Cylindrical Coordinate System Research Articles

Arbitrary manipulations of broadband terahertz focused Poincaré beams enabled by anisotropic silicon metasurfaces

Focused Poincaré beams (FPBs) carrying inhomogeneous polarization properties describe the paraxial solution of the vector Helmholz equation in cylindrical coordinate systems, with potential for classical and quantum applications. Here, we theoretically demonstrate a generalized strategy for generating FPBs based on the spin decoupling mechanism. Silicon metasurfaces with high degrees of freedom allow the proposed design to generate vector beams carrying arbitrary topological charges in a broadband terahertz (THz) frequency range. By adopting a purely phase-based modulation approach, the single-layer polarization-multiplexed metasurface can act as a bridge between the conventional Poincaré sphere (PS) and the higher-order Poincaré sphere (HOPS), preserving the corresponding mapping relations. Benefiting from the independence of the on-demand tailored helical phases, we further evaluate the coaxial evolutionary behavior of FPBs, generating structured THz fields with incremental topological charges. It is foreseeable that this work will provide an efficient meta-platform for the generation of broadband THz structured fields and facilitate their application in information encryption and quantum scientific information.

Heat Conduction in Cylindrical Coordinates with Time-Varying Conduction Coefficients: A Practical Engineering Approach

This research aims to develop a mathematical method for expressing the Laplace operator in cylindrical coordinates and applying it to solve heat conduction equations in various scenarios. The method commences by transforming Cartesian coordinates into cylindrical coordinates and identifying the necessary substitutions. The result is the expression of the Laplace operator in cylindrical coordinates, which is subsequently employed to address heat conduction equations within cylindrical coordinates. Various cases encompassing different initial and boundary conditions, as well as variations in the conduction coefficient over time, are meticulously considered. In each instance, precise mathematical solutions are determined and subjected to thorough analysis. This study carries substantial implications for comprehending heat transfer within cylindrical coordinate systems and finds relevance in a wide array of scientific and engineering contexts. The research's findings can be harnessed for technology development, heating system design, and heat transfer modeling across diverse applications, including mechanical engineering and materials science. Therefore, the research's contribution holds paramount significance in advancing our understanding of heat transfer within cylindrical coordinates and in devising more efficient and accurate solutions for an array of heat-related issues within the realms of science and engineering.

Controllability Analysis of Motion of Artificial Satellite Under the Effect of Oblateness of the Earth

In this article we have studied the controllability of artificial satellite under the effect of zonal harmonic J2 in cylindrical polar coordinates systems. Seven different cases of thrusters in various directions have been analyzed and it is found that the system is controllable if we apply thrusters in either r, θ and z or θ and z direction. The equations governing motion of satellite have been linearized and Kalman controllability test is applied to check the controllability of the system. We have also derived controller u for the linearized system. The trajectory of the system has been plotted to show the controllability of the system.

Analytical solution of non-Newtonian Williamson fluid under the effect of magnetohydrodynamics

In this paper, inscribed non-Newtonian Williamson fluid relaxation and contraction phenomena due to the effect of magnetohydrodynamics (MHD) is taken into account. This issue is modeled, and precise solutions are handled using a low Reynolds number and long wavelength approximation. Graphical depiction of the consequences of various parameters on temperature and velocity are provided. It becomes clear that the temperature reactions to increases in Biot numbers are decreased. Considering a cylindrical coordinate system, the impact of various physical parameters on the temperature distribution, velocity, the pressure gradient, and frictional force are secured from the flow for non-Newtonian Williamson fluid exposed from MATLAB. Additionally, it has been discovered that the velocity profile at the tube’s center increases for shear-thinning a fluid and decreases for thickening a fluid. Furthermore, the peristaltic walls show the opposite behavior.

Amplitude-Frequency Characteristics and Stability Regions of a Two-Layer Liquid Under Angular Vibrations of a Solid Body

The article considers the problem of nonlinear oscillations of the motion of liquids completely filling an axisymmetric cylindrical vessel moving around a horizontal axis \(O{\kern 1pt} *{\kern 1pt} Y\). The motion of each fluid is assumed to be potential and formulated in a cylindrical coordinate system. The influence of nonlinear coefficients on the characteristics of dynamic processes during finite rotational movements of the vessel is estimated and the case of forced angular oscillations of a vessel with liquids relative to a fixed axis is considered. The main nonlinear effects associated with the rotation of the diameter of the interface of liquids are also revealed. An approximate solution of the obtained non-linear equations, found by the Bubnov-Galerkin method, was used in the article. As a result of the transformation, the amplitude-frequency characteristics and stability regions of a two-layer liquid are constructed under forced angular oscillations of a round cylindrical vessel.

