Table of Contents
ISRN Mathematical Physics
Volume 2012 (2012), Article ID 831063, 17 pages
http://dx.doi.org/10.5402/2012/831063
Research Article

First Problem of Stokes for Generalized Burgers' Fluids

1Abdus Salam School of Mathematical Sciences, GC University, Lahore 54600, Pakistan
2Department of Mathematics, NED University of Engineering and Technology, Karachi 75270, Pakistan

Received 11 October 2011; Accepted 31 October 2011

Academic Editors: I. Radinschi and W.-H. Steeb

Copyright © 2012 Muhammad Jamil. 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. Z. S. Yu and J. Z. Lin, “Numerical research on the coherent structure in the viscoelastic second-order mixing layers,” Applied Mathematics and Mechanics, vol. 19, no. 8, pp. 717–723, 1998. View at Google Scholar · View at Scopus
  2. K. R. Rajagopal, “On the boundary conditions for fluids of differential type,” in Navier-Stokes Equations and Related Non-Linear Problems, A. Sequeira, Ed., Plenum Press, New York, NY, USA, 1995. View at Google Scholar
  3. K. R. Rajagopal and P. N. Kaloni, “Some remarks on boundary conditions for fluids of differential type,” in Continuum Mechanics and its Applications, G. A. C. Gram and S. K. Malik, Eds., Hemisphere, Washington, DC, USA, 1989. View at Google Scholar
  4. R. G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, NY, USA, 1999.
  5. K. R. Rajagopal, “A note on unsteady unidirectional flows of a non-Newtonian fluid,” International Journal of Non-Linear Mechanics, vol. 17, no. 5-6, pp. 369–373, 1982. View at Google Scholar · View at Scopus
  6. K. R. Rajagopal, “On the creeping flow of the second-order fluid,” Journal of Non-Newtonian Fluid Mechanics, vol. 15, no. 2, pp. 239–246, 1984. View at Google Scholar · View at Scopus
  7. T. Hayat, R. Ellahi, and F. M. Mahomed, “Exact solutions for thin film flow of a third grade fluid down an inclined plane,” Chaos, Solitons and Fractals, vol. 38, no. 5, pp. 1336–1341, 2008. View at Publisher · View at Google Scholar · View at Scopus
  8. W. Tan and T. Masuoka, “Stability analysis of a Maxwell fluid in a porous medium heated from below,” Physics Letters, Section A, vol. 360, no. 3, pp. 454–460, 2007. View at Publisher · View at Google Scholar · View at Scopus
  9. C. Fetecau, T. Hayat, and C. Fetecau, “Starting solutions for oscillating motions of Oldroyd-B fluids in cylindrical domains,” Journal of Non-Newtonian Fluid Mechanics, vol. 153, no. 2-3, pp. 191–201, 2008. View at Publisher · View at Google Scholar · View at Scopus
  10. Z. Zhang, C. Fu, and W. Tan, “Linear and nonlinear stability analyses of thermal convection for Oldroyd-B fluids in porous media heated from below,” Physics of Fluids, vol. 20, no. 8, Article ID 084103, 12 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus
  11. M. A. Rana, A. M. Siddiqui, and R. Qamar, “Hall effects on hydromagnetic flow of an Oldroyd 6-constant fluid between concentric cylinders,” Chaos, Solitons and Fractals, vol. 39, no. 1, pp. 204–213, 2009. View at Publisher · View at Google Scholar · View at Scopus
  12. C. Fetecau, M. Jamil, C. Fetecau, and I. Siddique, “A note on the second problem of Stokes for Maxwell fluids,” International Journal of Non-Linear Mechanics, vol. 44, no. 10, pp. 1085–1090, 2009. View at Publisher · View at Google Scholar · View at Scopus
  13. S. Hyder Ali Muttaqi Shah, M. Khan, and H. Qi, “Exact solutions for a viscoelastic fluid with the generalized Oldroyd-B model,” Nonlinear Analysis: Real World Applications, vol. 10, no. 4, pp. 2590–2599, 2009. View at Publisher · View at Google Scholar · View at Scopus
  14. C. Fetecau, M. Jamil, C. Fetecau, and D. Vieru, “The Rayleigh-Stokes problem for an edge in a generalized Oldroyd-B fluid,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 60, no. 5, pp. 921–933, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  15. T. Hayat, S. Najam, M. Sajid, M. Ayub, and S. Mesloub, “On exact solutions for oscillatory flows in a generalized Burgers fluid with slip condition,” Zeitschrift fur Naturforschung Section A, vol. 65, no. 5, pp. 381–391, 2010. View at Google Scholar
  16. C. Fetecau, J. Zierep, R. Bohning, and C. Fetecau, “On the energetic balance for the flow of an Oldroyd-B fluid due to a flat plate subject to a time-dependent shear stress,” Computers and Mathematics with Applications, vol. 60, no. 1, pp. 74–82, 2010. View at Publisher · View at Google Scholar · View at Scopus
  17. M. Jamil and C. Fetecau, “Helical flows of Maxwell fluid between coaxial cylinders with given shear stresses on the boundary,” Nonlinear Analysis: Real World Applications, vol. 11, no. 5, pp. 4302–4311, 2010. View at Publisher · View at Google Scholar · View at Scopus
  18. M. Jamil, C. Fetecau, and M. Imran, “Unsteady helical flows of Oldroyd-B fluids,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1378–1386, 2011. View at Publisher · View at Google Scholar
