Spherical Polar Coordinate System Research Articles

Friction tensor for an ellipsoid enclosed in a rarefied gas

This research renders calculations of rotranslational friction and diffusion tensors in a rarefied gas, considering both specular and diffuse reflection off ellipsoidal particles. Calculations were conducted in a spherical polar coordinate system, revealing the dynamic nature of these off-diagonal tensors. The tensor elements were determined analytically, intending to compute the surface integrals. All diagrams, and pictures of two-dimensional surfaces were obtained by the numerical integration process in MATLAB. Additionally, we performed calculations without using the time parameter and time derivatives in the relevant relations. Deviations from the state of spherical symmetry were observed in the results for an ellipsoid and show that the tensors are dynamic quantities. Therefore, it is necessary to provide average values. We established that the non-zero elements of the friction tensor for an ellipsoid represent the directions of linear momentum transfer. Also, the non-zero components of the diffusion tensor signify the extent of fluctuations in relevant momentum transfers in corresponding directions.

Mathematical modelling of couple stress fluid flow around a semi‐permeable sphere enclosing a solid core

AbstractThe focus of this article is to study theoretically the steady‐axisymmetric creeping flow dynamics of a couple stress fluid external to a semi‐permeable sphere containing a solid core. This problem is motivated by emulsion hydrodynamics in chemical engineering where rheological behaviour often arises in addition to porous media effects. The non‐Newtonian Stokes’ couple stress fluid model features couple stresses and body couples that are absent in the classical Navier–Stokes viscous model. It provides a robust framework for simulating emulsions, complex suspensions and other liquids which possess microstructure. The physical regime is delineated into two zones – the interior of the semi‐permeable zone (region II) which engulfs the solid core and the external couple stress fluid zone (region I). The model is formulated using a spherical polar coordinate system in terms of the stream function ψ. The Brinkman‐extended Darcy model is deployed for the porous medium hydrodynamics and isotropic permeability is considered. Analytical expressions are derived for dimensionless pressure, tangential stress and the couple stress components using the method of separation of variables and Gegenbauer functions of the first kind. The integration constants are evaluated with appropriate boundary conditions on the inner and outer boundary of the semi‐permeable zone with the aid of Mathematica symbolic software. Solutions for the drag force exerted by the couple stress fluid on the semi‐permeable sphere and volumetric flow rate are also derived with corresponding expressions for the drag coefficient and non‐dimensional volumetric flow rate. The influence of permeability (k), separation parameter (l), couple stress viscosity coefficient (η), couple stress inverse length dependent parameter (λ = √(μ/η)) and couple stress viscosity ratio (τ = η/η/) on all key variables is studied graphically. Additionally, streamline contours are computed for a range of parameters including inverse permeability parameter (α = 1/k). The computations show that increasing couple stress inverse length dependent parameter (λ) greatly reduces the dimensionless volumetric flow rate in particular at high values of separation parameter (l). Flow rate is, however, markedly enhanced with permeability (k) and also couple stress viscosity parameter (η). In the presence of the solid core, a much greater drag coefficient is observed at all values of couple stress inverse length dependent parameter (λ) relative to the case without a solid core. A significant distortion in streamlines is computed with increasing separation parameter (l) with a dual vortex structure emanating. With greater couple stress viscosity parameter (τ), no tangible modification is observed in the streamline contours. The present work generalizes the earlier study of Krishnan and Shukla to consider a semi‐permeable (porous medium) outer sphere and furthermore presents detailed streamline visualizations of the creeping flow for a range of emerging parameters of relevance to non‐Newtonian chemical engineering processes including emulsion droplet dynamics in porous materials.

Relationship between Hanbury Brown-Twiss effect and spectral degree of polarization in light scattering from a collection of particles of $\mathcal {L}$ types.

