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Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 627584, 13 pages
http://dx.doi.org/10.1155/2015/627584
Research Article

Approximation of a Class of Incompressible Third Grade Fluids Equations

1State Key Laboratory of Geomechanics and Geotechnical Engineering, Institute of Rock and Soil Mechanics, Chinese Academy of Sciences, Wuhan, Hubei 430071, China
2College of Science, Naval University of Engineering, Wuhan, Hubei 430033, China
3Power China Zhongnan Engineering Corporation Limited, Changsha, Hunan 410014, China

Received 21 October 2014; Revised 21 December 2014; Accepted 22 December 2014

Academic Editor: Ivan Area

Copyright © 2015 Zeqi Zhu et al. 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. R. S. Rivlin and J. L. Ericksen, “Stress-deformation relations for isotropic materials,” Journal of Rational Mechanics and Analysis, vol. 4, pp. 323–425, 1955. View at Google Scholar · View at MathSciNet
  2. T. Aziz and A. Aziz, “MHD flow of a third grade fluid in a porous half space with plate suction or injection: an analytical approach,” Applied Mathematics and Computation, vol. 218, no. 21, pp. 10443–10453, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. T. Aziz, F. M. Mahomed, and A. Aziz, “Group invariant solutions for the unsteady MHD flow of a third grade fluid in a porous medium,” International Journal of Non-Linear Mechanics, vol. 47, no. 7, pp. 792–798, 2012. View at Publisher · View at Google Scholar · View at Scopus
  4. T. Aziz, F. M. Mahomed, M. Ayub, and D. P. Mason, “Non-linear time-dependent flow models of third grade fluids: a conditional symmetry approach,” International Journal of Non-Linear Mechanics, vol. 54, pp. 55–65, 2013. View at Publisher · View at Google Scholar · View at Scopus
  5. V. Busuioc and D. Iftimie, “Global existence and uniqueness of solutions for the equations of third grade fluids,” International Journal of Non-Linear Mechanics, vol. 39, no. 1, pp. 1–12, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. D. Cioranescu and V. Girault, “Weak and classical solutions of a family of second grade fluids,” International Journal of Non-Linear Mechanics, vol. 32, no. 2, pp. 317–335, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. M. Hamza and M. Paicu, “Global existence and uniqueness result of a class of third-grade fluids equations,” Nonlinearity, vol. 20, no. 5, pp. 1095–1114, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. M. Paicu, G. Raugel, and A. Rekalo, “Regularity of the global attractor and finite-dimensional behavior for the second grade fluid equations,” Journal of Differential Equations, vol. 252, no. 6, pp. 3695–3751, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. A. Sequeira and J. Videman, “Global existence of classical solutions for the equations of third grade fluids,” Journal of Mathematical and Physical Sciences, vol. 29, no. 2, pp. 47–69, 1995. View at Google Scholar · View at MathSciNet
  10. C. Zhao, Y. Liang, and M. Zhao, “Upper and lower bounds of time decay rate of solutions to a class of incompressible third grade fluid equations,” Nonlinear Analysis: Real World Applications, vol. 15, pp. 229–238, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. R. A. Adams, Sobolev Spaces, Academic Press, New York, NY, USA, 1975. View at MathSciNet
  12. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer, Berlin, Germany, 2nd edition, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  13. R. Temam, Navier-Stokes Equations (Theory and Numerical Analysis), North-Holland, Amsterdam, Netherlands, 1984. View at MathSciNet
  14. V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, vol. 49 of AMS Colloquium Publications, American Mathematical Society, Providence, RI, USA, 2002.
  15. C. Zhao and Y. You, “Approximation of the incompressible convective Brinkman-Forchheimer equations,” Journal of Evolution Equations, vol. 12, no. 4, pp. 767–788, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. C. Zhao, “Approximation of the incompressible non-newtonian fluid equations by the artificial compressibility method,” Mathematical Methods in the Applied Sciences, vol. 36, no. 7, pp. 840–856, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. C. Zhao, “Pullback asymptotic behavior of solutions for a non-autonomous non-Newtonian fluid on two-dimensional unbounded domains,” Journal of Mathematical Physics, vol. 53, no. 12, Article ID 122702, pp. 1–21, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. C. Zhao, “Dynamics of non-autonomous equations of non-newtonian fluid on 2D unbounded domains,” Dynamics of Partial Differential Equations, vol. 10, no. 3, pp. 283–312, 2013. View at Publisher · View at Google Scholar · View at Scopus
  19. C. Zhao, G. Liu, and W. Wang, “Smooth pullback attractors for a non-autonomous 2D non-Newtonian fluid and their tempered behaviors,” Journal of Mathematical Fluid Mechanics, vol. 16, no. 2, pp. 243–262, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  20. C. Zhao, L. Kong, G. Liu, and M. Zhao, “The trajectory attractor and its limiting behavior for the convective Brinkman-Forchheimer equations,” Topological Methods in Nonlinear Analysis. In press.