Strongly conservative discretization of governing equations in cylindrical coordinates

Cylindrical coordinates are often used in computational fluid dynamics (CFD), particularly when one considers axisymmetric flow. Although cylindrical coordinates have several advantages in accurately describing the rotational flow as well as the complex boundary, there are limits to how far they can be taken. These are (1) the governing equations are mostly in a non-conservative form in current standard models, which cannot preserve the conservation of numerical simulations, (2) the singularity along the axis at the coordinate origin (z-axis), leading to difficulty reproducing the flow across the z-axis, and (3) the time step is extremely shortened to satisfy the CFL condition near the z-axis because the numerical cell thereof is narrow in the azimuthal direction for a given angular resolution. A strongly conservative model is proposed in this paper to overcome these difficulties. First, the strongly conservative form of the governing equations is derived by considering changes in the direction of the base vector within a cell, and a concept of direction error is defined to reveal the inherent error of the standard model. Second, inspired by the derivation of the first, a strongly conservative method for evaluating the flux across each cell surface is proposed, preserving the conservation of discretization and enlarging the time step. Finally, the proposed model is tested by three classical problems, i.e., uniform flow, Sod shock tube problem, and expanding outflows. The results imply that the proposed model is strongly conservative for the cylindrical coordinate system, and it can overcome the limits of the standard model and eliminate the inherent error in the standard model. Furthermore, the strongly conservative model and method are extended to arbitrary coordinates for the numerical conservation in general curvilinear coordinates of the problem with complex boundaries.

Nonlinear optical properties of coupled quantum dots in peanut configuration

ABSTRACT This paper presents a theoretical investigation of nonlinear properties of coupled quantum dots in the peanut configuration. A complete Hamiltonian is formulated using the adiabatic method and the confinement energy is represented in a cylindrical coordinate system. The effective energy for the slow subsystem is analysed and graphed as a function of a fixed value of axial coordinate for both the asymmetric and symmetric peanut QD cases. The electron motion in the presence of an electric field in the vertical direction is considered, and the eigenfunctions and energy spectrum of electron motion are determined for the same direction utilising the finite element method. The dependence of the first three energy levels on the electric field is shown, and the electron probability density for the ground state and the first excited state is plotted. In addition, calculations for the nonlinear optical properties are presented, particularly optical rectification, second harmonic generation and third harmonic generation. The results demonstrate that these properties can be effectively controlled by varying the external electric field. The findings suggest that coupled peanut QDs hold significant potential for applications in high-performance optoelectronic devices.

Eigenfunction expansion solution for wave diffraction from a uniform cylinder in front of a partially reflective wall

In this work, wave diffraction from a uniform circular cylinder in front of a partially reflective wall is solved with the aid of its mirror image cylinder arranged symmetrically about the wall. The mirror image cylinder generates a scattering potential with its value being decreased by the wave reflection from the partially reflective wall. The eigenfunction expansion method is applied to solve this wave diffraction problem, and the Graf's addition theorem is applied to transform the scattering potential back from the imaginary to the original cylindrical coordinate system. Numerical simulations were carried out to examine the effects of the wave reflection of the partially reflective wall, the wave incidence direction, and the distance between the cylinder and the wall. It shows that the non-dimensional wave forces oscillate with the incident wave number, and the oscillation amplitude increases with the increasing reflection coefficient of the wall. In addition, the wave numbers at which the force peaks/troughs occur change significantly with the wave incidence angle and the distance between the cylinder and the wall.

Inversion of radial formation velocity distribution based on monopole acoustic logging while drilling

Acoustic logging is an effective solution to obtain the radial formation velocity profile near a borehole, and it is critical for borehole stability evaluation and production performance monitoring. Herein, the responses of monopole acoustic logging while drilling (LWD) in a fluid-filled borehole surrounded by homogeneous and heterogeneous formations are numerically simulated using the finite difference algorithm in a cylindrical coordinate system. A stepwise inversion method is proposed to determine radial formation velocity distributions near the borehole based on the acoustic LWD data as follows. The heterogeneous formation around the borehole is approximated as multi-layered media with different elastic parameters, and then the slowness time correlation (STC) algorithm is applied to determine the compressional (P) and shear (S) wave velocities corresponding to different offsets. Finally, the thickness of each layer is determined according to the ray theory. Influences of measurement environment parameters, such as the central frequency of the source, thickness of the formation variation zone and elastic parameter changes of the formation variation zone, on the inversion results are evaluated to confirm the applicability of the proposed stepwise inversion method. The numerical simulation results demonstrate that refracted P-, refracted S- and Stoneley waves exhibit specific responses to the heterogeneity of the near-borehole formation, whereas collar waves are minimally influenced by the formation properties. The formation velocities do not change with the offset in the homogeneous formation models, although stepped changes are observed in the heterogeneous formation models. Short-offset waveforms are affected by the properties of the formation close to the borehole wall, whereas long-offset waveforms are primarily affected by the properties of the original formation. Relationships between the formation velocities and the offset can be converted into the radial formation velocity distributions by calculating the arrival times of waves propagating along different ray paths. Under common survey conditions, the inversion result of each model is similar to its real velocity distribution, which confirms the effectiveness of the proposed stepwise inversion method. This study thus provides a theoretical basis for developing 3D acoustic LWD tools.