We consider the vectorial extension of the recently developed matrix theory for the correlation between intensity fluctuations (CIF) of the scattered field generated by a collection of particles of $\mathcal {L}$ types [Y. Ding and D. M. Zhao, Opt. Express 30 46460, 2022]. In the spherical polar coordinate system, we establish a closed-form relation that connects the normalized CIF of the electromagnetic scattered field with the pair-potential matrix (PPM), the pair-structure matrix (PSM), and the spectral degree of polarization $\mathcal {P}$ of the incident field. Based on this, we pay much attention to the dependence of the normalized CIF of the scattered field on $\mathcal {P}$. It is found that the normalized CIF can be monotonically increasing or be nonmonotonic with $\mathcal {P}$ in the region [0, 1], determined by the polar angle θ and the azimuthal angle ϕ. Also, the distributions of the normalized CIF with $\mathcal {P}$ at polar angles and azimuthal angles are greatly different. These findings are explained mathematically as well as physically, and may be of interest to some related fields, especially where the CIF of the electromagnetic scattered field plays important roles.

Two‐phase modeling of evaporation characteristics of ethanol droplet under force‐convection environment

AbstractStudies on the evaporation phenomenon of a pure ethanol droplet have been mostly confined to the semianalytical modeling in stagnant ambient. Investigation into this aspect in a convective environment by considering the Navier–Stokes equation is also minimal. Hence, in this study we analyze and investigate the evaporation characteristics of a single‐component spherical‐shaped isolated pure ethanol droplet under force convective air environment by considering both gas‐ and liquid‐phase motions, nonunitary Lewis number in the interface, variable Stefan flow (blowing) effect, and the transient droplet heating. The finite difference method is utilized while solving the governing equations of the spherical polar coordinate system for species, momentum, and energy transfer. The maximum Reynolds number and ambient temperature are kept at 100 and 600 K, respectively. The present work is validated by comparing the normalized surface regression curve of the droplet with the earlier experimental and theoretical results. Using the current simulated data, flow and temperature profiles of both gas and liquid regions are visualized in streamline and isotherm contour plots at various instants of time. It is observed that at a moderate Reynolds number a detached vortex forms at the downstream location of the droplet. However, the detachment length increases with time. The temperature gradients along the droplet surface are observed at the initial stage. Moreover, the heat‐up period occupies about 20% of the total lifetime of the droplet. The droplet life and heat‐up period decrease with an increase in free‐stream velocity. In addition, the saturation temperature increases with ambient temperature.

CONFINEMENT ENERGY OF QUANTUM DOTS AND THE BRUS EQUATION

A review of the ground state confinement energy term in the Brus equation for the bandgap energy of a spherically shaped semiconductor quantum dot was made within the framework of effective mass approximation. The Schrodinger wave equation for a spherical nanoparticle in an infinite spherical potential well was solved in spherical polar coordinate system. Physical reasons in contrast to mathematical expediency were considered and solution obtained. The result reveals that the shift in the confinement energy is less than that predicted by the Brus equation as was adopted in most literatures.

Efficient solution of the anisotropic spherically aligned axisymmetric Jeans equations of stellar hydrodynamics for galactic dynamics

ABSTRACTI present a flexible solution for the axisymmetric Jeans equations of stellar hydrodynamics under the assumption of an anisotropic (three-integral) velocity ellipsoid aligned with the spherical polar coordinate system. I describe and test a robust and efficient algorithm for its numerical computation. I outline the evaluation of the intrinsic velocity moments and the projection of all first and second velocity moments, including both the line-of-sight velocities and the proper motions. This spherically aligned Jeans anisotropic modelling (JAMsph) method can describe in detail the photometry and kinematics of real galaxies. It allows for a spatially varying anisotropy, or stellar mass-to-light ratio gradients, as well as for the inclusion of general dark matter distributions and supermassive black holes. The JAMsph method complements my previously derived cylindrically aligned JAMcyl and spherical Jeans solutions, which I also summarize in this paper. Comparisons between results obtained with either JAMsph or JAMcyl can be used to assess the robustness of inferred dynamical quantities. As an illustration, I modelled the ATLAS3D sample of 260 early-type galaxies with high-quality integral-field spectroscopy, using both methods. I found that they provide statistically indistinguishable total density logarithmic slopes. This may explain the previously reported success of the JAM method in recovering density profiles of real or simulated galaxies. A reference software implementation of JAMsph is included in the publicly available jam software package.

