emission. Methods. First we solve the time harmonic Maxwell equations in vacuum by using oblate and prolate spheroidal coordinates adapted to the stellar boundary conditions. The solutions are expanded in series of radial and angular spheroidal wave functions. Particular emphasis is put on the magnetic dipole radiation. Second, we compute approximate solutions by integrating the time-dependent Maxwell equations in spheroidal coordinates numerically. Results. We show that the spin-down luminosity corrections compared to a perfect sphere are, to leading order, given by terms involving (a/rL)2 and (a/R)2 where a is the stellar oblateness or prolateness, R the smallest star radius, and rL the light-cylinder radius. The corresponding perturbations in the electromagnetic field are only perceptible close to the surface, deforming the polar cap rims. At large distances r ≫ a, the solution tends asymptotically to the perfect spherical case of a rotating dipole.

Analytic Solutions for Space-Charge-Limited Current Density From a Sharp Tip

While analytic equations for space-charge limited current density (SCLCD) have been derived for planar and nonplanar geometries, the SCLCD for emission from a sharp tip has not been derived. In this article, we use variational calculus (VC) to derive an exact analytic equation for SCLCD for a 1-D tip-to-tip geometry, represented by hyperboloids in the prolate spheroidal coordinates, and recover the SCLCD for a tip-to-plate geometry as a special case. We then consider circles in the extended Poincaré disk, which is the stereographic projection of hyperboloids onto a plane, as conformal transformations to derive the SCLCD for a misaligned tip-to-tip geometry, where the axes of rotation of the hyperboloids are displaced by a small distance. This mapping technique is also applied to study the effect of a small angle tilt in a tilted tip-to-tip geometry, where the axes of rotation of the hyperboloids meet at an angle.

Multiorbital impact on strong-field ionization of HCl molecules

We have put forward the time-dependent Hartree-Fock approach to investigate ionization dynamics of HCl molecules subjected to strong few-cycle laser pulses, which is based on the finite-element discrete-variable representation in prolate spheroidal coordinates and the Crank-Nicolson method. It is demonstrated that the electron is favored to be freed by the peak electric field pointing towards the H atom for the HCl molecule possessing the bonding length $R=2.5$ a.u., while the ionization prefers to occur for the maximum electric field pointing from the H nucleus to Cl nucleus with increasing internuclear separations. Analysis indicates that the ionization yield of $5\ensuremath{\sigma}$ electrons becomes significant with respect to that of $2\ensuremath{\pi}$ electrons with stretching bonding lengths, and the relative contributions of the ionization from $5\ensuremath{\sigma}$ and $2\ensuremath{\pi}$ electrons change as a function of internuclear separations, which gives rise to the above distinct oriented ionization of HCl molecules.

On the electrostatic interactions involving long-range Rydberg molecules

A ground state atom immersed in the wave function of the valence electron of a Rydberg atom can generate a long-range Rydberg molecule (LRRM). In this work, using the multipole expansion of the electrostatic interaction in prolate spheroidal coordinates, approximate and compact expressions of the electrostatic potential that determine the chemistry of trilobite and butterfly LRRM are explored. It is shown that even the prolate spheroidal monopole term can be used to describe general features of the potential generated by an LRRM at short distances. It is also shown that even at long separations that allow a perturbative description of the intermolecular interaction between two LRRM, the convergence of the multipole prolate spheroidal expansion is faster than that of its spherical analogue.

Analytical temperature profiles for laser-irradiated gold nanorod immersed in an optically non-absorbing medium

