Circular Cylindrical Coordinates Research Articles

Riemannian laplacian in rotational circular cylindrical coordinates for all gravitational fields in nature based upon the great metric tensor

Most mathematicians’ and physicists’ formulations are based upon Euclidean Geometry for applications in various fields. In this paper, I formulate a new Laplacian operator in Rotational Circular Cylindrical Coordinates which is built upon the Riemannian Geometry, based upon the Great metric tensor, which is more mathematically complete and unique in all gravitational fields for applications in mechanics.

The Effects of Fractional Relaxation Time and Magnetic Field on Blood Flow through Arteries along with Nanoparticles

In this paper, we report the effects of fractional relaxation time on the parameters of blood flow together with magnetic particles through straight circular cylindrical arterial segment. A mathematical model of blood flow subject to pulsatile pressure gradient in the axial direction with external magnetic field applied normal to the direction of flow is presented. Combining the momentum equation together with the Maxwell model parameter appropriately, leads to the governing fractional partial differential equation which permits to obtain the velocity profile of blood along with magnetic particles. By adopting the non-dimensionalized form of the new version of the governing fractional partial differential equation allowed us to obtain the dimensionless relaxation time parameter λ1 which controls blood flow conditions. Solving the fractional partial differential equations using Laplace and finite Hankel transforms we found that the influence of the order of Caputo's fractional time-derivative and fractional relaxation time on the blood flow parameters with magnetic particles are enormous. The graphical results plotted of different influential parameters are presented and discussed in details. The velocities of blood flow and that of magnetic particles are reduced under the influence of the external magnetic field and the relaxation time parameter. The magnetic particles are assumed to be uniformly distributed within the blood, since they are flowing in the same axial direction designated by along a circular cylindrical coordinates of radius. This is a very good indication that blood velocity can be controlled by the application of external magnetic field as well as the relaxation time parameter during treatment to avoid tissues damage. The present study has important applications in magnetic field control of biotechnological processes, bio magnetic device technology, biomedical engineering and pathology. Keywords: Arterial segment, Blood flow, Relaxation time, Magnetic field, Magnetic particles

Slow Rotation of a Soft Colloidal Sphere Normal to Two Plane Walls

The creeping flow of a viscous fluid around a soft colloidal sphere rotating about a diameter normal to two planar walls at an arbitrary position between them is theoretically investigated in the steady limit of small Reynolds numbers. The fluid velocity outside the particle consists of the general solutions of the Stokes equation in circular cylindrical and spherical coordinates, while the fluid velocity inside the porous surface layer of the particle is expressed by the general solution of the Brinkman equation in spherical coordinates. The boundary conditions are implemented first on the planar walls by means of the Hankel transforms and then at the particle and hard-core surfaces by a collocation technique. The torque exerted on the particle by the fluid is calculated as a function of the ratio of the core-to-particle radii, ratio of the particle radius to the flow penetration length of the porous layer, and relative particle-to-wall spacings over the entire range. The wall effect on the rotating soft particle can be significant. The hydrodynamic torque exerted on the confined soft sphere increases as the relative particle-to-wall spacings decrease and stays finite even when the soft sphere contacts the plane walls. It is smaller than the torque on a hard sphere (or soft one with a reduced thickness or penetration length of the porous layer), holding the other parameters constant. For a given relative wall-to-wall spacing, this torque is minimal when the particle is situated midway between the walls and rises as it locates closer to either wall.

Synergetic characteristics of three-dimensional transient heat transfer in geothermal reservoir combined with power conversion for enhanced geothermal system

This paper establishes a complete enhanced geothermal system (EGS) during 30 years system life cycle. The power generation system is established by an ORC (Organic Rankine Cycle) system. Then, the two-dimension numerical model with the injection well and production well is established on the basis of circular cylindrical coordinates by FVM (Finite volume method). Meanwhile, a geothermal reservoir with one primary fracture is established based on the porosity model by FEM (Finite element method). On the basis of this complete numerical model, the coupled relationship between heat transfer and power generation is investigated by temperature difference θ, mass flow rate, and the structure of primary fracture, respectively. The temperature difference θ is defined by the temperature difference between the outlet temperature of production well and the evaporating temperature to analyze the performance of power generation. Then, the flow rate of the injection well and the structure of fracture are investigated to analyze for the performance of power generation. For the structure of fracture, the plane angle α and the plane angle β are employed to describe the vertical fracture and the horizontal fracture, respectively. The results show that a higher temperature difference θ leads a higher net work output and exergetic efficiency. And the flow rate has a more significant impact on the performance of power generation than the temperature difference θ during the 30 years system life cycle. For the primary fracture, a vertical fracture has a stronger impact on the outlet temperature of production well than that of a horizontal fracture.

