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The Scientific World Journal
Volume 2015, Article ID 692494, 9 pages
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.

Linked References

  1. G. B. Arfken, Mathematical Methods for Physicists, Academic Press, New York, NY, USA, 2nd edition, 1970. View at MathSciNet
  2. J. M. Rassias, Counter Examples in Differential Equations and Related Topics, World Scientific, Singapore, 1994.
  3. O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Breach Science Publishers, New York, NY, USA, 2nd edition, 1975, Translated from the Russian by R. A. Silverman and J. Chu.
  4. P.-L. Lions, Mathematical Topics in Fluid Mechanics: Volume 1: Incompressible Models, vol. 3 of Oxford Lecture Series in Mathematics and Its Applications, Oxford University press, Oxford, UK, 1996. View at MathSciNet
  5. V. Christianto and F. Smarandache, “An exact mapping from Navier-Stokes equation to Schrödinger equation via Riccati equation,” Progress in Physics, vol. 1, pp. 38–39, 2008. View at Google Scholar · View at MathSciNet
  6. F. Kamran, Z.-C. Chen, X. Ji, and C. Yi, “Similarity reduction of a (3+1) Navier-Stokes system,” Engineering Computations, vol. 23, no. 6, pp. 632–643, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. G. Nugroho, A. M. S. Ali, and Z. A. A. Karim, “Toward a new simple analytical formulation of Navier-Stokes equations,” World Academy of Science, Engineering and Technology, vol. 39, pp. 197–201, 2009. View at Google Scholar · View at Scopus
  8. J. Jormakka, “Solutions to three-dimensional navier-stokes equations for incompressible fluids,” Electronic Journal of Differential Equations, vol. 1, no. 1, pp. 1–14, 2010. View at Google Scholar
  9. R. M. Temam, Navier Stokes Equations, Theory and Numerical Analysis, reprint of the 1984 edition, AMS Chelsea Publishing, Providence, RI, USA, 2001.
  10. A. Vasseur, “Higher derivatives estimate for the 3D Navier–Stokes equation,” Annales de l'Institut Henri Poincare (C) Non Linear Analysis, vol. 27, no. 5, pp. 1189–1204, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. V. V. Scheffer, “Partial regularity of solutions to the Navier-Stokes equations,” Pacific Journal of Mathematics, vol. 66, no. 2, pp. 535–552, 1976. View at Publisher · View at Google Scholar · View at MathSciNet
  12. V. V. Scheffer, “The Navier-Stokes equations on a bounded domain,” Communications in Mathematical Physics, vol. 73, no. 1, pp. 1–42, 1980. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. J. Serrin, “The initial value problem for the Navier—stokes equations,” in Nonlinear Problems, pp. 69–98, University of Wisconsin Press, Madison, Wis, USA, 1963. View at Google Scholar
  14. A. Vasseur, “Regularity criterion for 3D Navier-Stokes equations in terms of the direction of the velocity,” Applications of Mathematics, vol. 54, no. 1, pp. 47–52, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. G. R. Liu, Mesh Free Methods: Moving Beyond the Finite Element Method, CRC Press, New York, NY, USA, 2003. View at MathSciNet
  16. B. L. van der Waerden, Modern Algebra, vol. 1-2, Frederick Ungar Publishing Company, New York, NY, USA, 1964.
  17. C. Shu and L. F. Fan, “A new discretization method and its application to solve incompressible Navier-Stokes equations,” Computational Mechanics, vol. 27, no. 4, pp. 292–301, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. M. R. Spiegel, Vector Analysis, Schaum's Outline Series, McGraw Hill, New York, NY, USA, 1963.
  19. A. P. Wills, Vector Analysis with an Introduction to Tensor Analysis, Dover Publications, New York, NY, USA, 1958.
  20. S. C. Goldstein, Lectures on Fluid Mechanics, Intersciences Publishers, London, UK, 2000.
  21. Z. U. Warsi, Fluid Dynamics, CRC Press, New York, NY, USA, Second edition, 1999. View at MathSciNet
  22. H. Schlichting, Boundary Layer Theory, McGraw-Hill, New York, NY, USA, 1979. View at MathSciNet
  23. D. Kuzmin and O. Mierka, “On the implementation of the kε turbulence model in incompressible flow solvers based on a finite element discretization,” in International Conference on Boundary and Interior Layers, BAIL 2006, G. Lube and G. Rapin, Eds., University of Göttingen, Göttingen, Germany, 2006. View at Google Scholar
  24. P. B. Mucha, “The Eulerian limit and the slip boundary conditions—admissible irregularity of the boundary,” in Regularity and Other Aspects of the Navier-Stokes Equations, vol. 70 of Banach Center Publications, pp. 169–183, Polish Academy of Sciences, Warsaw, Poland, 2005. View at Google Scholar
  25. W. M. Rusin, “On the inviscid limit for the solutions of two-dimensional incompressible Navier-Stokes equations with slip-type boundary conditions,” Nonlinearity, vol. 19, no. 6, pp. 1349–1363, 2006. View at Google Scholar · View at MathSciNet · View at Scopus
  26. Y. Xiao and Z. Xin, “On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition,” Communications on Pure and Applied Mathematics, vol. 60, no. 7, pp. 1027–1055, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. R. Westermann, C. Johnson, and T. Ertl, “A level-set method for flow visualization,” in Proceedings of the Conference on Visualization (VIS '00), pp. 147–154, Salt Lake City, Utah, USA, October 2000. View at Scopus
  28. D. W. Riddle, Analytic Geometry, Brooks/Cole Publishing Company, 6th edition, 1995.
  29. E. C. Zachmanoglou and D. W. Thoe, Introduction to Partial Differential Equations with Applications, Williams & Wilkins, Baltimore, Md, USA, 1976.
  30. E. M. Zauderer, Partial Differential Equations of Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1989. View at MathSciNet
  31. A. D. Snider, Partial Differential Equations. Sources and Solutions, Prentice Hall, Upper Saddle River, NJ, USA, 1999.
  32. S. M. Buckley, Sobolev Spaces, Department of Mathematics, National University of Ireland, Maynooth Co., Kildare, Ireland, 2000.