Table of Contents
Chinese Journal of Engineering
Volume 2013, Article ID 751909, 21 pages
http://dx.doi.org/10.1155/2013/751909
Research Article

Influence of Temperature Gradient on the Stability of a Viscoelastic Film with Gradually Fading Memory over a Topography with Ridges and Furrows

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

Received 28 June 2013; Accepted 22 August 2013

Academic Editors: K. Ariyur, B.-Y. Cao, and B. Sun

Copyright © 2013 Mohammed Rizwan Sadiq Iqbal. 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. A. A. Nepomnyashchy, M. G. Velarde, and P. Colinet, Interfacial Phenomena and Convection, Chapman & Hall/ CRC, Boca Raton, Fla, USA, 2002.
  2. S. Kalliadasis, C. Ruyer-Quil, B. Scheid, and M. G. Velarde, Falling Liquid Films, Springer, London, UK, 2012.
  3. J. G. Oldroyd, “On the formulations of rheological equations of state,” Proceedings of the Royal Society A, vol. 200, no. 1063, p. 523, 1950. View at Google Scholar
  4. C. Truesdell and W. Noll, The Non-Linear Field Theories of Mechanics, Hanbook of Physics, Springer, Berlin, Germany, 1965.
  5. K. R. Rajagopal, “Mechanics of non-Newtonian fluids,” in Recent Developments in Theoretical Fluid Mechanics, vol. 291 of Pittman Research Notes in Mathematical Series, Longman Scientific & Technical, Harlow, UK, 1993. View at Google Scholar
  6. R. B. Bird, “Useful non-Newtonian models,” Annual Review of Fluid Mechanics, vol. 8, pp. 13–134, 1976. View at Publisher · View at Google Scholar
  7. R. P. Chhabra and J. F. Richardson, Non-Newtonian Flow in the Process Industries: Fundamentals and Engineering Applications, Butterworth-Heinemann, Oxford, UK, 1999.
  8. G. {\L}ukaszewicz, Micropolar Fluids: Theory and Applications, Birkhäauser, Basel, Switzerland, 1999.
  9. J. D. Ferry, Viscoelastic Properties of Polymers, John Wiley & Sons, New York, NY, USA, 1961.
  10. T. Altan, S. I. Oh, and H. Gegel, Metal Forming: Fundamentals and Applications, American Society of Metals, Metals park, Ohio, USA, 1983.
  11. K. Walters, “Non-Newtonian effects on an elastico-viscous liquid contained between coaxial cylinders (II),” Quarterly Journal of Mechanics and Applied Mathematics, vol. 13, no. 4, pp. 444–461, 1960. View at Publisher · View at Google Scholar · View at Scopus
  12. H. I. Andersson and E. N. Dahl, “Gravity-driven flow of a viscoelastic liquid film along a vertical wall,” Journal of Physics, vol. 32, pp. 1557–1562, 1999. View at Publisher · View at Google Scholar · View at Scopus
  13. I. M. R. Sadiq and R. Usha, “Linear instability in a thin viscoelastic liquid film on an inclined, non-uniformly heated wall,” International Journal of Engineering Science, vol. 43, pp. 1435–1449, 2005. View at Publisher · View at Google Scholar · View at Scopus
  14. M. C. Lin and C. K. Chen, “Finite amplitude long-wave instability of a thin viscoelastic fluid during spin coating,” Applied Mathematical Modelling, vol. 36, no. 6, pp. 2536–2549, 2012. View at Publisher · View at Google Scholar · View at Scopus
  15. M. A. Sirwah and K. Zakaria, “Nonlinear evolution of the travelling waves at the surface of a thin viscoelastic falling film,” Applied Mathematical Modelling, vol. 37, pp. 1723–1752, 2013. View at Google Scholar
  16. H.-C. Chang, “Wave evolution on a falling film,” Annual Review of Fluid Mechanics, vol. 26, pp. 103–136, 1994. View at Publisher · View at Google Scholar
  17. A. Oron, S. H. Davis, and S. G. Bankoff, “Long-scale evolution of thin liquid films,” Reviews of Modern Physics, vol. 69, no. 3, pp. 931–980, 1997. View at Google Scholar · View at Scopus