Particle-in-Cell Investigation on Temporal Flow and Expansion Behaviors of Deuterium-Titanium Plasmas From Ion Source to Expansion Cup

By using a self-programed circular cylindrical coordinates 2-D, three velocity component, particle-in-cell (2D3V-PIC) code, the temporal dynamic flow and expansion behaviors of deuterium–titanium plasmas from the ion source to expansion cup are numerically studied in this article. The simulation results demonstrate as follows. There is a notable plasma sheath near the wall of the ion source with the electric field intensity on the order of MV/m in the radial direction; this plasma sheath will accelerate ions and decelerate electrons. Compared with Ti <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2+</sup> ion, the radial dispersion phenomenon of D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> ion is more obvious. The electric field intensity of plasma sheath is first increased and then decreased, which consistently influences the radial dispersion behavior of ions in the expansion cup. Due to the plasma sheath near the wall of the ion source, the ion radial-accelerating effect always exists. Due to the separation between ion and electron, there is also an ion axial-accelerating phenomenon. Moreover, the ion axial-accelerating effect will decrease and, finally, disappear due to the decrease in the separation effect between electron and ion. In the steady phase, the ion current on the extraction wall is higher than the loss of ion current on the wall of the ion source, and the ion extraction efficiency of Ti <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2+</sup> is much higher than D <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">+</sup> .

Comparisons of Paraboloidal Shells and Sinusoidal-Shaped Shells in Natural Frequencies

Natural frequencies and mode shapes are obtained for a sinusoidal-shaped shell of revolution by using the Ritz method from a three-dimensional (3-D) analysis instead of a mathematically two-dimensional (2-D) thin shell theory or high order thick shell theory. The present analysis uses circular cylindrical coordinates instead of 3-D shell coordinates, which have been used in traditional shell analyses. Convergence studies can analyze the first five frequencies to four-digit exactitude. Results are given for a variety of shallow and deep sinusoidal-shaped shells with different boundary conditions. The sinusoidal-shaped shells are very similar to paraboloidal shells in shape. The frequencies of the sinusoidal-shaped shells from the present 3-D method are compared with those from 2-D thin shell theories for paraboloidal shells. The present 3-D method is applicable to very thick as well as thin shells.

Comparisons of Paraboloidal Shells and Sinusoidal-Shaped Shells in Natural Frequencies

Natural frequencies and mode shapes are obtained for a sinusoidal-shaped shell of revolution by using the Ritz method from a three-dimensional (3-D) analysis instead of a mathematically two-dimensional (2-D) thin shell theory or high order thick shell theory. The present analysis uses circular cylindrical coordinates instead of 3-D shell coordinates, which have been used in traditional shell analyses. Convergence studies can analyze the first five frequencies to four-digit exactitude. Results are given for a variety of shallow and deep sinusoidal-shaped shells with different boundary conditions. The sinusoidal-shaped shells are very similar to paraboloidal shells in shape. The frequencies of the sinusoidal-shaped shells from the present 3-D method are compared with those from 2-D thin shell theories for paraboloidal shells. The present 3-D method is applicable to very thick as well as thin shells.

