Elliptical Coordinate System Research Articles

UNSTEADY HEAT TRANSFER INSIDE AN ELLIPTIC CYLINDER

Considering the importance of transfer phenomena in chemical engineering and environmental protection, in particular heat transfer modelling, numerical methods are applied to investigate the transient heat transfer inside an elliptic cylinder. The temperature of the surrounding medium was considered constant and equal to the temperature on the surface of the cylinder. The heat balance equation was solved numerically in elliptic coordinate system. The computations were focused on the influence of the axis ratio on the heat transfer.

Unsteady heat transfer from an elliptic cylinder

Numerical methods are used to investigate the transient heat transfer from an elliptic cylinder to a steady stream of viscous, incompressible fluid. The temperature of the cylinder is considered spatially uniform but not constant in time. The momentum and heat balance equations were solved numerically in elliptic coordinate system. The solutions span the parameter ranges 5 ⩽ Re ⩽ 40, 1 ⩽ Pr ⩽ 100 and axis ratio ε, 0.1 ⩽ ε ⩽ 0.75. The computations were focused on the influence of the axis ratio and volume heat capacity ratio on the heat transfer rate.

Propagation Characteristics of an Elliptically-cored Highly Birefringent Fiber using an Analytical Method

By using the method of separation of variables in the elliptical coordinate system, a representative formula in terms of Mathieu and modified Mathieu functions is derived and longitudinal field components for the even and odd modes are presented for the proposed fiber. The eigen value equation for the fiber to be considered is obtained by solving the fourth order determinantal equation under weak guidance condition and is given in this paper. The cutoff frequency of several lower orders modes for different eccentricities have been obtained and their propagation characteristics have been plotted. The effect of the eccentricity on the modal cutoff values and mode transmissions is discussed. Finally, the modal birefringence in the proposed fiber is also estimated.

Analytical study of stress distributions due to semi-elliptic notches in shafts under torsion loading

Closed form solutions for the stress fields created by a semi-elliptic circumferential notch in an axisymmetric shaft under torsional loading are developed. The boundary value problem has been formulated by an approach using complex potential functions and the natural elliptic coordinate system. The solutions obtained for the shear stresses have a wide range of applicability, both in terms of the size and shape of the notches and the diameter of the shafts.

TOWARDS THE CHARACTERISTIC DISPERSION RELATION FOR STEP-INDEX HYPERBOLIC WAVEGUIDE WITH CONDUCTING HELICAL WINDING

Astudy is presented of the light wave propagation in a new type of dielectric optical waveguide with hyperbolic kind of cross- section. Further, the waveguide is assumed to have a conducting helical winding. The analysis essentially requires the use of elliptical coordinate system, which finally results into Mathieu and modified Mathieu functions as the representatives of the electromagnetic fields within the lightguide. Field components in the different sections of the guide are deduced, and the characteristic dispersion equation for the system is derived. The preliminary investigation on such type of waveguide throws the idea that the presence of helix pitch angle (which serves the purpose of additional controlling parameter for the guide) in the dispersion relation would greatly affect the propagation characteristics of the guide, and this can be of great practical importance.

Numerical studies of time-independent and time-dependent scattering by several elliptical cylinders

A numerical solution to the problem of time-dependent scattering by an array of elliptical cylinders with parallel axes is presented. The solution is an exact one, based on the separation-of-variables technique in the elliptical coordinate system, the addition theorem for Mathieu functions, and numerical integration. Time-independent solutions are described by a system of linear equations of infinite order which are truncated for numerical computations. Time-dependent solutions are obtained by numerical integration involving a large number of these solutions. First results of a software package generating these solutions are presented: wave propagation around three impenetrable elliptical scatterers. As far as we know, this method described has never been used for time-dependent multiple scattering.

Some aspects of theoretical analysis of the light propagation in an elliptical optical fiber

The paper presents the results of theoretical analysis of the propagation of the electromagnetic field in optical fiber with an elliptic core. Due to the elliptic symmetry of the core the analysis had to be carried out in an elliptic coordinate system. This analysis has shown that the functions describing the z-component of the electric field satisfy the Mathieu equations. The presented results of the theoretical analysis are original and have not published so for.