Laser heating of gold nanorods immersed in an optically non-absorbing medium is important in both therapeutic and diagnostic medical procedures. For both continuous wave and pulsed lasers, we derive analytical temperature solutions by modeling a nanorod as a prolate spheroid. For the continuous wave laser irradiation, an exact analytical solution for the steady state heat transfer case using integer-degree Legendre function expansions in prolate spheroidal coordinates is presented. The steady state solution reveals that for the case of gold nanorod immersed in water, there is insignificant temperature variations on any single spheroidal surface that is coincident or confocal to the prolate spheroid surface, and the temperature only changes across spheroidal surfaces within the immediate surrounding water. This is found to be due to the very high thermal conductivity of gold which is more than a hundred-fold higher than that of water. The insight that the temperature rise is nearly invariant on any single spheroidal surface, along with a dimensional analysis, helps reduce the complexity of the heat equation in the unsteady regime of a pulsed laser illumination, and leads to a simple, lower dimensional model of unsteady heat equation, which takes into account the temporal dependency and spatial dependency in the direction across spheroidal surfaces. For the reduced unsteady heat equation, a simple, efficient, and insightful solution algorithm based on the Laplace transform and complex-degree Legendre functions is proposed. The maximum temperature and temperature profiles as functions of laser power, and nanorod geometries are obtained, revealing that for a fixed laser fluence and fixed nanorod mass, the more stretched nanorods yield higher temperatures in the spheroid and its immediate surroundings. In other words, more stretched nanorods are more efficient in converting the laser energy into localized, elevated temperatures. The obtained temperature solutions are useful in designing a parameter space for optimum therapeutic procedures for ablating cancer cells using high temperatures generated in the immediate vicinity of nanoparticles. The unsteady temperature solutions can also be used to obtain solutions to thermo-elastic models which are used to compute the photoacoustic signals that can, in turn, be used for cellular imaging and diagnostics.

Generalized Sturmian Functions in prolate spheroidal coordinates

With the aim of describing bound and continuum states for diatomic molecules, we develop and implement a spectral method that makes use of Generalized Sturmian Functions (GSF) in prolate spheroidal coordinates. In order to master all computational issues, we apply it here to one--electron molecular ions and compare with benchmark data for both ground and excited states. We actually propose two different schemes to solve the two coupled differential equations. The first one is an iterative 1d procedure in which one solves alternately the angular and the radial equations, the latter yielding the state energy. The second, named direct 2d method, consists in representing the Hamiltonian matrix in a two-imensional GSF basis set, and its further diagonalization. Both spectral schemes are timewise computationally efficient and very accurate results are obtained with minimal basis sets. This is related on one side to the use of the natural coordinate system and, on the other, to the intrinsic good property of all GSF basis elements that are constructed as to obey appropriate physical boundary conditions. The present implementation for bound states paves the way for the study of continuum states involved in ionization of one or two-electron diatomic targets.

Multi-domain spectral approach with Sommerfeld condition for the Maxwell equations

We present a multi-domain spectral approach with an exterior compactified domain for the Maxwell equations for monochromatic fields. The Sommerfeld radiation condition is imposed exactly at infinity being a finite point on the numerical grid. As an example, axisymmetric situations in spherical and prolate spheroidal coordinates are discussed, as well as the interaction of a radiating dipole with a nano-particle.

Mathematical modeling of triode system on the basis of field emitter with ellipsoid shape

In this paper the mathematical modeling of the triode emission axially symmetric system on the basis of field emitter is considered. Emitter is an ellipsoid of revolution, anode is a confocal ellipsoidal surface of revolution. Modulator is a part of the ellipsoidal surface of revolution, confocal with the cathode and anode surfaces. The boundary-value problem for the Laplace's equation in the prolate spheroidal coordinates with the boundary conditions of the first kind is solved. The variable separation method is applied to calculate the axisymmetrical electrostatic potential distribution. The potential distribution is represented as the Legendre functions expansion. The expansion coefficients are the solution of the system of linear equations. All geometrical dimensions of the system are the parameters of the problem.

Stream Function Solution of the Brinkman Equation in Parabolic Cylindrical Coordinates

The present work concerns the general stream function solution of the Brinkman equation in parabolic cylindrical coordinates, arising in the study of fluid flow through porous medium. Analytical stream function solutions of this equation are available in the coordinates (Cartesian, cylindrical polar, spherical polar and prolate spheroidal coordinates). Stream function solution of the Stokes equation in parabolic cylindrical coordinates is also investigated analytically. The parabolic cylinder functions are a class of functions which are the solution of Weber differential equation. A transformation of parabolic cylinder function into the Whittaker function is used. Method of inverse operator is applied to obtain particular integral in solving the Stokes equation. Explicit expressions of velocity components and vorticity are also reported.

