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The Scientific World Journal
Volume 2015, Article ID 692494, 9 pages
http://dx.doi.org/10.1155/2015/692494
Research Article

On a Modified Form of Navier-Stokes Equations for Three-Dimensional Flows

Section of Mechanics, School of Applied Mathematics and Physical Sciences, NTUA, Greece

Received 1 August 2014; Revised 7 March 2015; Accepted 8 March 2015

Academic Editor: Ajda Fošner

Copyright © 2015 J. Venetis. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A rephrased form of Navier-Stokes equations is performed for incompressible, three-dimensional, unsteady flows according to Eulerian formalism for the fluid motion. In particular, we propose a geometrical method for the elimination of the nonlinear terms of these fundamental equations, which are expressed in true vector form, and finally arrive at an equivalent system of three semilinear first order PDEs, which hold for a three-dimensional rectangular Cartesian coordinate system. Next, we present the related variational formulation of these modified equations as well as a general type of weak solutions which mainly concern Sobolev spaces.