Cylindrical Coordinate System Research Articles

Sturm–Liouville problem for a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems

The problem of constructing eigenfunctions of a one-dimensional thermoelastic operator in Cartesian, cylindrical, and spherical coordinate systems is considered. The corresponding Sturm–Liouville problem is formulated using Fourier’s separation of variables applied to a coupled system of thermoelasticity equations, assuming that the heat transfer rate is finite. It is shown that the eigenfunctions of the one-dimensional thermoelastic operator are expressed in terms of well-known trigonometric, cylinder, and spherical functions. However, coupled thermoelasticity problems are solved analytically only under certain boundary conditions, whose form is determined by the properties of the eigenfunctions.

Hydrodynamic parameters affecting flow-accelerated corrosion in elbows

Flow-accelerated corrosion (FAC) is a serious degradation mechanism affecting carbon-steel piping in the secondary coolant systems of nuclear power plants. Many researchers have investigated the correlation between wall thinning and hydrodynamic parameters; however, their relationship remains unclear. In this study, the hydrodynamic parameters influencing FAC in elbow sections were investigated for identifying the locations highly susceptible to FAC. Accelerated FAC tests and computational fluid dynamics (CFD) analyses were conducted on a specimen containing elbow sections. The results revealed that the side of the elbow pipe was the most susceptible to FAC because it experienced the most severe wall thinning. The velocity and vorticity vectors obtained from the CFD analysis were decomposed into axial, circumferential, and radial components within a cylindrical coordinate system to evaluate their effects. We found that all velocity and vorticity components contribute to the rate of wall thinning induced by FAC. Therefore, a combination of velocity and vorticity components should be used as hydrodynamic parameters to characterize FAC-induced wall thinning. Specifically, the circumferential velocity and axial vorticity were identified as dominant parameters influencing FAC in the elbows.

Nonlinear evolution of vortical disturbances entrained in the entrance region of a circular pipe

The nonlinear evolution of free-stream vortical disturbances entrained in the entrance region of a circular pipe is investigated using asymptotic and numerical methods. Attention is focused on the low-frequency disturbances that induce streamwise elongated structures. A pair of vortical modes with opposite azimuthal wavenumbers is used to model the free-stream disturbances. Their amplitude is assumed to be intense enough for nonlinear interactions to occur inside the pipe. The formation and evolution of the perturbation flow are described by the nonlinear unsteady boundary-region equations in the cylindrical coordinate system, derived and solved herein for the first time. Matched asymptotic expansions are employed to construct appropriate initial conditions and the initial–boundary value problem is solved numerically by a marching procedure in the streamwise direction. Numerical results show the stabilising effect of nonlinearity on the intense algebraic growth of the disturbances and an increase of the wall-shear stress due to the nonlinear interactions. A parametric study is carried out to evince the effect of the Reynolds number, the streamwise and azimuthal wavelengths, and the radial length scale of the inlet disturbance on the nonlinear flow evolution. Elongated pipe-entrance nonlinear structures (EPENS) occupying the whole pipe cross-section are discovered. EPENS with $h$ -fold rotational symmetry comprise $h$ high-speed streaks positioned near the wall, and $h$ low-speed streaks centred around the pipe core. These distinct structures display a striking resemblance to nonlinear travelling waves found numerically and observed experimentally in fully developed pipe flow. Good agreement of our mean-flow and root mean square data with experimental measurements is obtained.

Simulation of silicon conical field effect GAA nanotransistors with stack SiO2/HfO2 dielectric of gate

The issues of modeling the electrophysical characteristics of a silicon conical field effect GAA nanotransistor are discussed. An analytical model of the drain current of a transistor with a fully enclosing conical gate with a stack sub-gate oxide SiO2/HfO2 has been developed, taking into account the effect of the charge of the interphase trap at the Si/SiO2 interface. To simulate the potential distribution in a conical working area under the condition of constant trap density, an analytical solution of the Poisson equation was obtained using the method of parabolic approximation in a cylindrical coordinate system with appropriate boundary conditions. The potential model was used to develop an expression for the GAA drain current of a nanotransistor with a stack gate oxide. The key electrophysical characteristics are numerically investigated depending on the density of traps and the thicknesses of SiO2 and HfO2 layers.

Smart analysis of sandwich foldable cylinders as gymnastic accessories

A novel smart analysis of sandwich composite cylindrical shell is presented in this article. It is assumed that the sandwich shell is manufactured from the graphene origami reinforced copper core between two intelligent layers. The proposed structure in this article can be used as an idealized model for cylinder shapes in gymnastic sport and physical therapy units. The virtual work principle in the piezo elasticity framework and the cylindrical coordinate system is extended to derive governing equations. To arrive at more accurate results, a third order shear deformable model is extended for kinematic relations. The electro elastic responses are explored using the analytical method to seek impact of main parameters of the composite structure such as foldability parameter, graphene content percent, and applied electric potential on the electro elastic strain, deformation and stress results.

Research on the high-performance computing method for the neutron diffusion spatiotemporal kinetics equation based on the convolutional neural network

Due to the uncertainty of computational results and the lack of interpretability of models in solving physical field equations in current deep learning, this paper designs a convolutional neural network that can be used to solve the neutron diffusion spatiotemporal kinetics equation in polar and cylindrical coordinate systems. This algorithm directly utilizes the macroscopic cross-section of the material without using the lattice homogenization method, replaces the finite volume method with the extended matrices, and solves the extended matrices using the convolutional kernels instead of the iterative algorithms. Taking the simplified Tsinghua High Flux Reactor (THFR) as an example, the feasibility of the algorithm is verified on the PyTorch platform and compared with the calculation results of the source iteration method running on the GPU. The calculation results show that when the number of grids in the radial and axial sections of the simplified THFR model is 804,600 and 3,576,000, respectively, and the algorithm is iterated 3000 times, the normalized power of the convolutional neural network and the source iteration method converges to 10−10, and the maximum point by point error of the neutron flux density of the above two algorithms converges to 10−5. The computational time consumed by the convolutional neural network is approximately 880.64 s and 3729.62 s, which reduces the computational time by 4.66% and 5.05% compared to the GPU parallel accelerated source iteration method, and the former consumes 43.75% less memory compared to the latter. The convolutional neural network is mainly used as the virtual physics engine for the THFR digital twin system, in addition to solving the neutron diffusion spatiotemporal kinetics equation and further improving computational speed. The algorithm directly utilizes the neutron macroscopic cross-section of the material to calculate the neutron flux density distribution without using the lattice homogenization, providing theoretical guidance and algorithm support for developing the high-precision multi-physical field coupling model.

Inverse Problem for a Distributed System from Pulse Technology

An inverse problem with distributed parameters for the process of the self-focusing of cylindrical X-ray pulses in a plasma is considered, and a mathematical model of the studied process in a cylindrical coordinate system is described, taking into account the symmetry of the pulse relative to the direction of its propagation. A similar process in the case of plane pulses is compared, a computational method for solving the direct problem of interaction between the plasma and pulse for the given parameter values is presented, the second order of approximation and the asymptotic stability of the constructed difference scheme are proved. It is proposed to use the equivalence set method to solve the inverse problem of determining the initial parameters of the plasma and pulse from the shape of a cylindrical X-ray pulse passing through it and the dynamics of its maximum intensity. Using this problem as an example, an algorithm for using the equivalence set method to solve inverse problems is described.

Differential Equation of Thermal Conductivity and Convective Heat Transfer in a Cylindrical Coordinate System

Differential Equation of Thermal Conductivity and Convective Heat Transfer in a Cylindrical Coordinate System

Nonlinear electrodynamics and its possible connection to relativistic superconductivity: an example

Abstract This work presents an illustrative example suggesting a potential connection between the Heisenberg–Euler (HE) model of nonlinear electrodynamics (NLED) and relativistic superconductivity. Within a cylindrical coordinate system, we derive nonlinear Maxwell’s equations from the HE Lagrangian to the fourth order in the electromagnetic (EM) field. By considering static electric and magnetic fields with Gaussian radial profiles, we compute the corresponding HE current density. In parallel, we explore relativistic type-II superconductivity in analogy with the Ginzburg–Landau (GL) theory. We calculate the vortex-supercurrent density of the condensate and propose a link between this current and the HE current. This connection enables the derivation of the relativistic order parameter state. The problem considered in this article may contribute to the understanding of superconductivity in strong EM environments within the broader framework of NLED.

Optimization of The Influence of Temperature on The Electrical Distribution of Structures with Radial p-n Junction Structures

In recent years, advances in optoelectronics and electronics have prioritized optimizing semiconductor device performance and reducing power consumption by modeling new semiconductor device geometries. One such innovative structure is the radial p-n junction structure. In this work, we present a concept that submicron three-dimensional simulations were conducted on radial p-n junction structures based on GaAs material to investigate the influence of temperature ranging from 250K to 500K with a step of 50K on the electrophysical distribution, such as space charge, electro-potential, and electric field, in radial p-n junction structures, as well as various forward voltages. In particular, we focus on the shell radius within the structure: 0.5 μm and 1 μm for the shell. The modeling results were compared with the results obtained from solving the theoretical Poisson equation in the cylindrical coordinate system.

Circular Clustering With Polar Coordinate Reconstruction.

There is a growing interest in characterizing circular data found in biological systems. Such data are wide-ranging and varied, from the signal phase in neural recordings to nucleotide sequences in round genomes. Traditional clustering algorithms are often inadequate due to their limited ability to distinguish differences in the periodic component θ. Current clustering schemes for polar coordinate systems have limitations, such as being only angle-focused or lacking generality. To overcome these limitations, we propose a new analysis framework that utilizes projections onto a cylindrical coordinate system to represent objects in a polar coordinate system optimally. Using the mathematical properties of circular data, we show that our approach always finds the correct clustering result within the reconstructed dataset, given sufficient periodic repetitions of the data. This framework is generally applicable and adaptable to most state-of-the-art clustering algorithms. We demonstrate on synthetic and real data that our method generates more appropriate and consistent clustering results than standard methods. In summary, our proposed analysis framework overcomes the limitations of existing polar coordinate-based clustering methods and provides an accurate and efficient way to cluster circular data.

Long-lived Equilibria in Kinetic Astrophysical Plasma Turbulence

Turbulence in classical fluids is characterized by persistent structures that emerge from the chaotic landscape. We investigate the analogous process in fully kinetic plasma turbulence by using high-resolution, direct numerical simulations in two spatial dimensions. We observe the formation of long-lived vortices with a profile typical of macroscopic, magnetically dominated force-free states. Inspired by the Harris pinch model for inhomogeneous equilibria, we describe these metastable solutions with a self-consistent kinetic model in a cylindrical coordinate system centered on a representative vortex, starting from an explicit form of the particle velocity distribution function. Such new equilibria can be simplified to a Gold–Hoyle solution of the modified force-free state. Turbulence is mediated by the long-lived structures, accompanied by transients in which such vortices merge and form self-similarly new metastable equilibria. This process can be relevant to the comprehension of various astrophysical phenomena, going from the formation of plasmoids in the vicinity of massive compact objects to the emergence of coherent structures in the heliosphere.

Analytical solution to the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system

The inverse heat conduction problem involves estimating an unknown cooling or heating action based on internal temperature histories in a body. Such a problem is ill-posed because the estimated results are highly sensitive to input data noise. An analytical solution is developed for the two-dimensional inverse heat conduction problem in an axisymmetric cylindrical coordinate system using the Laplace transform technique. On the basis of the known temperatures at multiple measuring points distributed on two cylindrical measuring surfaces along the axial direction inside the solid, the expressions for the surface temperature and heat flux are explicitly obtained in the form of a power series of time and eigenfunctions of the space variable. A comprehensive estimation error assessment is conducted using the internal temperature distribution that is obtained using a direct heat conduction analysis under the prescribed stationary and moving boundary conditions. The constant thermal condition is well reproduced using the present inverse solution, except in the vicinity of the discontinuity point. In addition, the effective heat flux and effective heat transfer coefficient are introduced to obtain the best estimation of the transient heat transfer in the major cooling area of the rotating cylinder.

Semi-analytical vibration modeling of complex axisymmetric shells using shifted Legendre series

This paper presents a semi-analytical approach based on the shifted Legendre series for vibration modeling of complex axisymmetric shells. This method divides the structure along its generator into complex axisymmetric shells comprised solely of conical and spherical shell segments. A global cylindrical coordinate system is established to unify the coordinate system of complex axisymmetric shells. Coupling between segments and the simulation of arbitrary boundary conditions are achieved using artificial spring technology. Based on the first-order shear deformation theory (FSDT), the energy functional of each segment is derived. The circumferential displacement is expanded using Fourier series, while the meridional displacement is expanded using the shifted Legendre series at shifted Gauss-Lobatto nodes. Utilizing the inner product matrix and differential matrix of the shifted Legendre series, the integration and differentiation processes in the vibration characteristic equation are simplified into a product form, significantly enhancing computational efficiency. Through convergence analysis and comparison of the computational results of the present method with numerical simulation results, experimental results and literature results demonstrate the advantages of high convergence, high accuracy and fast solution speed of this solution method. On this basis, the vibration characteristics of ring-stiffened double-layer hemispherical shells are investigated. The results demonstrate the great applicability of this method for vibration modeling of complex axisymmetric shells in engineering.

Entropy generation and heat transfer analysis of unsteady micropolar magnetized hybrid-nanofluid flow over a radially stretchable permeable rotating disk with viscous and joule dissipation effects

PurposeThis study investigates the unsteady, three-dimensional radiative, micropolar hybrid nanofluid flow across the radially rotating disc in a Darcy-Forchheimer porous medium, under the influence of an applied magnetic field. The effects of viscous dissipation and Joule heating are also considered. Detailed analyses of entropy generation and the Bejan number are provided. Copper (Cu) and Titanium dioxide (TiO2) nanoparticles are modeled mathematically using a cylindrical coordinate system with engine oil as the base fluid. Both Copper (Cu) and Titanium dioxide (TiO2) nanoparticles are consider 50 %-50 %. MethodologyThe Navier-Stokes equations governing momentum, microrotation, and energy are transformed into ordinary differential equations using similarity variables. These modified equations are solved numerically using shooting approach (bvp4c). The study includes graphical representations and discussions on the impact of various controlling parameters. FindingsThe considered range for the parameters are Mn = 0.1 to 3.1, β = 0.1 to 1.3, π = 0.5 to 2.0, St = 0.1 to 1.3, K1 = S0 = A3 = 0.1 to 1.0, Bi = 0.1 to 0.4, Ec = 1.0 to 1.4, . Br = 0.10 to 0.25, α1 = 0.1 to 2.5. Results indicate that the unsteady parameter decay the velocity profile (both radial and tangential direction) and reduce the microrotation velocity in r and z direction, while enhancing the microrotational velocity in θ direction. Additionally, an increase in the stretching parameter raises the radial velocity but lowers the tangential velocity profile. The temperature profile and entropy generation increases for large values of magnetic parameter, whereas the Bejan number decreases. Similarly, tables were analyzed to illustrate the impact of varying physical parameters on the skin friction local Nusselt number. It is noted that the magnetic parameter enhance the skin friction coefficient while decline the local Nusselt number. The concentration of Copper (Cu) and Titanium dioxide (TiO2) nanoparticles is (0-6)%.

Design and study of the shape parameters for the electrode plates of the electron gun in the four-dimensional transmission electron microscope.

The accelerating electrode of a four-dimensional transmission electron microscope electron gun is modeled. The general expression of the electric-field distribution is derived for any point on the axis in a cylindrical coordinate system, and equations for the shape parameters of the electrode plate are obtained. The accuracy of the field expression is determined for different electrode plate parameters, and the shape parameters of the electron gun electrode are further investigated. This work can provide a theoretical basis for the initial design of a transmission electron microscope electron gun and the retrofit design of a four-dimensional electron gun.

Analysis of hemodynamic parameters on two-layered blood flow in a curved artery

Arterial stenosis is the thickening of the arterial wall due to the growth of aberrant tissues that prevent adequate blood flow in the human circulatory system and induces cardiovascular diseases. Mild stenosis, may lead to serious or permanent damage if remains uncured. There are differences in the curvature response and material composition between the outer layers and the core. There are several locations in blood vessels where they are curved, which affects the blood flow and shear stress. The Navier-Stokes equation in the cylindrical polar coordinate system has been extended by incorporating curvature term in two-layered blood flow along the axial direction with appropriate boundary conditions. Mathematical expressions for hemodynamic parameters such as velocity profile, volumetric flow rate, pressure drop, and shear stress have been calculated analytically in the case of a curve artery with stenosis. Moreover, we have analyzed the effect of stenosis on different hemodynamic parameters with the variation of core and peripheral-layer viscosity and curvature. Flow quantities are affected by the habitancy of stenosis and stipulate different blood flow behavior in both layers in the case of curved artery. This modeling technique may help researchers in medicine, mathematical biology, and bio-engineering.

Feedrate-preserved motion planning for five-axis directed energy deposition of freeform metal parts

Laser-powder DED (LP-DED) has been investigated extensively for the rapid prototype of complex and customized metallic parts and in-situ repairing tasks. However, unlike the material extrusion type which allows a relatively large overhang angle, the conventional 2.5-axis LP-DED faces a severe overhang issue that deteriorates the printing stability and quality. While five-axis LP-DED is able to effectively resolve the overhang problem, it however introduces new kinematic issues and limited nozzle orientation range due to physical restrictions of DED. The existing computer numerical control systems are mainly designed for machining, where the above issues can be mostly solved by feedrate scheduling, which, however, is not applicable for LP-DED processes, as the feedrate must be preserved as programmed lest deposition height would be altered since in most cases it is not practical to vary the powder flow rate in real-time. In this paper, a feedrate-preserved motion planning method is proposed to optimize the rotational angles of the machine tool in the machine coordinate system (MCS) under the constraints of MCS and workpiece coordinate system (WCS). Given a five-axis DED path as well as its accompanying initial nozzle orientation in WCS, we first map them via inverse kinematics to MCS, which is represented by a cylindrical coordinate system. Then, the motion path in the cylindrical coordinate system is optimized to avoid the singularity. Finally, the physical constraint of nozzle orientation in WCS is introduced into the time-weighted constrained Laplacian smoothing in MCS with the axial speed limits upheld. The experimental results confirm that the actual feedrate is successfully preserved as the theoretical one under the constraints of the axial kinematic limits, while the singularity is avoided.

Assisting Directional Drilling by Calculating a Safe Operating Envelope

Summary Nowadays, complex 3D trajectories are executed with a succession of circular arcs (CAs). Although they have constant curvature, their tool face is not constant. Consequently, directional drillers must adjust the tool face regularly to reach the target entry within its tolerances. This paper investigates the use of the constant curvature and constant tool face (CTC in short) curve as an alternative to the CA to assist the directional drilling work to reach the target entry within its boundaries. The problem is addressed by calculating a safe operating envelope (SOE) to reach the boundaries of the target entry and provide a tolerance window for the curvature and tool face to support directional drilling decisions. The target entry tolerance is discretized as a polygon. From the current bit position and its direction, the possible choices of curvatures and tool faces are obtained to reach the edges of the target entry shape. The SOE can be calculated with the CA or with the CTC curve. It is, therefore, possible to compare the advantages and disadvantages of both types of curves to attain the target entry and stay within its boundaries. The CA is shorter than the CTC curve. However, it requires adjusting the tool face during the navigation, which is not the case with the CTC curve. As a result, the directional driller can control the bottomhole assembly (BHA) direction such that the well lands within the target entry limits by using set points for tool face and curvature inside the calculated SOE. Furthermore, a new way to represent the SOE is introduced. It makes use of a 3D cylindrical representation where the curvature is mapped as the height of a cylinder, while the tool face corresponds to the azimuth in the cylindrical coordinate system, and the length is linked to the radial distance. This provides a visual aid to understand the SOE. Moreover, this visualization helps to appreciate the relationship between the initial bit location and direction in the construction of the SOE and how the margins increase in a particular manner as the bit approaches the target entry polygon. The CTC curve is the natural one followed by directional positive displacement motors (PDMs) or rotary steerable systems (RSS). Potentially, the CTC curve may be a more straightforward solution to automated directional drilling control because it is easier to be followed by both PDM and RSS.

Simplified Tunnel–Soil Model Based on Thin-Layer Method–Volume Method–Perfectly Matched Layer Method

In order to analyze the ground vibration responses induced by the dynamic loads in a tunnel, this paper proposes a new simplified tunnel–soil model. Specifically, based on the basic theory of the thin-layer method (TLM), the basic solution of three-dimensional layered foundation soil displacement was derived in the cylindrical coordinate system. The perfectly matched layer (PML) boundary condition was applied to the TLM. Subsequently, a tunnel–soil dynamic interaction analysis model was established using the volume method (VM) in conjunction with the TLM-PML method. The displacement frequency response function of the foundation soil around the tunnel foundation was derived. Finally, a ground vibration test under an impact load in a tunnel was carried out. The test and calculated results were compared. The comparison results show that the ground vibration acceleration response values within 25 m from the load are similar. Compared with the test results, the theoretical calculation results exhibit a decreasing trend in the range of 40–80 Hz between 25 and 60 m, with the maximum reduction being approximately one order of magnitude. In addition, the experimental comparison demonstrates that the model can be used to analyze the ground vibrations caused by underground loads.