Showing papers on "Elliptic coordinate system published in 2016"

Free convection enhancement in an annulus between horizontal confocal elliptical cylinders using hybrid nanofluids

Abstract: In the present paper, natural convection in an annulus between two confocal elliptic cylinders filled with a Cu-Al2O3/water hybrid nanofluid is investigated numerically. The inner cylinder is heated at a constant surface temperature while the outer wall is isothermally cooled. The basic equations are formulated in elliptic coordinates and developed in terms of the vorticity-stream function formulation using the dimensionless form for 2D, laminar and incompressible flow under steady-state condition. The governing equations are discretized using the finite volume method and solved by an in-house FORTRAN code. Numerical simulations are performed for various volume fractions of nanoparticles (0 ≤ ϕ ≤ 0.12) and Rayleigh numbers (103 ≤ Ra ≤ 3 × 105). The eccentricity of the inner and outer ellipses and the angle of orientation are fixed at e1 = 0.9, e2 = 0.6 and γ = 0° respectively. It is found that employing a Cu-Al2O3/water hybrid nanofluid is more efficient in heat transfer rate compared to the simil...

Particle capture efficiency of elliptic cylinder matrices for high-gradient magnetic separation

Abstract: High-gradient magnetic separation (HGMS) is an effective method to recover fine weakly magnetic minerals or remove ferromagnetic and paramagnetic particles from aqueous solution. The most commonly used matrices are circular cylinders of high magnetic permeability. However, special cross-section matrices may present better magnetic characteristics and improve the separation efficiency. In this paper, the magnetic field and flow field characteristics of elliptic cross-section matrix were studied in the elliptic coordinate system. The particle capture of the elliptic matrix in the longitudinal configuration of HGMS was modelled and the motion equations of the magnetic particles were derived. The particle capture radius and efficiency were calculated and were compared with those of the circular matrix. Two circumstances were considered to investigate the particle capture of elliptic matrix: the short axis of the elliptic matrix is equal to the diameter of the circular matrix and the cross-section area...

Eigenfunction Expansions for the Stokes Flow Operators in the Inverted Oblate Coordinate System

Abstract: When studying axisymmetric particle fluid flows, a scalar function, , is usually employed, which is called a stream function. It serves as a velocity potential and it can be used for the derivation of significant hydrodynamic quantities. The governing equation is a fourth-order partial differential equation; namely, , where is the Stokes irrotational operator and is the Stokes bistream operator. As it is already known, in some axisymmetric coordinate systems, such as the cylindrical, spherical, and spheroidal ones, separates variables, while in the inverted prolate spheroidal coordinate system, this equation accepts -separable solutions, as it was shown recently by the authors. Notably, the kernel space of the operator does not decompose in a similar way, since it accepts separable solutions in cylindrical and spherical system of coordinates, while semiseparates variables in the spheroidal coordinate systems and it -semiseparates variables in the inverted prolate spheroidal coordinates. In addition to these results, we show in the present work that in the inverted oblate spheroidal coordinates, the equation also -separates variables and we derive the eigenfunctions of the Stokes operator in this particular coordinate system. Furthermore, we demonstrate that the equation -semiseparates variables. Since the generalized eigenfunctions of cannot be obtained in a closed form, we present a methodology through which we can derive the complete set of the generalized eigenfunctions of in the modified inverted oblate spheroidal coordinate system.

Polarizability of radially anisotropic elliptic inclusion

Abstract: This article discusses a two-dimensional electrostatic scattering problem where an elliptic inclusion is suspended in a homogeneous background and impinged by an electric field which is uniform and static. The novelty of the discussion stems from the inclusion's material parameters. The material of the inclusion is assumed to be axially anisotropic, so that the axis of anisotropy aligns itself with the radial unit vector of the elliptic coordinate system. Similar varieties of anisotropy have been formerly referred to as radial anisotropy, and the same term is employed herein. The radially anisotropic elliptic inclusions are studied with an analytic method. The validation is likewise analytic. The validation method compares the new results with the results for radially anisotropic circles and homogeneous two-dimensional needles. The elliptic inclusion is found to facilitate both cloaking and field concentration.

The geometrical description of electromagnetic radiation

Abstract: The paper describes the relationship between the solutions of Maxwell’s equations which can be considered at least locally as plane waves and the curvilinear coordinates of geometrical optics; it generalizes the results achieved by Luneburg, concerning the evolution of surfaces of electromagnetic fields discontinuities If vectors and are orthogonal to each other and their directions do not change with time t, but may vary from point to point in the domain G, then under some conditions there is an orthogonal coordinate system in which -lines represent rays of geometrical optics, -lines point out -direction and, -lines point out -direction This coordinate system will be called phase-ray coordinate system In the article, it will be proved that field under study can be represented by two scalar functions The article will also specify the necessary and sufficient conditions for the existence of a coordinate system, generated by the solution of Maxwell’s equations with the holonomic field of the Poynting ve

Boundary methods for mixed boundary problems of Laplace׳s equation in elliptic domains with elliptic holes☆