  18. S. W. Joo, S. H. Davis, and S. G. Bankoff, “Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers,” Journal of Fluid Mechanics, vol. 230, pp. 117–146, 1991. View at Google Scholar · View at Scopus
  19. S. Miladinova, S. Slavtchev, G. Lebon, and E. Toshev, “Long-wave instabilities of non-uniformly heated falling films,” Journal of Fluid Mechanics, vol. 453, pp. 153–175, 2002. View at Publisher · View at Google Scholar · View at Scopus
  20. B. Scheid, Evolution and stability of falling liquid films with thermocapillary effects [Ph.D. thesis], Université Libre de Bruxelles, Brussels, Belgium, 2004.
  21. I. M. R. Sadiq, R. Usha, and S. W. Joo, “Instabilities in a liquid film flow over an inclined heated porous substrate,” Chemical Engineering Science, vol. 65, no. 15, pp. 4443–4459, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. M. Vlachogiannis and V. Bontozoglou, “Experiments on laminar film flow along a periodic wall,” Journal of Fluid Mechanics, vol. 457, pp. 133–156, 2002. View at Publisher · View at Google Scholar · View at Scopus
  23. A. Wierschem, C. Lepski, and N. Aksel, “Effect of long undulated bottoms on thin gravity-driven films,” Acta Mechanica, vol. 179, no. 1-2, pp. 41–66, 2005. View at Publisher · View at Google Scholar · View at Scopus
  24. K. Argyriadi, M. Vlachogiannis, and V. Bontozoglou, “Experimental study of inclined film flow along periodic corrugations: the effect of wall steepness,” Physics of Fluids, vol. 18, no. 1, Article ID 012102, 2006. View at Publisher · View at Google Scholar · View at Scopus
  25. L. A. Dávalos-Orozco, “Nonlinear instability of a thin film flowing down a smoothly deformed surface,” Physics of Fluids, vol. 19, no. 7, Article ID 074103, 2007. View at Publisher · View at Google Scholar · View at Scopus
  26. A. Oron and C. Heining, “Weighted-residual integral boundary-layer model for the nonlinear dynamics of thin liquid films falling on an undulating vertical wall,” Physics of Fluids, vol. 20, no. 8, Article ID 082102, 2008. View at Publisher · View at Google Scholar · View at Scopus
  27. T. Häcker and H. Uecker, “An integral boundary layer equation for film flow over inclined wavy bottoms,” Physics of Fluids, vol. 21, no. 9, Article ID 092105, 2009. View at Publisher · View at Google Scholar · View at Scopus
  28. S. J. D. D'Alessio, J. P. Pascal, and H. A. Jasmine, “Instability in gravity-driven flow over uneven surfaces,” Physics of Fluids, vol. 21, no. 6, Article ID 062105, 2009. View at Publisher · View at Google Scholar · View at Scopus
  29. C. Heining, V. Bontozoglou, N. Aksel, and A. Wierschem, “Nonlinear resonance in viscous films on inclined wavy planes,” International Journal of Multiphase Flow, vol. 36, pp. 78–90, 2010. View at Publisher · View at Google Scholar · View at Scopus
  30. C. Heining and N. Aksel, “Effects of inertia and surface tension on a power-law fluid flowing down a wavy incline,” International Journal of Multiphase Flow, vol. 36, no. 11-12, pp. 847–857, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. S. Saprykin, P. M. J. Trevelyan, R. J. Koopmans, and S. Kalliadasis, “Free-surface thin-film flows over uniformly heated topography,” Physical Review, vol. 75, no. 2, Article ID 026306, 2007. View at Publisher · View at Google Scholar · View at Scopus
  32. N. Tiwari and J. M. Davis, “Stabilization of thin liquid films flowing over locally heated surfaces via substrate topography,” Physics of Fluids, vol. 22, no. 4, Article ID 042106, 2010. View at Publisher · View at Google Scholar · View at Scopus
