Showing papers on "Elliptic coordinate system published in 1977"
TL;DR: In this article, a method for automatic generation of boundary-fitted curvilinear coordinate systems, where the transformed coordinates are solutions of an elliptic differential system in the physical plane and where the coordinate lines are coincident with all boundaries of a general multiply-connected, two-dimensional region containing any number of arbitrarily shaped bodies, is described along with a suitable computer code for implementing the method.
246 citations
TL;DR: A detailed classification of orthogonal coordinate systems for which the wave equation ψtt−Δ3ψ=0 admits an R−separable solution is given in this paper.
Abstract: A detailed classification is made of orthogonal coordinate systems for which the wave equation ψtt−Δ3ψ=0 admits an R‐separable solution. Only those coordinate systems are given which are not conformally equivalent to coordinate systems that have been found in previous articles. We find 106 new coordinates to give a total of 367 conformally inequivalent orthogonal coordinates for which the wave equation admits an R‐separation of variables.
14 citations
01 Jun 1977
TL;DR: In this article, a generalized curvilinear orthogonal coordinate system is presented which can be used for approximating various axisymmetric and two-dimensional body shapes of interest to aerodynamicists.
Abstract: A generalized curvilinear orthogonal coordinate system is presented which can be used for approximating various axisymmetric and two-dimensional body shapes of interest to aerodynamicists. Such body shapes include spheres, ellipses, spherically capped cones, flat-faced cylinders with rounded corners, circular disks, and planetary probe vehicles. A set of transformation equations is also developed whereby a uniform velocity field approaching a body at any angle of attack can be resolved in the transformed coordinate system. The Navier-Stokes equations are written in terms of a generalized orthogonal coordinate system to show the resultant complexity of the governing equations.
3 citations
TL;DR: In this paper, exterior differential forms are used to represent the time dependent diffusion equation suitable for neutronic calculations, which allows one to express the equation straightforwardly and easily in any orthogonal coordinate system.
Abstract: This paper illustrates the applicability of exterior differential forms to reactor theory by using them to represent the time dependent diffusion equation suitable for neutronic calculations. This formalism allows one to express the equation straightforwardly and easily in any orthogonal coordinate system. Because of present interest in CTR designs an application is made to a toroidal geometry with a general elliptical cross section.
1 citations
TL;DR: In this article, a non-linear partial differential equation controlling the temperature distribution in a burning solid propellent in a rectangular, cylindrical or spherical coordinate system is transformed from a fixed coordinate system to a moving Lagrangian coordinate system, and then, using a similarity variable is further transformed to a nonlinear, second order, ordinary differential equation.
Abstract: The non-linear partial differential equation controlling the temperature distribution in a burning solid propellent in a rectangular, cylindrical or spherical coordinate system is transformed from a fixed coordinate system to a moving Lagrangian coordinate system, and then, using a similarity variable is further transformed to a non-linear, second order, ordinary differential equation. The burning wall temperature is time dependent, and the burning rate is an explicit function of the wall surface temperature. The boundary conditions for the resulting non-linear differential equation are given, and the necessary form of the burning rate is determined. Additionally, the linear partial differential equation for this burning solid propellent problem is also treated in all three coordinate systems. A non-linear example is given and solved, in all three coordinate systems, to illustrate the method.
TL;DR: In this article, the potential energy curves of the two-centre HeH2+, LiH3+ and HeLi4+ heteronuclear systems in the lowest lying π and δ states, that is in the 2pπ and 3dδ excited states, were determined with a simple analytical non-exponential wave function in elliptical coordinates using the variational method.
Abstract: The potential energy curves of the two-centre HeH2+, LiH3+ and HeLi4+ heteronuclear systems in the lowest lying π and δ states, that is in the 2pπ and 3dδ excited states are determined with a simple analytical non-exponential wave function in elliptical coordinates using the variational method. All considered states are repulsive. The energy values calculated in the appropriate approximation agree well with the exact ones, where they are available in the considered range of the internuclear distance (0.5a0 ≤R ≤ 6a0).