Navier Stokes Theorem in Hydrology

HR: 1340h
AN: H23F-1489 [Abstracts]
TI: Navier Stokes Theorem in Hydrology
AU: * NARAYANAN, M
EM: narayam@muohio.edu
AF: MIAMI UNIVERSITY, UNIVERSITY BOULEVARD, HAMILTON, OH 45011 United States
AB: In a paper presented at the 2004 AGU International Conference, the author outlined and stressed the importance of studying and teaching certain important mathematical techniques while developing a course in Hydrology and Fluid Mechanics. The Navier-Stokes equations are the foundation of fluid mechanics, and Stokes' theorem is used in nearly every branch of mechanics as well as electromagnetics. Stokes' Theorem also plays a vital role in many secondary theorems such as those pertaining to vorticity and circulation. Mathematically expressed, Stokes' theorem can be expressed by considering a surface S having a bounding curve C. Here, V is any sufficiently smooth vector field defined on the surface and its bounding curve C. In an article entitled "Corrections to Fluid Dynamics" R. F. Streater, (Open Systems and Information Dynamics, 10, 3-30, 2003.) proposes a kinetic model of a fluid in which five macroscopic fields, the mass, energy, and three components of momentum, are conserved. The dynamics is constructed using the methods of statistical dynamics, and results in a non-linear discrete-time Markov chain for random fields on a lattice. In the continuum limit he obtains a non-linear coupled parabolic system of field equations, showing a correction to the Navier-Stokes equations. In 2001, David Hoff published an article in JournＦs Ｒuations aux dＳivＦs partielles. (Art. No. 7, 9 p.). His paper is entitled : Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions. In his paper, David Hoff proves the global existence of solutions of the Navier-Stokes equations of compressible, barotropic flow in two space dimensions with piecewise smooth initial data. These solutions remain piecewise smooth for all time, retaining simple jump discontinuities in the density and in the divergence of the velocity across a smooth curve, which is convected with the flow. The strengths of these discontinuities are shown to decay exponentially in time, more rapidly for larger acoustic speeds and smaller viscosities. This is an extremely useful paper that helps instructors to develop creative techniques for the classroom. References : Streater, R. F. Corrections to Fluid Dynamics : Open Systems and Information Dynamics, 10, 3-30, 2003. Hoff, David.: Dynamics of Singularity Surfaces for Compressible Navier-Stokes Flows in Two Space Dimensions. JournＦs Ｒuations aux dＳivＦs partielles, Art. No. 7, 9 p. 2001. Arfken, G. "Gauss's Theorem." 1.11 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 57-61, 1985. Morse, P. M. and Feshbach, H. "Gauss's Theorem." In Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 37-38, 1953. Eric W. Weisstein. "Divergence Theorem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DivergenceTheorem.html
UR: http://www.muohio.edu
DE: 1847 Modeling
DE: 1849 Numerical approximations and analysis
SC: Hydrology [H]
MN: Fall Meeting 2005