  19. M. Jamil, A. Rauf, C. Fetecau, and N. A. Khan, “Helical flows of second grade fluid due to constantly accelerated shear stresses,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 1959–1969, 2011. View at Publisher · View at Google Scholar
  20. J. E. Dunn and K. R. Rajagopal, “Fluids of differential type: critical review and thermodynamic analysis,” International Journal of Engineering Science, vol. 33, no. 5, pp. 689–729, 1995. View at Google Scholar · View at Scopus
  21. K. R. Rajagopal, “Mechanics of non-Newtonian fluids in recent development in theoretical fluid Mechanics,” Pitman Research Notes in Mathematics Series, vol. 291, pp. 129–162, 1993. View at Google Scholar
  22. J. M. Burgers, “Mechanical considerations-model systems-phenomenological theories of relaxation and of viscosity,” in First Report on Viscosity and Plasticity, J. M. Burgers, Ed., Nordemann Publishing Company, New York, NY, USA, 1935. View at Google Scholar
  23. J. Murali Krishnan and K. R. Rajagopal, “Thermodynamic framework for the constitutive modeling of asphalt concrete: theory and applications,” Journal of Materials in Civil Engineering, vol. 16, no. 2, pp. 155–166, 2004. View at Publisher · View at Google Scholar · View at Scopus
  24. A. R. Lee and A. H. D. Markwick, “The mechanical properties of bituminous surfacing materials under constant stress,” Journal of the Indian Chemical Society, vol. 56, pp. 146–156, 1937. View at Google Scholar
  25. B. H. Tan, I. Jackson, and J. D. Fitz Gerald, “High-temperature viscoelasticity of fine-grained polycrystalline olivine,” Physics and Chemistry of Minerals, vol. 28, no. 9, pp. 641–664, 2001. View at Publisher · View at Google Scholar · View at Scopus
  26. W. R. Peltier, P. Wu, and D. A. Yuen, “The viscosities of the earth mantle,” in Anelasticity in the Earth, F. D. Stacey, M. S. Paterson, and A. Nicholas, Eds., American Geophysical Union, Colo, USA, 1981. View at Google Scholar
  27. P. Ravindran, J. M. Krishnan, and K. R. Rajagopal, “A note on the flow of a Burgers' fluid in an orthogonal rheometer,” International Journal of Engineering Science, vol. 42, no. 19-20, pp. 1973–1985, 2004. View at Publisher · View at Google Scholar · View at Scopus
  28. M. Khan, S. Hyder Ali, and C. Fetecau, “Exact solutions of accelerated flows for a Burgers' fluid. I. The case γ < λ2 / 4,” Applied Mathematics and Computation, vol. 203, no. 2, pp. 881–894, 2008. View at Publisher · View at Google Scholar · View at Scopus
  29. T. Hayat, S. B. Khan, and M. Khan, “Influence of hall current on rotating flow of a burgers' fluid through a porous space,” Journal of Porous Media, vol. 11, no. 3, pp. 277–287, 2008. View at Publisher · View at Google Scholar · View at Scopus
  30. M. Khan, R. Malik, C. Fetecau, and C. Fetecau, “Exact solutions for the unsteady flow of a Burgers' fluid between two sidewalls perpendicular to the plate,” Chemical Engineering Communications, vol. 197, no. 11, pp. 1367–1386, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. D. Tong, “Starting solutions for oscillating motions of a generalized Burgers' fluid in cylindrical domains,” Acta Mechanica, vol. 214, no. 3-4, pp. 395–407, 2010. View at Publisher · View at Google Scholar · View at Scopus
  32. M. Jamil and C. Fetecau, “Some exact solutions for rotating flows of a generalized Burgers' fluid in cylindrical domains,” Journal of Non-Newtonian Fluid Mechanics, vol. 165, no. 23-24, pp. 1700–1712, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. M. Jamil, A. A. Zafar, C. Fetecau, and N. A. Khan, “Exact analytic solutions for the flow of a generalized burgers fluid induced by an accelerated shear stress,” Chemical Engineering Communications, vol. 199, no. 1, pp. 17–39, 2012. View at Publisher · View at Google Scholar
  34. R. I. Tanner, “Note on the Rayleigh problem for a visco-elastic fluid,” Zeitschrift für Angewandte Mathematik und Physik, vol. 13, pp. 573–580, 1962. View at Google Scholar
  35. P. N. Srivastava, “Non-steady helical flow of a visco-elastic liquid,” Archives of Mechanics, vol. 18, pp. 145–150, 1966. View at Google Scholar
  36. I. C. Christov, “Stokes' first problem for some non-Newtonian fluids: results and mistakes,” Mechanics Research Communications, vol. 37, no. 8, pp. 717–723, 2010. View at Publisher · View at Google Scholar · View at Scopus
  37. I. N. Sneddon, Fourier Transforms, McGraw-Hill, New York, NY, USA, 1951.
  38. A. Kuros, Cours d’Algebre Superieure, Edition Mir Moscow, 1973.
  39. F. R. Gantmacher, Applications of the Theory of Matreices, Wiley, New York, NY, USA, 1959.
  40. C. Fetecau, T. Hayat, M. Khan, and C. Fetecau, “Erratum: Unsteady flow of an Oldroyd-B fluid induced by the impulsive motion of a plate between two side walls perpendicular to the plate (Acta Mechanica, vol. 198 no. 21–33, 2008,” Acta Mechanica, vol. 216, no. 1–4, pp. 359–361, 2011. View at Publisher · View at Google Scholar