Design Method for N-Lobed Noncircular Bevel Gears

As a type of spatial transmission mechanism, noncircular bevel gears (NBGs) can transfer power and motion between two intersecting axes with variable transmission following a suitable program of motion. Utilizing the spherical triangle theorem and meshing principle, parametric equations are established in the spherical polar coordinate system for the driving and driven gears, for the pitch curves, and for the addendum and dedendum curves of a NBG for a given transmission ratio and axis angle. A formulation of the tooth profile of a NBG is deduced using an analytic method. Three-dimensional models of the 3- and 4-lobed NBGs are derived in verifying this method.

Numerical solution of three-dimensional time-dependent Schrödinger equation based on graphic processing unit acceleration

In the field of quantum mechanics, the theoretical study of the interaction between intense laser field and atoms and molecules depends very much on the numerical solution of the time-dependent Schrödinger equation. However, solving the three-dimensional time-dependent Schrödinger equation is not a simple task, and the analytical solution cannot be obtained, so it can only be solved numerically with the help of computer. In order to shorten the computing time and obtain the results quickly, it is necessary to use parallel methods to speed up computing. In this paper, under the background of strong field ionization, the three-dimensional time-dependent Schrödinger equation of hydrogen atom is solved in parallel, and the suprathreshold ionization of hydrogen atom under the action of linearly polarized infrared laser electric field is taken for example. Based on the spherical polar coordinate system, the time-dependent Schrödinger equation is discretized by the splitting operator-Fourier transform method, and the photoelectron continuous state wave function under the length gauge can be obtained. In Graphics processing unit (GPU) accelerated applications, the sequential portion of the workload runs on central processing unit (CPU) (which is optimized for single-threaded performance), while the compute-intensive part of the application runs in parallel on thousands of GPU cores. The GPU can make full use of the advantage of fine-grained parallelism based on multi-thread structure to realize parallel acceleration of the whole algorithm. Two accelerated computing modes of CPU parallel and GPU parallel are adopted, and their parallel acceleration performance is discussed. Compared with the results from the existing physical laws, the calculation error is also within an acceptable range, and the result is also consistent with the result from the existing physical laws of suprathreshold ionization, which also verifies the correctness of the program. In order to obtain a relatively accurate acceleration ratio, many different experiments are carried out. Computational experiments show that under the condition of ensuring accuracy, the GPU parallel computing speeds by up to about 60 times maximally based on the computational performance of CPU. It can be seen that the accelerated numerical solution of three-dimensional time-dependent Schrödinger equation based on GPU can significantly shorten the computational time. This work has important guiding significance for rapidly solving the three-dimensional time-dependent Schrödinger equation by using GPU.

Effects of fractional and two-temperature parameters on stress distributions for an unbounded generalized thermoelastic medium with spherical cavity

Effects of fractional and two-temperature parameters on the distribution of stresses of an unbounded isotropic thermoelastic medium with spherical cavity are studied in the context of the theory of two-temperature generalized thermoelasticity based on the Green-Naghdi model III using fractional order heat conduction equation. The surface of the cavity is considered to be free from traction and is subjected to a smooth and time-dependent-heating effect. A spherical polar coordinate system has been used to describe the problem and the resulting governing equations are solved in Laplace transform domain. Numerical Laplace transform inversion method has been then applied to get the stresses in time domain. The numerical estimates of the distributions of stresses are obtained and are presented graphically to study the effects of fractional and two-temperature parameters.