Analytical Solutions for Pore-Fluid Flow Focusing Within Inclined Elliptic Inclusions in Pore-Fluid-Saturated Porous Rocks: Solutions Derived in an Elliptical Coordinate System

Exact analytical solutions have been derived rigorously for the pore-fluid velocity, pore-fluid-flow focusing factor, stream function and excess pore-fluid pressure around and within a buried inclined elliptic inclusion in pore-fluid-saturated porous rocks. The geometric characteristics of the buried inclined elliptic inclusion are represented by the aspect ratio and dip angle of the inclusion, while the hydrodynamic characteristic is represented by the permeability ratio of the elliptic inclusion to its surrounding rock. Since an elliptic inclusion of any aspect ratio can be used to approximately represent geological faults and cracks, the present analytical solutions can be used to investigate the pore-fluid-flow patterns around buried faults and cracks within the crust of the Earth. Therefore, the present analytical solution not only provides a better understanding of the physics behind the pore-fluid-flow focusing problem around and within buried faults and cracks, but also provides a valuable benchmark solution for validating any numerical method in dealing with this kind of pore-fluid-flow focusing problem. The pore-fluid-flow focusing factor of a buried elliptic inclusion is demonstrated to be dependent on the aspect ratio, the permeability ratio and the dip angle.

Mixed-mode I/II fields around a crack with a cohesive zone ahead of the crack tip

Considering a cohesive zone ahead of the crack tip, the linear elastic crack problem under Mode I, Mode II or mixed-mode conditions is formulated in an elliptic coordinate system, so that the cohesive surfaces are conveniently represented by straight line segments. It is shown that the displacement and stress fields around the crack tip and the cohesive zone, expressed in terms of elliptic coordinates, have a simple mathematical form, which does not contain a stress singularity at the crack tip due to the existence of the cohesive zone.

Theoretical and numerical analyses of pore-fluid flow patterns around and within inclined large cracks and faults

Large cracks and faults play an important and diverse role in controlling pore-fluid flow patterns in pore-fluid-saturated porous rocks. Examples of fault-related fluid flow include (1) geological structural controls on groundwater flow and contaminant transport, (2) formation and localization of some valuable mineral deposits around and within fault zones, (3) impacts of heterogeneous fault zone hydraulic properties on the formation and location of petroleum reservoirs through fault sealing, compartmentalization and variability in pore-fluid flow pathways and (4) induced rupture and failure processes by the interaction between material deformation and pore-fluid flow around faults through cycles of brittle deformation and seismicity. In this paper, theoretical and numerical methods are used to investigate pore-fluid flow patterns around and within large cracks and faults in pore-fluid-saturated porous rocks. Although considerable numerical investigations have been carried out on the flow pattern around large cracks and faults, a full set of exact analytical solutions in the case of a fault having any finite but non-zero permeability is not yet available due to mathematical difficulties in solving the related partial differential equations in a conventional Cartesian coordinate system. For the purpose of deriving a full set of exact analytical solutions, large cracks and faults are treated as elliptic inclusions, in which both the aspect ratio of an elliptic crack and the permeability ratio of the crack to its surrounding rock are used to represent the hydrodynamic property contrast of the elliptic crack. It is assumed that the pore-fluid flow in the far field away from the elliptic crack is uniform and that the long axis of the elliptic crack is inclined to the inflow direction in the undisturbed far field. Under these assumptions, a full set of analytical solutions has been derived for the pore-fluid velocity, stream function and excess pore-fluid pressure around and within an inclined elliptic crack in pore-fluid-saturated porous rocks. To overcome mathematical difficulties in describing and satisfying interface conditions between the elliptic crack and its surrounding rocks, analytical solutions are firstly derived in an elliptical coordinate system, and then explicitly expressed, through rigorous inverse mappings, in the conventional Cartesian coordinate system using elementary functions. This allows the present analytical solutions not only to be used to produce theoretical understanding of the pore-fluid flow pattern around and within a large crack, but also to be used as valuable benchmark solutions for validating any numerical methods. After a finite element computational model is validated by comparing the numerical solutions with the present analytical ones, it is used to investigate pore-fluid flow patterns around and within two parallel inclined cracks in pore-fluid-saturated porous rocks. Some conclusions in relation to the effects of inclined large cracks on pore-fluid flow patterns are made through both the theoretical and the numerical analyses.