Abstract: The particular solutions (PS) and fundamental solutions (FS) in polar coordinates can be found in many textbooks, but with much less coverage in elliptic coordinates (Chen et al., 2010 [5] , Chen et al., 2012 [6] , Morse and Feshbach, 1953 [20] , Li et al., 2015 [18] ). Since the elliptic domains with elliptic holes may be found in some engineering problems, the PS and the FS expansions in elliptic coordinates are essential for numerical computations. For Dirichlet problems of Laplace׳s equation in elliptic domains, the null field method (NFM), the interior field method (IFM) and the collocation Trefftz method (CTM) are reported in [18] . There seems to exist few reports for mixed problems, where the Dirichlet and Neumann conditions are assigned on the exterior and the interior boundaries, simultaneously. This paper is devoted to such mixed problems by the NFM and the IFM, and the explicit algebraic equations are derived for elliptic domains. Besides, other effective particular solutions (PS) are sought, and the collocation Trefftz method (CTM) [16] is employed. The CTM may be used for Robin problems in elliptic domains. The effective algorithms for the mixed problems of Laplace׳s equation on elliptic domains are the main goal of this paper. The techniques of the mixed techniques in this paper can be applied to Dirichlet problems, the dual techniques are called in Chen and Hong (1999 [4] ), Hong and Chen (1988 [8] ), and Portela et al. (1992 [21] ). A preliminary study for the dual techniques is one goal of this paper.

Diffraction of a plane wave by a multiserial lattice located in a dielectric layer

Abstract: A modified zero field method is applied to solve the 2D problem of scattering by a multiserial lattice. The lattice consists of perfectly conducting elements and is situated in a homogeneous dielectric layer. The system of integral equations is derived and formulas for calculating the reflection and transmission coefficients are obtained for various geometries of the lattice elements. To increase the efficiency of the algorithm work, both polar and elliptic coordinates are applied to choose the auxiliary contour on which the zero field condition is stipulated.

A least-squares method for the inverse reflector problem in arbitrary orthogonal coordinate systems

Abstract: In this article we solve the inverse reflector problem for a light source emitting a parallel light bundle and a target in the far-field of the reflector by use of a least-squares method. We derive the Monge-Ampère equation, expressing conservation of energy, while assuming an arbitrary coordinate system. We generalize a Cartesian coordinate least-squares method presented earlier by C.R. Prins et al. [13] to arbitrary orthogonal coordinate systems. This generalized least-squares method provides us the freedom to choose a coordinate system suitable for the shape of the light source. This results in significantly increased numerical accuracy. Decrease of errors by factors up to 104 is reported. We present the generalized least-squares method and compare its numerical results with the Cartesian version for a disk-shaped light source.

Velocity and Acceleration in Rotational Oblate Spheroidal Coordinates

Abstract: we had established the velocity and Acceleration in some of orthogonal curvilinear coordinates such as oblate spheroidal, prolate spheroidal, parabolic cylindrical, parabolic and second torridal coordinates for application in Mechanics, in this paper we proceed to derive expression for the instantaneous velocity and acceleration in rotational oblate spheroidal coordinates

A Topography-Fitted Coordinate System

Abstract: This chapter is devoted to the presentation of the curvilinear coordinates that we use for the description of a shallow mass flow down arbitrary topography.

Riemannian Acceleration in Oblate Spheroidal Coordinate System

Abstract: The planetary bodies are more of a spheroid than they are a sphere thereby making it necessary to describe motions in a spheroidal coordinate system. Using the oblate spheroidal coordinate system, a more approximate and realistic description of motion in these bodies can be realized. In this paper, we derive the Riemannian acceleration for motion in oblate spheroidal coordinate system using the golden metric tensor in oblate spheroidal coordinates. The Riemannian acceleration in the oblate spheroidal coordinate system reduces to the pure Newtonian acceleration in the limit of c0 and contains post-Newtonian correction terms of all orders of c-2. The result obtained thereby opens the way for further studies and applications of the motion of particles in oblate spheroidal coordinate system.

The notes on thin shells

Abstract: Geometry of the spacetime with a spherical shell embedded in it is studied in two coordinate systems - in Kodama-Schwarzschild coordinates and in Gaussian normal coordinates. We consider transformations between the coordinate systems as in the 4D spacetime so as at the surface $§$ swept in the spacetime by the spherical shell. Extrinsic curvatures of the surface swept by the shell are calculated in both coordinate systems. Applications to the Israel junction conditions are discussed.

Porcupic concentrators and bulbic cloaks in planar configuration

Abstract: This contribution concerns anisotropic elliptic inclusions in a static and uniform excitation field. In particular, the discussion considers anisotropy which is axial in a way that is defined by the elliptic coordinate system. This type of axial anisotropy has been referred to as “radial anisotropy”, although other varieties of radial anisotropy do exist for other geometries. In case of an ellipse, there are at least two methods to create radial anisotropy, and the two methods lead to qualitatively different outcomes. The terms “porcupic” and “bulbic” have been used to refer to these two classes of radial anisotropy. The porcupic inclusions are found to exhibit field concentration and the bulbic ones are found to exhibit cloaking. The contribution discusses both types of inclusions first in isolation and then as layered structures.