A balanced model of a hurricane vortex coupled to a boundary layer

AbstractThe boundary layer plays a key role in several aspects of hurricane dynamics. Here we focus on its contribution to the balanced circulation. Previous studies, whilst including centrifugal terms, have not included an explicit balance in the boundary layer. Here, we improve the balanced theory to include an Ekman balance, the so‐called frictional axisymmetric vortex (FAV). This approach is analogous to semigeostrophic theory that includes realistic boundary‐layer diffusion: semigeotriptic (SGT) theory. We formulate the FAV for an axisymmetric system in cylindrical polar coordinates. We then derive a Sawyer–Eliassen equation for the vertical circulation. Example solutions for idealised hurricane‐scale and synoptic‐scale vortices are compared.

Advancing Tropical Cyclone Precipitation Forecast Verification Methods and Tools

Abstract Tropical cyclone (TC) forecast verification techniques have traditionally focused on track and intensity, as these are some of the most important characteristics of TCs and are often the principal verification concerns of operational forecast centers. However, there is a growing need to verify other aspects of TCs as process-based validation techniques may be increasingly necessary for further track and intensity forecast improvements as well as improving communication of the broad impacts of TCs including inland flooding from precipitation. Here we present a set of TC-focused verification methods available via the Model Evaluation Tools (MET) ranging from traditional approaches to the application of storm-centric coordinates and the use of feature-based verification of spatially defined TC objects. Storm-relative verification using observed and forecast tracks can be useful for identifying model biases in precipitation accumulation in relation to the storm center. Using a storm-centric cylindrical coordinate system based on the radius of maximum wind adds additional storm-relative capabilities to regrid precipitation fields onto cylindrical or polar coordinates. This powerful process-based model diagnostic and verification technique provides a framework for improved understanding of feedbacks between forecast tracks, intensity, and precipitation distributions. Finally, object-based verification including land masking capabilities provides even more nuanced verification options. Precipitation objects of interest, either the central core of TCs or extended areas of rainfall after landfall, can be identified, matched to observations, and quickly aggregated to build meaningful spatial and summary verification statistics.

The polar-Cartesian mild-slope equation

A novel mild-slope equation is derived based on a manipulation of the cylindrical and Cartesian coordinate reference systems. The vertical profile of the velocity field is constructed by solving an approximate problem in cylindrical coordinates. This allows us to address the local derivatives on the bottom profile along a constant-slope line. This formulation is as opposed to the Cartesian-based mild-slope equations in terms of which the profile is constructed by assuming a constant depth. An angular profile is derived for the three-dimensional case on a sloping plane beach. For the two-dimensional case, a mild-slope polar-Cartesian equation is derived, for which an improved linear dispersion relation is reconstructed. This is accomplished due to the inclusion of first-order derivatives of the local bottom profile. The coefficients of the polar-Cartesian mild-slope equation contain the derivatives of the bottom profile up to third order as opposed to second-order derivatives in the Cartesian-based equations. The equation is derived by applying the variational principle to the Cartesian Lagrangian when formulated as a function of the profile in polar coordinates. It is then compared with existing models of the mild-slope equation for simulations of two-dimensional test cases and a quasi-three-dimensional case, which have known analytical solutions. Our modified equation exhibits better matching to the exact solutions for a majority of the investigated cases.

Parabolic Cylindrical Description of Velocity and Acceleration Tensor in Bipolar Coordinate Using Tensor Analysis

Instantaneous velocity and acceleration are often studied and expressed in Cartesian, Circular Cylindrical, and Spherical Coordinate systems but it is a well-known fact that some bodies cannot be perfectly described in these coordinate systems, so they required some other curvilinear systems such as oblate spheroidal, parabolic cylindrical, elliptic cylindrical bipolar, and others. It is of important interest in theoretical physics to establish equations of motion in Bipolar Coordinate system, which is essentially suitable to accurately describe the motion of bipolar bodies such as hyperbolas, ellipses and other curves in the universe. In this work, we derived a new expression for the instantaneous velocity and acceleration vector of bodies and test particles in the bipolar coordinate system using tensor analysis. The Laplacian of a scalar field in this bipolar coordinate system was also derived which has applications in mechanics.