Heat transfer in laminar flow of non-Newtonian fluids in ducts of elliptical section

Laminar forced convection inside tubes of various cross-section shapes is of interest in the design of a low Reynolds number heat exchanger apparatus. Heat transfer to thermally developing, hydrodynamically developed forced convection inside tubes of simple geometries such as a circular tube, parallel plate, or annular duct has been well studied in the literature and documented in various books, but for elliptical duct there are not much work done. The main assumption used in this work is a laminar flow of a power flow inside elliptical tube, under a boundary condition of first kind with constant physical properties and negligible axial heat diffusion (high Peclet number). To solve the thermally developing problem, we use the generalized integral transform technique (GITT), also known as Sturm–Liouville transform. Actually, such an integral transform is a generalization of the finite Fourier transform where the sine and cosine functions are replaced by more general sets of orthogonal functions. The axes are algebraically transformed from the Cartesian coordinate system to the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall. The GITT is then applied to transform and solve the problem and to obtain the once unknown temperature field. Afterward, it is possible to compute and present the quantities of practical interest, such as the bulk fluid temperature, the local Nusselt number and the average Nusselt number for various cross-section aspect ratios.

Finite-difference method for incompressible Navier–Stokes equations in arbitrary orthogonal curvilinear coordinates

A finite-difference method for solving three-dimensional time-dependent incompressible Navier–Stokes equations in arbitrary curvilinear orthogonal coordinates is presented. The method is oriented on turbulent flow simulations and consists of a second-order central difference approximation in space and a third-order semi-implicit Runge–Kutta scheme for time advancement. Spatial discretization retains some important properties of the Navier–Stokes equations, including energy conservation by the nonlinear and pressure-gradient terms. Numerical tests cover Cartesian, cylindrical-polar, spherical, cylindrical elliptic and cylindrical bipolar coordinate systems. Both laminar and turbulent flows are considered demonstrating reasonable accuracy and stability of the method.

Generation of an elliptic Bessel beam

In an elliptic coordinate system the solution for a diffraction-free beam is given by a Mathieu beam, which is mathematically complicated and therefore takes considerable computational time and a large memory space. A technique is reported for the generation of elliptic hollow beams, which are actually scaled versions of higher-order Bessel beams. The analysis is based on the fact that stretching coordinates in the space domain results in a contraction of the coordinates in the frequency domain, along with a change in the overall amplitude of the spectrum. The beam, produced holographically, is structurally stable and retains its shape up to approximately 1 m.

Virtual boundary method for the analysis of elliptically cross-sectional photonic-crystal fibers with elliptical pores

This paper introduces a new semianalytical method for the analysis of propagation characteristics of elliptically cross-sectional photonic-crystal fibers (PCFs) with elliptical pores. This method known as a virtual boundary method (VBM) is based on the equivalency between an actual PCF and a three-layered, transversely inhomogeneous waveguide. The complicated refractive-index profile of the PCF is written as a double Fourier series, and an approximate separable wave equation is found in an elliptical coordinate system for the longitudinal field components. The exact solution to the derived equation is expressed in terms of higher order transcendental functions, such as regular and irregular Coulomb-wave functions and Mathieu functions. After having expressed all the field components, boundary conditions are imposed on the boundaries, and then, a transcendental equation for the propagation constant is derived, which is solved numerically. The validity of the method is ensured by comparing various quantities, such as effective indexes, modal birefringences, and electromagnetic field distributions, with those from an accurate full-vector finite-element method (FEM) simulator, showing relatively good agreement between the results. The method correctly confirms some of the unique PCFs' properties, such as strong localization of light within the fiber and enhancement of modal birefringence as a function of the topology of hole arrangement.

Dynamic stress concentration around elliptic cavities in saturated poroelastic soil under harmonic plane waves

Potential function and complex function in the elliptic coordinate system are employed to solve the problem of scattering harmonic plane waves by multiple elliptic cavities in water saturated soil medium. The steady state Biot’s dynamic equations of poroelasticity are uncoupled into Helmholtz equations via given potentials. The stresses and pore water pressures are obtained by using complex functions in elliptic coordinates with certain boundary conditions. Finally, the dynamic stresses for the case of two interacting elliptic cavities are obtained and discussed in details via a numerical example.

Propagation Characteristics of Confocal Elliptical Coaxial Lines Filled with Multilayered Media

By using the method of separation of variables in the elliptical coordinate system, a recursive formula for the electromagnetic fields in a confocal elliptical coaxial line filled with multilayered homogeneous isotropic mediums is derived; and the eigenequation is given in this paper. When the confocal elliptical coaxial line becomes circular coaxial line, the fields and the eigenequation for the circular coaxial line can be carried out from the eigenequation of the confocal elliptical coaxial line using the asymptotic formulae of Mathieu and the modified Mathieu functions with a large radial variable in elliptical system. The results show that the eigenequation of a circular coaxial line filled with multilayered media can be treated as a special case of a confocal elliptical coaxial line. In addition, some numerical examples will reveal the influence of the permittivity and the permeability of the media in the confocal elliptical coaxial line on the propagation characteristics.

Thermally Developing Forced Convection of Non-Newtonian Fluids Inside Elliptical Ducts

Laminar-forced convection inside tubes of various cross-section shapes is of interest in the design of a low Reynolds number heat exchanger apparatus. Heat transfer to thermally developing, hydrodynamically developed forced convection inside tubes of simple geometries such as a circular tube, parallel plate, or annular duct has been well studied in the literature and documented in various books, but for elliptical duct there are not much work done. The main assumptions used in this work are a non-Newtonian fluid, laminar flow, constant physical properties, and negligible axial heat diffusion (high Peclet number). Most of the previous research in elliptical ducts deal mainly with aspects of fully developed laminar flow forced convection, such as velocity profile, maximum velocity, pressure drop, and heat transfer quantities. In this work, we examine heat transfer in a hydrodynamically developed, thermally developing laminar forced convection flow of fluid inside an elliptical tube under a second kind of a boundary condition. To solve the thermally developing problem, we use the generalized integral transform technique (GITT), also known as Sturm-Liouville transform. Actually, such an integral transform is a generalization of the finite Fourier transform, where the sine and cosine functions are replaced by more general sets of orthogonal functions. The axes are algebraically transformed from the Cartesian coordinate system to the elliptical coordinate system in order to avoid the irregular shape of the elliptical duct wall. The GITT is then applied to transform and solve the problem and to obtain the once unknown temperature field. Afterward, it is possible to compute and present the quantities of practical interest, such as the bulk fluid temperature, the local Nusselt number, and the average Nusselt number for various cross-section aspect ratios.

System of Equations of the Dynamic Problem of Thermoelasticity in Stresses for an Elliptic Cylinder

On the basis of the equations of motion, Cauchy formulas, generalized Hooke’s law, and compatibility conditions for the Saint-Venant strains, a system of determining equations of the dynamic problem of thermoelasticity in stresses is deduced for a homogeneous isotropic cylinder in an elliptic cylindrical coordinate system. This system is reduced to a system of consecutively correlated wave equations in which the equation for the first invariant of the stress tensor is independent. The initial conditions for the resolving functions are presented.

Scattering by Two Infinitely Long Dielectric-Coated Confocal Conducting Elliptic Cylinders

Analytical solution of the problem of a plane electromagnetic wave scattering by two infinitely long dielectric coated confocal conducting elliptic cylinders is derived. The incident, scattered and transmitted fields in every region are expressed in terms of Mathieu functions. To impose the boundary conditions, the translation addition theorem for Mathieu functions is implemented to express the scattered field by one cylinder in terms of the elliptic coordinate system of the other cylinder. The resulting system of equations is written in matrix form while the scattered and transmitted field coefficients are obtained by using matrix inversion. Numerical results are obtained for the scattered field in the far zone for different axial ratios, electrical separations, relative dielectric constants, and angles of incidence.

ITERATIVE SOLUTION TO THE TM SCATTERING BY TWO INFINITELY LONG LOSSY DIELECTRIC ELLIPTIC CYLINDERS

An analytic solution to the problem of a plane electromagnetic wave scattering by two infinitely long lossy dielectric elliptic cylinders is presented using an iterative procedure to account for the multiple scattered field between the cylinders. To compute the higher order terms of the scattered fields, the translation addition theorem for Mathieu functions is implemented to express the field scattered by one lossy dielectric cylinder in terms of the elliptic coordinate system of the other cylinder in order to impose the boundary conditions. Scattered field coefficients of various orders are obtained and written in matrix form. Numerical results are obtained for the scattered field in the far zone for different axial ratios, electrical separations, relative dielectric constants, and angles of incidence.