A new modified ECM approach on the identification of plastic anisotropic properties by spherical indentation

This paper proposes a new approach on the identification of plastic anisotropic properties (σY, n, m), which is based on new modified expanding cavity model (ECM). The new ECM incorporates Hill's an anisotropic yield criterion in 1948 by equivalent transformation of stress expression form from spherical polar coordinate system to space rectangular coordinate system, and considers piling-up or sinking-in effect by introducing volume correction factor t as a semi-theoretical function. Through the indentation analysis for new ECM, the representative stress-strain method can characterize indentation mean pressure and uniaxial flow properties. Meanwhile, nonlinear functional relations between indentation response and plastic anisotropic properties are established. The effectiveness and reliability of new developed ECM approach are verified by the comparisons between identified properties and input properties in spherical indentation simulations, the comparisons between plastic anisotropic properties identified from instrumented indentation experiments and uniaxial compression experiments, and the comparisons between representative flow stress-strain points and compression experimental curves. The identified results are obtained by iterative calculations using the trust-region techniques based on the nonlinear least-square method. Excellent agreement is achieved. Finally, it is demonstrated that the proposed approach is reliable and useful for identification of plastic anisotropic properties of various anisotropic materials.

Finite slice analysis (FINA) of sliced and velocity mapped images on a Cartesian grid.

Although time-sliced imaging yields improved signal-to-noise and resolution compared with unsliced velocity mapped ion images, for finite slice widths as encountered in real experiments there is a loss of resolution and recovered intensities for the slow fragments. Recently, we reported a new approach that permits correction of these effects for an arbitrarily sliced distribution of a 3D charged particle cloud. This finite slice analysis (FinA) method utilizes basis functions that model the out-of-plane contribution of a given velocity component to the image for sequential subtraction in a spherical polar coordinate system. However, the original approach suffers from a slow processing time due to the weighting procedure needed to accurately model the out-of-plane projection of an anisotropic angular distribution. To overcome this issue we present a variant of the method in which the FinA approach is performed in a cylindrical coordinate system (Cartesian in the image plane) rather than a spherical polar coordinate system. Dubbed C-FinA, we show how this method is applied in much the same manner. We compare this variant to the polar FinA method and find that the processing time (of a 510 × 510 pixel image) in its most extreme case improves by a factor of 100. We also show that although the resulting velocity resolution is not quite as high as the polar version, this new approach shows superior resolution for fine structure in the differential cross sections. We demonstrate the method on a range of experimental and synthetic data at different effective slice widths.

Strongly magnetized rotating dipole in general relativity

Electromagnetic waves arise in many area of physics. Solutions are difficult to find in the general case. In this paper, we numerically integrate Maxwell equations in a 3D spherical polar coordinate system. Straightforward finite difference methods would lead to a coordinate singularity along the polar axis. Spectral methods are better suited to deal with such artificial singularities related to the choice of a coordinate system. When the radiating object is rotating like for instance a star, special classes of solutions to Maxwell equations are worthwhile to study such as quasi-stationary regimes. Moreover, in high-energy astrophysics, strong gravitational and magnetic fields are present especially around rotating neutron stars. In order to study such systems, we designed an algorithm to solve the time-dependent Maxwell equations in spherical polar coordinates including general relativity as well as quantum electrodynamical corrections to leading order. As a diagnostic, we compute the spindown luminosity expected from these stars and compare it to the classical i.e. non relativistic and non quantum mechanical results. It is shown that quantum electrodynamics leads to an irrelevant change in the spindown luminosity even for magnetic field around the critical value of $\numprint{4.4e9}$~\si{\tesla}. Therefore the braking index remains close to its value for a point dipole in vacuum namely $n=3$. The same conclusion holds for a general-relativistic quantum electrodynamically corrected force-free magnetosphere.

A deep non-hydrostatic compressible atmospheric model on a Yin-Yang grid

The singularity in the traditional spherical polar coordinate system at the poles is a major factor in the lack of scalability of atmospheric models on massively parallel machines. Overset grids such as the Yin-Yang grid introduced by Kageyama and Sato [1] offer a potential solution to this problem. In this paper a three-dimensional, compressible, non-hydrostatic atmospheric model is developed and tested on the Yin-Yang grid building on ideas previously developed by the authors on the solution of Elliptic boundary value problems and conservation on overset grids. Using several tests from the literature, it is shown that this model is highly stable (even with little off-centering), accurate, and highly efficient in terms of computational cost. The model also incorporates highly efficient and accurate approaches to achieve positivity, monotonicity and conservative transport, which are paramount requirements for any atmospheric model. The parallel scalability of this model, using in excess of 212 million unknowns and more than 6000 processors, is also discussed and shown to compare favourably with a highly optimised latitude–longitude model in terms of scalability and actual run times.