Convective and Microwave Assisted Drying of Wet Porous Materials with Prolate Spheroidal Shape: A Finite-Volume Approach

Convective heating is a traditional method used for the drying of wet porous materials. Currently, microwave drying has been employed for this purpose, due to its excellent characteristics of uniform moisture removal and heating inside the material, higher drying rate, and low energy demand. This paper focuses on the study of the combined convective and microwave drying of porous solids with prolate spheroidal shape. An advanced mathematical modeling based on the diffusion theory (mass and energy conservation equations) written in prolate spheroidal coordinates was derived and the numerical solution using the finite-volume method is presented. Here, we evaluated the effect of the heat and mass transport coefficients and microwave power intensity on the moisture removal and heating of the solid. Results of the drying and heating kinetics and the moisture and temperature distribution inside the solid are presented and discussed. It was verified that the higher the convective heat and mass transfer coefficients and microwave power intensity, the faster the solid will dry and heat up.

Geometric Analysis of the Doppler Frequency for General Non-Stationary 3D Mobile-to-Mobile Channels Based on Prolate Spheroidal Coordinates

Mobile-to-mobile channels often exhibit time-variant Doppler frequency shifts due to the movement of transmitter and receiver. An accurate description of the Doppler frequency turns out to be very difficult in Cartesian coordinates and any subsequent algebraic analysis of the Doppler frequency is intractable. In contrast to other approaches, we base our investigation on a geometric description of the Doppler frequency with the following three mathematical pillars: prolate spheroidal coordinate system, algebraic curve theory, and differential forms. The prolate spheroidal coordinate system is more appropriate to algebraically investigate the problem. After the transformation into the new coordinate system, the theory of algebraic curves is needed to resolve the ambiguities. Finally, the differential forms are required to derive the joint delay Doppler probability density function. This function is normalized by the equivalent ellipsoidal area of the scattering plane bounded by the delay ellipsoid. The results generalize in a natural way our previous model to a complete 3D description. Our solutions enable insight into the geometry of the Doppler frequency and we were able to derive a Doppler frequency that is dependent on the delay and the scattering plane. The presented theory allows describing any time-variant, single-bounce, mobile-to-mobile scattering channel.

Quantitative phase-mode electrostatic force microscopy on silicon oxide nanostructures.

Phase-mode electrostatic force microscopy (EFM-Phase) is a viable technique to image surface electrostatic potential of silicon oxide stripes fabricated by oxidation scanning probe lithography, exhibiting an inhomogeneous distribution of localized charges trapped within the stripes during the electrochemical reaction. We show here that these nanopatterns are useful benchmark samples for assessing the spatial/voltage resolution of EFM-phase. To quantitatively extract the relevant observables, we developed and applied an analytical model of the electrostatic interactions in which the tip and the surface are modelled in a prolate spheroidal coordinates system, fitting accurately experimental data. A lateral resolution of ∼60 nm, which is comparable to the lateral resolution of EFM experiments reported in the literature, and a charge resolution of ∼20 electrons are achieved. This electrostatic analysis evidences the presence of a bimodal population of trapped charges in the nanopatterned stripes.

Advantages of Slater-type spinor orbitals in the Dirac–Hartree–Fock method. Results for hydrogen-like atoms with super-critical nuclear charge

This work presents the formalism for evaluating molecular SCF equations, adapted to four-component Dirac spinors, which in turn reduce to Slater-type orbitals with non-integer principal quantum numbers in the non-relativistic limit. If Slater-type spinor orbitals are used in the algebraic approximation to solve the Dirac equation, the “catastrophe” previously noted for atomic numbers $$Z>137$$, in Dirac equation resolution with a potential corresponding to a point-charge no longer applies. It is observed that, ground-state energy of hydrogen-like atoms reaches the negative-energy continuum $$\left( -mc^2 \right)$$ with super-critical nuclear charge $$Z_{c}$$, about $$Z_{c}=160$$. The difficulty associated with finding relations for molecular integrals over Slater-type spinors which are non-analytic in the sense of complex analysis at $$r = 0$$, is eliminated. Unique numerical accuracy is provided by solving the molecular integrals through Laplace expansion of the Coulomb interaction and prolate spheroidal coordinates. New convergent series representation formulae are derived. The technique draws on previous work by the author. The general formalism is presented in this paper.