Elliptic coordinate system

About: Elliptic coordinate system is a research topic. Over the lifetime, 670 publications have been published within this topic receiving 11135 citations. The topic is also known as: elliptical coordinate system & elliptic coordinates.

Use of numerically generated body-fitted coordinate systems for solution of the Navier-Stokes equations

Abstract: A procedure for numerical solution of the time-dependent, two-dimensional incompressible Navier-Stokes equations that can treat the unsteady laminar flow about bodies of arbitrary shape, such as two-dimensional airfoils, multiple airfoils, and submerged hydrofoils, as naturally as it can deal with the flow about simple bodies. The solution is based on a method of automatic numerical generation of a general curvilinear coordinate system with coordinate lines coincident with all boundaries of a general multiconnected region containing any number of arbitrarily shaped bodies. The curvilinear coordinates are generated as the solution of two elliptical partial differential equations with Dirichlet boundary conditions, one coordinate being specified to be constant on each of the boundaries, and a distribution of the other being specified along the boundaries. The solution compares excellently with the Blasius boundary layer solution for the flow past a semiinfinite flat plate.

Potential energy surfaces and coordinate dependence

Abstract: For coordinate systems with a nontrivial metric tensor covariant derivatives must be used to obtain properties that are coordinate independent. Applications to instantaneous normal mode theory and bifurcation points are presented as illustrations.

Electronic Bases of Molecular Vibrations. I. General Theory for Diatomic Molecules

Abstract: The force constant of a diatomic molecule in a given electronic state is the second derivative with respect to the internuclear distance R of the molecular electronic energy, evaluated at the equilibrium distance Re. In principle it is determined by the electronic‐charge distribution ψ*ψ, where ψ is the appropriate solution of the electronic‐wave equation. Expressions for the force constant in terms of the electronic‐charge distribution are derived which differ depending upon (1) what coordinate representation is used to describe the wavefunction, (2) whether the virial theorem or the Hellmann—Feynman theorem is employed to define the first energy derivative. Correspondingly, if the wavefunction is expressed in confocal elliptic coordinates, two different expressions for the force constant are obtained. These expressions give identical results for exact wavefunctions or for approximate wavefunctions which are constructed to satisfy the virial theorem for all R. If this condition is not satisfied, the two expressions will give, in general, different results. It is shown that if the Hellmann—Feynman theorem is used to define the first energy derivative, then the force constant assumes a particularly simple form which is suggestive of the force constant of a classical harmonic oscillator. The utility of the force constant expressions as probes to explore molecular electronic‐charge distributions is discussed and related problems are suggested.

An internal coordinate invariant reaction pathway

Abstract: In this work we show that some properties of a potential energy surface are not independent of the choice of the coordinate frame. So the reaction pathway often described as steepest descent way does not correspond to an invariant curve under coordinate transformations. We propose an internal intrinsic reaction pathway by using some quasi-dynamical considerations (like instantaneous internal acceleration). Our work precises the intrinsic-reaction coordinates of Fukui to any set of 3N-6 internal parameters. Finally, from the equations of motion we deduce the form of the normal reaction coordinates frame anywhere along the postulated reaction pathway.