In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows...In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg-Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the <em>rz</em>-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of tornado-like flows that develop as swirl increases. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.展开更多
A two-center correlated orbital approach was used to calculate the electronic ground state energy for the HeH+ molecular ion. The wavefunctions were constructed from the exact solution of the Schrdinger equation for...A two-center correlated orbital approach was used to calculate the electronic ground state energy for the HeH+ molecular ion. The wavefunctions were constructed from the exact solution of the Schrdinger equation for the HeH++ problem in prolate-spheroidal coordinates taken together with a Hylleraas type correlation factor. With a simple single term wavefunction,we obtained ground state energy of -2.95308691 hartree without any variational parameters in the calculation. When a two-configuration-state wavefunction was used and effective charges were allowed to be adjusted,we found an energy of -2.97384868 hartree,which is to be compared with -2.97869074 hartree obtained by an 83 term configuration interaction wavefunction or -2.97364338 hartree by an ab initio calculation (at the MP4(SDQ)/6-311++G(3df,3dp) level) using the well-known "canned" code.展开更多
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 a...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 c<sup>0</sup> and contains post-Newtonian correction terms of all orders of c<sup>-2</sup>. The result obtained thereby opens the way for further studies and applications of the motion of particles in oblate spheroidal coordinate system.展开更多
文摘In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg-Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the <em>rz</em>-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of tornado-like flows that develop as swirl increases. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.
基金Supported by the National Natural Science Foundation of China (Grant No. 20773618)the Research Fund of Chongqing University of Posts and Telecom-munications (Grant No. A2006-81)
文摘A two-center correlated orbital approach was used to calculate the electronic ground state energy for the HeH+ molecular ion. The wavefunctions were constructed from the exact solution of the Schrdinger equation for the HeH++ problem in prolate-spheroidal coordinates taken together with a Hylleraas type correlation factor. With a simple single term wavefunction,we obtained ground state energy of -2.95308691 hartree without any variational parameters in the calculation. When a two-configuration-state wavefunction was used and effective charges were allowed to be adjusted,we found an energy of -2.97384868 hartree,which is to be compared with -2.97869074 hartree obtained by an 83 term configuration interaction wavefunction or -2.97364338 hartree by an ab initio calculation (at the MP4(SDQ)/6-311++G(3df,3dp) level) using the well-known "canned" code.
文摘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 c<sup>0</sup> and contains post-Newtonian correction terms of all orders of c<sup>-2</sup>. The result obtained thereby opens the way for further studies and applications of the motion of particles in oblate spheroidal coordinate system.