North–south component of galactic cosmic ray anisotropy at 1 AU

The galactic cosmic ray (GCR) solar diurnal anisotropy (SDA) may be represented by a vector (A; Ar, Aφ, Aθ) in a spherical polar coordinate system centered on the sun. We reported (elsewhere) the results of a detailed study of time variations of yearly radial (Ar) and east–west (Aφ) components recorded by the global network of the neutron monitors (NMs) with a long track record for 1963–2013, for four sunspot number (SSN) cycles (20–23) and the rising phase of cycle 24. A powerful new technique is used to compute and study time variations of the transverse component (Aθ) due to off-ecliptic GCR contributions, with the same NM data; GCR radial particle density gradient (Gr) drives all three components. The north–south anisotropy (Aθ) is computed from yearly NM data (Gϕ=0), a flat heliospheric current sheet (HCS) model and the concept of GCR isotropic hard sphere scattering in the solar wind plasma. Relationships to SSN, rigidity and solar polarity intervals are studied. For a positive (p-) polarity the solar magnetic field in the northern hemisphere points outward and GCRs drift from polar regions toward equatorial plane and out along HCS, setting up a symmetric gradient (Gθs) pointing away from HCS (there is a local GCR density minimum on HCS); for n-polarity interval Gθs points towards HCS (there is a local GCR density maximum on HCS). Also, there exists a heliospheric asymmetric density gradient (Gθa) perpendicular to the ecliptic plane, it is the main contributor to Aθ for the period of our analysis. This is the most interesting and significant insight.

Numerical relativity in spherical polar coordinates: Off-center simulations

We have recently presented a new approach for numerical relativity simulations in spherical polar coordinates, both for vacuum and for relativistic hydrodynamics. Our approach is based on a reference-metric formulation of the BSSN equations, a factoring of all tensor components, as well as a partially implicit Runge-Kutta method, and does not rely on a regularization of the equations, nor does it make any assumptions about the symmetry across the origin. In order to demonstrate this feature we present here several off-centered simulations, including simulations of single black holes and neutron stars whose center is placed away from the origin of the coordinate system, as well as the asymmetric head-on collision of two black holes. We also revisit our implementation of relativistic hydrodynamics and demonstrate that a reference-metric formulation of hydrodynamics together with a factoring of all tensor components avoids problems related to the coordinate singularities at the origin and on the axes. As a particularly demanding test we present results for a shock wave propagating through the origin of the spherical polar coordinate system.

Dynamic fisheye grids for binary black hole simulations

We present a new warped gridding scheme adapted to simulating gas dynamics in binary black hole spacetimes. The grid concentrates grid points in the vicinity of each black hole to resolve the smaller scale structures there, and rarefies grid points away from each black hole to keep the overall problem size at a practical level. In this respect, our system can be thought of as a ‘double’ version of the fisheye coordinate system, used before in numerical relativity codes for evolving binary black holes. The gridding scheme is constructed as a mapping between a uniform coordinate system—in which the equations of motion are solved—to the distorted system representing the spatial locations of our grid points. Since we are motivated to eventually use this system for circumbinary disc calculations, we demonstrate how the distorted system can be constructed to asymptote to the typical spherical polar coordinate system, amenable to efficiently simulating orbiting gas flows about central objects with little numerical diffusion. We discuss its implementation in the Harm3d code, tailored to evolve the magnetohydrodynamics equations in curved spacetimes. We evaluate the performance of the system’s implementation in Harm3d with a series of tests, such as the advected magnetic field loop test, magnetized Bondi accretion, and evolutions of hydrodynamic discs about a single black hole and about a binary black hole. Like we have done with Harm3d, this gridding scheme can be implemented in other unigrid codes as a (possibly) simpler alternative to adaptive mesh refinement.