Vibration of Hemispherical-Cylindrical-Hemispherical Shells and Complete Hollow Spherical Shells with Variable Thickness

The natural frequencies and mode shapes of enclosed shell typed structures with variable thickness (hemispherical-cylindrical-hemispherical shells and complete hollow spherical shells) are determined by the Ritz method using a three-dimensional (3D) analysis. However, in the conventional shell analysis, mathematically two-dimensional (2D) thin shell theories or higher order thick shell theories are often employed, which adopt limiting assumptions about the displacement variation through the shell thickness. While most researchers have adopted the 3D shell coordinates that are normal and tangential to the shell mid-surface, the present analysis is based upon the circular cylindrical coordinates. By the Ritz method, the Legendre polynomials, which are mathematically orthonormal and minimal, are used as the admissible functions, instead of the ordinary algebraic polynomials. The strain and kinetic energies of the combined shell structures are formulated, and upper bound solutions of the frequencies are obtained by minimizing the solution for frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. The frequencies from the present 3D method are compared with those from other 3D approach and 2D thin and thick shell theories existing in the literature. The present 3D analysis is applicable to both very thick shells and very thin shells.

3D Vibration Analysis of Combined Shells of Revolution

A three-dimensional (3D) method of analysis is presented for determining the natural frequencies and the mode shapes of combined hemispherical–cylindrical shells of revolution with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2D) conventional thin shell theories or higher-order thick shell theories, the present method is based upon the 3D dynamic equations of elasticity. Mathematically, minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangent to the shell mid-surface. Strain and kinetic energies of the combined shell of revolution with and without a top opening are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for the combined shells of revolution with or without a top opening, which are completely free and fixed at the bottom of the combined shells. The frequencies from the present 3D Ritz method are compared with those from 2D thin shell theories by previous researchers. The present analysis is applicable to very thick shells as well as very thin shells.

Three-dimensional localized chirped Airy-circular wave packets in free space.

By solving the (3+1)-dimensional free-space Schrödinger equation in circular cylindrical coordinates, we have systematically analyzed the propagation of chirped Airy-circular (CAiCi) wave packets. The complex amplitude of the CAiCi wave packets is constructed by the Airy function, the Gaussian function, and the confluent hypergeometric function. We find that the CAiCi wave packets are some coaxial ring pulses stacked along the temporal domain in the initial position, which are modulated by the chirped factor, the initial velocity, the distribution factor, and the propagation distance. Meanwhile, the wave packets will appear to undergo intensity attenuation, diffusion, convergence, and so on. We can also modulate the shape of the wave packets and change their optical properties by altering the mode numbers. Furthermore, we analyze the evolution properties of the wave packets in detail from the aspects of the gradient force, the scattering force, phase, the Poynting vector, and the angular momentum, and find some interesting phenomena.

Vibrations of a Torus with Hollow Elliptical Cross Section Having Variable Thickness

The natural frequencies of toroidal shells of revolution with a hollow elliptical cross section and variable thickness are determined by the Ritz method from a three-dimensional theory, whereas traditional shell theories are mathematically two-dimensional. Instead of ordinary algebraic polynomials, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions. The present analysis is based upon the circular cylindrical coordinates, whereas toroidal coordinates have been used in general. The potential and kinetic energies of the torus are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. Convergence to a four-digit exactitude is demonstrated for the first five frequencies of the torus. Comparisons are made between the frequencies from the present three-dimensional method, a two-dimensional thin-shell theory, and thin- and thick-ring theories. The present method is applicable to very thick shells as well as thin shells.

Vibrations of a composite shell of hemiellisoidal-cylindrical shell having variable thickness with and without a top opening

Natural frequencies and mode shapes of a composite shell of revolution consisting of a circular cylindrical shell and a deep or shallow hemi-ellipsoidal shell with variable thickness having a circular cylindrical hole or not are determined by the Ritz method using a mathematically three-dimensional analysis instead of two-dimensional thin shell theories or higher order thick shell theories. The present analysis is based upon the circular cylindrical coordinates while in the traditional shell analyses three-dimensional shell coordinates have been commonly used. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. Natural frequencies are presented for different boundary conditions. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the composite shells. The frequencies from the present three-dimensional method are compared with those from three types of two-dimensional thin shell theories (finite element method, finite difference method, and numerical integration) by previous researchers. The present analysis is applicable to very thick shells as well as thin shells; and to shallow shells as well as deep shells.

Free vibration analysis of shallow and deep ellipsoidal shells having variable thickness with and without a top opening

A three-dimensional (3-D) method of analysis is presented for determining the natural frequencies and the mode shapes of hemi-ellipsoidal domes having non-uniform thickness with and without a top opening by the Ritz method. Instead of mathematically two-dimensional (2-D) conventional thin shell theories or higher-order shell theories, the present method is based upon the 3-D dynamic equations of elasticity by the Ritz method. Mathematically minimal or orthonormal Legendre polynomials are used as admissible functions in place of ordinary simple algebraic polynomials which are usually applied in the Ritz method. The analysis is based upon the circular cylindrical coordinates instead of the shell coordinates which are normal and tangential to the shell mid-surface. Potential (strain) and kinetic energies of the hemi-ellipsoidal dome having variable thickness with and without a top opening are formulated, and the Ritz method is used to solve the eigenvalue problem, thus yielding upper bound values of the frequencies by minimizing the frequencies. As the degree of the Legendre polynomials is increased, the frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. Numerical results are presented for a variety of shallow and deep hemi-ellipsoidal domes having variable thickness of five values of aspect ratios with and without a top opening, which are completely free and fixed at the bottom. The frequencies from the present 3-D analysis are compared with those from other 3-D analysis and a 2-D thin shell theory.

Three-Dimensional Vibration Analysis of a Hermetic Capsule with Variable Thickness

The natural frequencies of a hermetic capsule of a circular cylinder closed with hemispheroidal caps at both ends and having variable thickness are determined by the Ritz method using a three-dimensional analysis. However, in the traditional shell analysis, mathematically two-dimensional thin-shell theories or higher-order thick-shell theories, which make very limiting assumptions about the displacement variation through the shell thickness, have usually applied. Although most researchers have used three-dimensional shell coordinates that are normal and tangent to the shell midsurface, the present analysis is based upon the circular cylindrical coordinates. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. The potential and kinetic energies of the hermetic capsule are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. As the degree of the Legendre polynomials is increased, frequencies converge to the exact values. Convergence to four-digit exactitude is demonstrated for the first five frequencies. The frequencies from the present three-dimensional method are compared with those from other three-dimensional approaches and two-dimensional shell theories by previous researchers. The present three-dimensional analysis is applicable to very thick shells as well as thin shells.

Odd version Mathieu-Gaussian beam based on Green function

Like the theoretical pattern of non-diffracting Bessel beams, ideal non-diffracting Mathieu beams also carry infinite energy, but cannot be generated as a physically realizable entity. Mathieu-Gaussian beams can be experimentally generated by modulating ideal Mathieu beams with a Gaussian function, and thus they are a kind of pseudo-non-diffracting beams with finite energy and finite transverse extent. The research of Mathieu-Gaussian beam propagating characteristics in free space is of great significance. In order to analytically study the propagation of Mathieu-Gaussian beams, the Mathieu function is expanded into the superposition of a series of Bessel functions in polar coordinates based on the superposition principle of light waves. It means that the Mathieu-Gaussian beam can be converted into accumulation of the infinite terms of the Bessel beams with different orders. According to the properties of the Bessel function, the free-space propagation properties of Mathieu-Gaussian beams can be studied in the circular cylindrical coordinates. Thus, a group of virtual optical sources are introduced to generate the odd Mathieu-Gaussian beams of the first kind, i.e., (2n+2)th-order, which is a family of Mathieu-Gaussian beams. Using the virtual source technique and the Green function, we derive the rigorous integral formula for the odd Mathieu-Gaussian beams of the first kind. Taking for example the first three orders with non-paraxial corrections, the analytical solution of the on-axis field of odd Mathieu-Gaussian beams of the first kind is further obtained from the integral formula. The axial intensity distribution of the odd Mathieu-Gaussian beams of the first kind is numerically calculated by the integral formula. The simulation results show that the calculation results obtained with the paraxial theory and the rigorous integral expressions of non-paraxial Mathieu-Gaussian beams are obviously different when the propagation distance of the odd Mathieu-Gaussian beams of the first kind is small. The calculation results of the two methods are coming closer and closer with the increasing propagation distance. The results indicate that the correct results can be obtained with the paraxial theory when we study the propagation of Mathieu-Gaussian beams in the far-field, but the non-paraxial theory must be used to obtain correct results when we study the propagation of Mathieu-Gaussian beams in the near-field. Owing to the complexity of the non-paraxial theory, it is difficult to obtain the exact analytic solutions of Mathieu-Gaussian beams in the near-field with the classical diffraction theory. Based on the superposition principle of light waves, by introducing the virtual source technique and the Green function, the complex Mathieu-Gaussian function can be expanded into the superposition of a series of simple Bessel functions, and the axial intensity distributions of Mathieu-Gaussian beams in the far-field and the near-field can be studied well. It will also provide a feasible method to study other complex beams propagating in free space.

Vibration analysis of a circular cylinder closed with a hemi-spheroidal cap having a hole

The natural frequencies and mode shapes of a circular cylindrical shell closed with a hemi-spheroidal dome are determined by the Ritz method using a three-dimensional (3-D) analysis instead of two-dimensional (2-D) thin shell theories or higher-order thick shell theories. The present analysis is based upon the circular cylindrical coordinates, while in the traditional shell analyses 3-D shell coordinates have been usually used. Using the Ritz method, the Legendre polynomials, which are mathematically orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. Natural frequencies are presented for different boundary conditions. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the combined shell. The frequencies from the present 3-D method are compared with those from 2-D thin shell theories. The present method is applicable to very thick shells as well as thin shells.

Vibrations of a cylindrical shell closed with a hemi-spheroidal dome from a three-dimensional analysis

The natural frequencies of a circular cylindrical shell closed with a hemi-spheroidal dome are determined by the Ritz method using a three-dimensional (3-D) analysis instead of two-dimensional (2-D) thin shell theories or higher-order thick shell theories. The present analysis is based upon the circular cylindrical coordinates,while in the traditional shell analyses 3-D shell coordinates have been usually used. Using the Ritz method, the Legendre polynomials, which are mathematically minimal or orthonormal, are used as admissible functions instead of ordinary simple algebraic polynomials. Natural frequencies are presented for different boundary conditions. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the combined shell. The frequencies from the present 3-D method are compared with those from 2-D thin shell theories. The present method is applicable to very thick shells as well as thin shells.

Vibrations of tori with hollow elliptical cross-section from a three-dimensional theory

Natural frequencies of a toroidal shells of revolution with hollow elliptical cross-section are determined by the Ritz method from a three-dimensional (3-D) theory while traditional shell theories are mathematically two-dimensional (2-D). The Legendre polynomials, which are mathematically orthonomal, are used instead of ordinary algebraic polynomials as admissible functions. The present analysis is based upon the circular cylindrical coordinates while the toroidal coordinates have been used in general. Potential and kinetic energies of the torus are formulated, and upper bound values of the frequencies are obtained by minimizing the frequencies. Convergence to four-digit exactitude is demonstrated for the first five frequencies of the torus. Comparisons are made between the frequencies from the present 3-D method, a 2-D thin shell theory, and thin and thick ring theories. The present method is applicable to very thick toroidal shells as well as thin ones.

J1050502 非コンパクトグリーン関数に関するモデリング

This paper describes verification and examination in applying Mathieu functions for constructing Non-compact Green functions for wing-like body. First the paper reviews composition of plane wave and scattering wave by Bessel functions for cylindrical body. Key idea is introduction of Bessel functions and Hankel functions in the composition. Similarity of Mathieu functions in elliptic cylindrical coordinates with Bessel functions in circular cylindrical coordinates is utilized for constructing the Non-compact Green's function for wing-like bodies. Adequacy of Mathieu Functions are examined in their behavior in neighborhood of body surface. Lastly the paper discusses application of the new Non-compact Green's functions to designing low-noise wing with porous surface.

Investigation of Characteristics of the Reflection of Stationary and Pulse Sound from Visco-Elastic Cylindrical Layers

The scattering task of stationary and impulse sound signals by the viscous-elastic and elastic cylindrical layers is studied with the help of Fourier series and fundamental solutions of the Helmholtz equation in circular cylindrical coordinates system. The reflection’s characteristics of sound (harmonic and impulse) are calculated in the large range of angles (including inverse direction).