- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review E

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

First-order energy-integral model for thin Newtonian liquids falling along sinusoidal furrows

I. Mohammed Rizwan Sadiq*

  • Department of Mathematics and Statistics, University of Konstanz, 78457 Konstanz, Germany

  • *rizwan.sadiq.i@gmail.com
PDF

Phys. Rev. E 85, 036309 – Published 21 March, 2012

DOI: https://doi.org/10.1103/PhysRevE.85.036309

Abstract

An average modeling methodology under the lubrication approach is used to formulate a set of three coupled nonlinear partial differential equations based on the Nusselt scales. This system, known as the energy-integral method in literature, simplifies the Navier-Stokes equation at the first order and analyzes the dynamics of a thin sheet of fluid flowing over a topography with sinusoidally varying longitudinal furrows. Limiting cases of the linear stability results are mathematically discussed and the complete linear system is numerically handled by means of finite differences to approximate the eigenfunctions and their derivatives in a periodic domain. In a geometry which resembles a vertical shift of a topography, with the amplitude being equal to the shift length, it is found that such a geometry stabilizes the flow compared to its counterpart with no shift, such that the wave characteristics get affected. To confirm the stability results, a numerical investigation is performed.

View figure in article

Article Text

References (58)

  1. R. Usha and B. Uma, Phys. Fluids 16, 2679 (2004).
  2. A. F. Elkouh, Int. J. Mech. Sci. 9, 253 (1967).
  3. S. H. Crandall and A. El-Shafei, J. Appl. Mech. 60, 728 (1993).
  4. V. Ya. Shkadov, Fluid Dyn. 2, 29 (1967).
  5. J. J. Lee and C. C. Mei, J. Fluid Mech. 307, 191 (1996).
  6. B. Uma and R. Usha, Phys. Fluids 20, 032102 (2008).
  7. H. A. Abderrahmane and G. H. Vatistas, Phys. Fluids 19, 098106 (2007).
  8. T. B. Benjamin, J. Fluid Mech. 2, 554 (1957).
  9. E. Novabri and A. Oron, Phys. Fluids 21, 062107 (2009).
  10. E. Novabri and A. Oron, Phys. Fluids 23, 012105 (2011).
  11. H. Tougou, J. Phys. Soc. Jpn. 44, 1014 (1978).
  12. C. Pozrikidis, J. Fluid Mech. 188, 275 (1988).
  13. V. Bontozoglou and G. Papapolymerou, Int. J. Multiphase Flow 23, 69 (1997).
  14. M. Vlachogiannis and V. Bontozoglou, J. Fluid Mech. 457, 133 (2002).
  15. C. Bielarz and S. Kalliadasis, Phys. Fluids 15, 2512 (2003).
  16. R. Usha and B. Uma, Int. J. Non-Linear Mech. 39, 1589 (2004).
  17. A. Wierscheim, C. Lepski, and N. Aksel, Acta Mech. 179, 41 (2005).
  18. K. Argyriadi, M. Vlachogiannis, and V. Bontozoglou, Phys. Fluids 18, 012102 (2006).
  19. Y. O. Kabova, A. Alexeev, T. Gambaryan-Roisman, and P. Stephan, Phys. Fluids 18, 012104 (2006).
  20. Y. Y. Trifonov, Int. J. Multiphase Flow 33, 1186 (2007).
  21. S. Saprykin, R. J. Koopmans, and S. Kalliadasis, J. Fluid Mech. 578, 271 (2007).
  22. A. Oron and C. Heining, Phys. Fluids 20, 082102 (2008).
  23. A. Wierscheim, T. Pollak, C. Heining, and N. Aksel, Phys. Fluids 22, 113603 (2010).