Design of noncircular bevel gear with concave pitch curve

Noncircular bevel gear can achieve variable transmission between intersecting axes. This article proposes the design method of high-order noncircular bevel gear with concave pitch curve. The parametric equations for driving and driven gear, addendum and dedendum surface under given transmission ratio and axis angle are established in a spherical polar coordinate system. The tooth profile equation of noncircular bevel gear is derived according to the generation method by using bevel gear cutter. The three-dimensional model for noncircular bevel gear is developed to verify the correctness of the design method.

Natural convection heat transfer between inner sphere and outer vertically eccentric cylinder

The effects of eccentricity and geometric configuration with a Newtonian fluid have been investigated numerically to determine heat transfer by natural convection between the sphere and vertical cylinder with isothermal boundary conditions. The inner sphere and outer vertical cylinder were heated and cooled in a steady change of temperature. Calculations were carried out systematically for a range of the Rayleigh numbers to determine the average Nusselt numbers which are affected by the geometric ratio ( HR: RR) and eccentricity ( ε) parameters on the flow and temperature fields. The governing equations, in terms of vorticity, stream function and temperature are expressed in a spherical polar coordinate system. Results of the parametric study conducted further reveal that the heat and flow fields are primarily dependent on the Rayleigh number, eccentricity and geometric configuration, for a Prandtl number of 0.7, with the Rayleigh number ranging from 10 3 to 10 6, the three eccentricities and two geometric configurations. Above all, the specification of different convective configurations has a significant effect on the average heat transfer rate across the composite annulus gap.

THE TILT OF THE HALO VELOCITY ELLIPSOID AND THE SHAPE OF THE MILKY WAY HALO

A sample of roughly 1,800 halo subdwarf stars with radial velocities and proper motions is assembled, using the repeated multi-band Sloan Digital Sky Survey photometric measurements in Stripe 82. Our sample of halo subdwarfs is extracted via a reduced proper motion diagram and distances are obtained using photometric parallaxes, thus giving full phase space information. The tilt of the velocity ellipsoid with respect to the spherical polar coordinate system is computed and found to be consistent with zero for two of the three tilt angles, and very small for the third. We prove that if the inner halo is in a steady-state and the triaxial velocity ellipsoid is everywhere aligned in spherical polar coordinates, then the potential must be spherically symmetric. The detectable, but very mild, misalignment with spherical polars is consistent with the perturbative effects of the Galactic disk on a spherical dark halo. Banana orbits are generated at the 1:1 resonance (in horizontal and vertical frequency) by the disk. They populate Galactic potentials at the typical radii of our subdwarf sample, along with the much more dominant short-axis tubes. However, on geometric grounds alone, the tilt cannot vanish for the banana orbits and this leads to a slight, but detectable, misalignment. We argue that the tilt of the stellar halo velocity ellipsoid therefore provides a hitherto largely neglected but important line of argument that the Milky Way's dark halo, which dominates the potential, must be nearly spherical.

Visualization of spherical data by Yin–Yang grid

Abstract Visualization in the spherical geometry is ubiquitous in geophysical data processing. For the spherical visualization, the commonly used spherical polar coordinate system is not ideal due to its grid convergence nature near the poles. We propose to use a spherical overset grid system called Yin–Yang grid as the base grid system of the spherical visualization. The convergence-free nature of the Yin–Yang grid leads to a balanced data distribution and effective visualization processing in a sphere. The Yin–Yang grid is already used in various geophysical simulations including the geodynamo and mantle convection in the spherical geometry. Data produced by the Yin–Yang grid can be, and should be, visualized directly on the same Yin–Yang grid system without any data remapping. Since the component grid of the Yin–Yang grid is a part of (or low latitude region of) the standard spherical polar coordinate system, it is straightforward to convert an existing spherical visualization tool based on the spherical polar coordinates into a tool based on the Yin–Yang grid.