Determination of the electric field strength of high-voltage substations

The electric field strength is one of the main factors influencing sensitive microprocessor equipment and personnel on power stations and substations, power lines. Determining its level is an important applied task for ensuring the safe operation of electrical installations. The aim is to develop calculation relationships for determining the electric field strength created by the busbar of high-voltage substations in the working areas of personnel. The solution of the problem was based on the use of the method of equivalent charges to determine the strength of the electric field created by the complex busbar of high-voltage substations. Methodology. The development was based on solving the problem of the potential of the electric field of a point charge located in a dielectric half-space for a cylindrical coordinate system. By representing the electrode in the form of a set of point charges and subsequent integration, an expression for calculating the potential is obtained, created by a busbar of arbitrary orientation of finite length in an analytical form. Using the principle of superposition of fields and the definition of the derivative, expressions were obtained for calculating the vertical component of the electric field strength at given heights. Results. Based on the obtained expressions, using Visual Basic, the simulation of the distribution of the electric field strength under a three-phase power line with a voltage of 150 kV was performed. Comparison with the known calculation results obtained on the basis of analytical expressions for infinitely long conductors showed that the obtained expressions have an error of no more than 7%. The scientific novelty lies in the fact that for the first time expressions were obtained for determining the electric field strength created by a system of electrodes of finite length, based on the analytical method for solving differential equations. Practical significance. The proposed technique is implemented as a test module of the LiGro specialized software package, which allows modeling complex busbar systems typical for power stations and substations and power lines. A test calculation was carried out for an operating substation of regional electric networks with a voltage class of 110 kV. By comparing the duration of the calculation of switchgears with a diagonal of about 500 m, it was found that the calculation time in the LiGro complex based on the analytical method is several tens of times less than the calculation based on the finite element method. In addition, a more powerful computer was used for the end element simulation.

A Computational Study of an Underexpanded Supersonic Jet

A general code has been developed for solving the wave equation in cylindrical co-ordinate system with source distribution due to various causes to obtain a predictive capability. The advantage of this approach is that source distributions due to various causes like combustion noise, jet noise etc. can be added as and when they become available without having to change the program. One of the source distribution needed is that due to jet noise. In this work volume source distribution due to jet mixing was approximately estimated using turbulence theory. Numerical results were compared with the experimental results for C-D nozzle designed for a Mach number of two.

Thermodynamic study of hybrid nanofluid to explore synergistic effects of multiple ferromagnetic nanoparticles in co-axial disks for magnetized fluid

We Consider the physical model of two-dimensional hybrid nanofluids flow, which is time dependent, laminar, incompressible inside the two co-axial stretchable disks. The length of upper and lower disks is2s(t). The two axial and radial velocity components (u,w) depends upper the radial of disks r and normal of direction z. There exists temperature T1>T2 which is constant for upper and lower disks. The continuity, momentum and energy equations are modeled in cylindrical coordinates system. The finite element technique is used to evaluate numerical solutions. It is observed that the velocity near upper and lower disks decreases by increasing the values of magnetic parameter. Increased the values of shape factorNs of (Nanoparticles) then the thermal conductivity (HDnf) decreased. The shear stress and heat transfer rate decreased by influence α,ϕ1,ϕ2 but an opposite reaction observed when increase the values of M,Re andNs.

A Station-Keeping Control Strategy for a Symmetrical Spacecraft Utilizing Hybrid Low-Thrust Propulsion in the Heliocentric Displaced Orbit

The solar sail spacecraft utilizing a hybrid approach of solar sail and solar electric propulsion in the heliocentric displaced orbit is affected by external disturbances, internal unmodeled dynamics, initial injection errors, and input saturation. To solve the station-keeping control problem under such complex conditions, an adaptive control strategy is proposed. First, the dynamical equations of the spacecraft utilizing hybrid low-thrust propulsion in the cylindrical coordinate system are derived. Second, the combined disturbance acceleration introduced by external disturbances and internal unmodeled dynamics is constructed, and a radial basis function neural network estimator is designed to estimate it online in real time. Third, an adaptive high-performance station-keeping controller based on an improved integral sliding surface and multivariate super-twisting sliding mode approaching law is designed. Then, stability analysis is conducted using Lyapunov theory, adaptive laws are designed, and the introduced virtual control accelerations are converted into actual control variables. Finally, simulations are conducted under different simulation conditions based on the disturbance sources. The results show that although the use of hybrid low-thrust propulsion breaks the symmetry of the solar sail in configuration, the proposed control strategy can effectively achieve the station-keeping and disturbance estimation of the spacecraft with only a small amount of propellant consumed and position tracking errors up to decimeters.

ANALYSIS OF ONE-DIMENSIONAL NON-NEWTONIAN MODELS FOR SIMULATION OF BLOOD FLOW IN ARTERIES

The paper is devoted to the theoretical analysis of one-dimensional (1D) models of blood flow. The non-Newtonian nature of blood is taken into account. The generalized Newtonian models, in which the dynamic viscosity is dependent only on the shear rate, are considered. The models are constructed by averaging the simplified Navier–Stokes system in cylindrical coordinates. The 1D models, corresponding to Newtonian, Power Law, Carreau, Carreau–Yasuda, Cross, Simplified Cross, Yeleswarapu, Modified Yeleswarapu, and Quemada models, are compared in this paper. The comparison is performed in order to estimate the influence of non-Newtonian frictional terms, velocity profiles and hematocrit values on the deviations of the solutions from the solution for the Newtonian case. For this purpose, two problems that can be solved analytically are considered: problems for the infinite interval and finite interval with periodic conditions. For the quantitative comparison of solutions, the non-Newtonian factor, which characterizes the relative deviations of solutions from the solution for the Newtonian model, is introduced. The analytical solutions of nonlinear problems are obtained by the perturbation method. As it is demonstrated for both solved problems, the relative deviations increase with the flattening of the velocity profile. For the models that depended on hematocrit, the non-Newtonian factor values increased with the increase in hematocrit value. The solutions obtained in the paper can be used as a tool for the comparison of different 1D models of blood flow and for the testing of programs that implement numerical algorithms.

Frequency‐dependent response of a rigid disk on a transversely isotropic soil layer subjected to horizontal motion

AbstractThe transverse isotropy of soil induced by geological processes over a long period has attracted considerable attention, which can lead to significant differences in the origination, propagation and attenuation of the stand waves compared with purely isotropic soil. Considering that the circular geometry is most common in many engineering, an analytical investigation of the dynamic behavior of a rigid disk resting on the transversely isotropic soil layer overlying the rigid bedrock under forced horizontal displacements is presented in this paper. Firstly, the governing equations of soil motion are established based on the elasticity theory of cylindrical coordinate system. Secondly, a set of partial differential equations for this class of problems are decoupled to ordinary differential forms by virtue of complete potential functions, Fourier expansion method and Hankel transform technique. Furthermore, the displacements, strains and stresses formulation in term of the spatial domain under general boundary conditions are obtained with success. For validation, dynamic boundary value problems in the finite‐layered soil containing transversely isotropic materials are degenerated to the isotropic solution corresponding to elastic half‐space. After that, various examples are also included to investigate the influence of the elastic modulus ratio, shear modulus ratio and thickness of soil layer on the dynamic response of the system. The numerical results suggest that the soil anisotropy and soil layer thickness have a non‐neglectful effect on the horizontal dynamic vibration of the rigid disk.

Marangoni instability of an evaporating binary mixture droplet

Evaporation of a binary mixture droplet (BMD) is a common natural phenomenon and widely applied in many industrial fields. For the case of a sessile BMD being the only contact-line pinning throughout an entire evaporation, a theoretical model describing the evaporating dynamics is established when considering the comprehensive effect of evaporative cooling, the thermal Marangoni effect, the solutal Marangoni effect, the convection effect, and the Stefan flow. The dynamics of a binary ethanol–water droplet on a heated substrate is simulated using a cylindrical coordinate system. The reasons for Marangoni instability-driven flow (MIF) are discussed, and the influence of initial ethanol concentration and substrate heating temperature are examined. An evaporating BMD first forms a MIF at the contact line and quickly affects the whole droplet. Under the influence of the Marangoni instability, the BMD presents a complex internal flow structure with multiple-vortex and nonlinear temperature and ethanol concentration distributions. The positive feedback induced by vortices and the nonlinear distribution of concentration and temperature promotes the development of a MIF. At low initial ethanol concentrations, the MIF loses its driving force and turns into a stable counterclockwise single-vortex flow as ethanol evaporates completely. However, at high initial ethanol concentrations, the MIF exists in the entire evaporation. Increasing ethanol concentration and substrate heating temperature can delay the appearance of the MIF; ethanol concentration affects the MIF duration time, and heating temperature alters the MIF intensity. To enhance flow intensity and mass transfer of BMDs, the temperature difference should first be increased, followed by increased ethanol concentration.