  24. S. J. D. D’Alessio, J. P. Pascal, and H. A. Jasmine, Phys. Fluids 21, 062105 (2009).
  25. T. Häcker and H. Uecker, Phys. Fluids 21, 092105 (2009).
  26. C. Heining and N. Aksel, Int. J. Multiphase Flow 36, 847 (2010).
  27. S. J. D. D’Alessio, J. P. Pascal, H. A. Jasmine, and K. A. Ogden, J. Fluid Mech. 665, 418 (2010).
  28. P. H. Gaskell, Y. C. Lee, and H. M. Thompson, Comput. Model. Eng. Sci. 62, 77 (2010).
  29. T. Gambaryan-Roisman and H. Yu, Heat Transf. Eng. 32, 705 (2011).
  30. T. Gambaryan-Roisman and P. Stephan, J. Enhanc. Heat Transfer 10, 445 (2003).
  31. K. Helbig, R. Nasarek, and T. Gambaryan-Roisman, ASME J. Heat Transfer 131, 011601 (2009).
  32. K. Helbig, A. Alexeev, T. Gambaryan-Roisman, and P. Stephan, Turbulence Combust. 75, 85 (2005).
  33. I. M. R. Sadiq, T. Gambaryan-Roisman, and P. Stephan, Phys. Fluids 24, 014104 (2012).
  34. P. Stephan and C. A. Busse, Int. J. Heat Mass Transf. 35, 383 (1992).
  35. C. Ruyer-Quil and P. Manneville, Eur. Phys. J. B 6, 277 (1998).
  36. C. Ruyer-Quil and P. Manneville, Eur. Phys. J. B 15, 357 (2000).
  37. B. Scheid, S. Kalliadasis, C. Ruyer-Quil, and P. Colinet, Europhys. Lett. 84, 64002 (2008).
  38. P. Trevelyan, B. Scheid, S. Kalliadasis, and C. Ruyer-Quil, J. Fluid Mech. 592, 295 (2007).
  39. A. Oron and O. Gottlieb, J. Eng. Math. 50, 121 (2004).
  40. B. Scheid, C. Ruyer-Quil, U. Thiele, O. Kabov, and J. C. Legros, J. Fluid Mech. 527, 303 (2005).
  41. S. Kalliadasis, A. Kiyashko, and E. A. Demekhin, J. Fluid Mech. 475, 377 (2003).
  42. P. Trevelyan and S. Kalliadasis, J. Eng. Math. 50, 177 (2004).
  43. L. M. Chamra, R. L. Webb, and M. R. Randlett, Int. J. Heat Mass Transf. 39, 1827 (1996).
  44. L. M. Chamra, R. L. Webb, and M. R. Randlett, Int. J. Heat Mass Transf. 39, 1839 (1996).
  45. M. R. J. Charest, J. E. D. Gauthier, and X. Huang, ASME Conf. Proc. 1, 781 (2006).
  46. I. M. R. Sadiq and R. Usha, Phys. Fluids 20, 022105 (2008).
  47. I. M. R. Sadiq, R. Usha, and S. W. Joo, Chem. Eng. Sci. 65, 4443 (2010).
  48. B. Scheid, S. Kalliadasis, C. Ruyer-Quil, and P. Colinet, Phys. Rev. E 78, 066311 (2008).
  49. B. Scheid, C. Ruyer-Quil, and P. Manneville, J. Fluid Mech. 562, 183 (2006).
  50. C. Heining, V. Bontozoglou, N. Aksel, and A. Wierschem, Int. J. Multiphase Flow 35, 78 (2009).
  51. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (Courier Dover Publications, New York, 1964).
  52. R. Cooke, Classical Algebra: Its Nature, Origins, and Uses (John Wiley and Sons, Hoboken, New Jersey, 2008).
  53. C. S. Yih, Phys. Fluids 6, 321 (1963).
  54. J. Liu, J. D. Paul, and J. P. Gollub, J. Fluid Mech. 250, 69 (1993).
  55. S. Kalliadasis, E. A. Demekhin, C. Ruyer-Quil, and M. Velarde, J. Fluid Mech. 492, 303 (2003).
  56. S. W. Joo, S. H. Davis, and S. G. Bankoff, J. Fluid Mech. 321, 279 (1996).
  57. L. N. Trefthen, Spectral Methods in MATLAB (SIAM, Philadelphia, 2000).
  58. M. K. Smith, J. Fluid Mech. 217, 469 (1990).

Outline

Information

Phys. Rev. E 85, 036309– Published 21 March, 2012
  • Received 30 September 2011

©2012 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation