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Riemannian Acceleration in oblate spheroidal Coordinate System

Journal of Applied Mathematics and Physics 2016年第2期4卷 279-285页

作者： N. E. J. Omaghali S. X. K. Howusu Department of Physics University of Jos Jos Nigeria Theoretical Physics Program National Mathematical Centre Abuja Nigeria

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.

关键词： Riemannian Acceleration Golden Metric Tensor oblate spheroidal coordinates Christoffel’s Symbols

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On the Axisymmetric Steady Incompressible Beltrami Flows

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Open Journal of Fluid Dynamics 2020年第3期10卷 208-238页

作者： Pavel Bělík Xueqing Su Douglas P. Dokken Kurt Scholz Mikhail M. Shvartsman Department of Mathematics Statistics and Computer Science Augsburg University Minneapolis MN USA Department of Statistics University of Michigan Ann Arbor MI USA Department of Mathematics University of St. Thomas Saint Paul MN USA

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 rz-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.

关键词： Axisymmetric Beltrami Flow Trkalian Flow Bragg-Hawthorne Equation Cylindrical coordinates Spherical coordinates Paraboloidal coordinates Prolate spheroidal coordinates oblate spheroidal coordinates Vorticity

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