  33. S. J. D. D'Alessio, J. P. Pascal, H. A. Jasmine, and K. A. Ogden, “Film flow over heated wavy inclined surfaces,” Journal of Fluid Mechanics, vol. 665, pp. 418–456, 2010. View at Publisher · View at Google Scholar · View at Scopus
  34. K. A. Ogden, S. J. D. D'Alessio, and J. P. Pascal, “Gravity-driven flow over heated, porous, wavy surfaces,” Physics of Fluids, vol. 23, Article ID 122102, 2011. View at Publisher · View at Google Scholar · View at Scopus
  35. T. Gambaryan-Roisman and P. Stephan, “Falling films in micro- and minigrooves: Heat transfer and flow stability,” Thermal Science & Engineering, vol. 11, p. 43, 2003. View at Google Scholar
  36. K. Helbig, R. Nasarek, T. Gambaryan-Roisman, and P. Stephan, “Effect of longitudinal minigrooves on flow stability and wave characteristics of falling liquid films,” Journal of Heat Transfer, vol. 131, no. 1, Article ID 011601, 8 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  37. I. M. R. Sadiq, T. Gambaryan-Roisman, and P. Stephan, “Falling liquid films on longitudinal grooved geometries: integral boundary layer approach,” Physics of Fluids, vol. 24, no. 1, Article ID 014104, 2012. View at Publisher · View at Google Scholar · View at Scopus
  38. I. M. R. Sadiq, “First-order energy-integral model for thin Newtonian liquids falling along sinusoidal furrows,” Physical Review, vol. 85, no. 3, Article ID 036309, 2012. View at Publisher · View at Google Scholar · View at Scopus
  39. C. Heining, T. Pollak, and N. Aksel, “Pattern formation and mixing in three-dimensional film flow,” Physics of Fluids, vol. 24, no. 4, Article ID 042102, 2012. View at Publisher · View at Google Scholar · View at Scopus
  40. K.-T. Kim and R. E. Khayat, “Transient coating flow of a thin non-Newtonian fluid film,” Physics of Fluids, vol. 14, no. 7, pp. 2202–2215, 2002. View at Publisher · View at Google Scholar · View at Scopus
  41. E. I. Mogilevskii and V. Y. Shkadov, “Effect of bottom topography on the flow of a non-newtonian liquid film down an inclined plane,” Moscow University Mechanics Bulletin, vol. 62, no. 3, pp. 76–83, 2007. View at Publisher · View at Google Scholar · View at Scopus
  42. R. Usha and B. Uma, “Long waves on a viscoelastic film flow down a wavy incline,” International Journal of Non-Linear Mechanics, vol. 39, no. 10, pp. 1589–1602, 2004. View at Publisher · View at Google Scholar · View at Scopus
  43. H. Tougou, “Long waves on a film flow of a viscous fluid down an inclined uneven wall,” Journal of the Physical Society of Japan, vol. 44, no. 3, pp. 1014–1019, 1978. View at Google Scholar · View at Scopus
  44. S. Saprykin, R. J. Koopmans, and S. Kalliadasis, “Free-surface thin-film flows over topography: influence of inertia and viscoelasticity,” Journal of Fluid Mechanics, vol. 578, pp. 271–293, 2007. View at Publisher · View at Google Scholar · View at Scopus
  45. M. Pavlidis, Y. Dimakopoulos, and J. Tsamopoulos, “Steady viscoelastic film flow over 2D topography: I. The effect of viscoelastic properties under creeping flow,” Journal of Non-Newtonian Fluid Mechanics, vol. 165, no. 11-12, pp. 576–591, 2010. View at Publisher · View at Google Scholar · View at Scopus
  46. F. Bouchut and S. Boyaval, “A new model for shallow viscoelastic fluids,” Mathematical Models and Methods in Applied Sciences, vol. 23, no. 8, pp. 1479–1526, 2013. View at Publisher · View at Google Scholar
  47. E. S. G. Shaqfeh, R. G. Larson, and G. H. Fredrickson, “The stability of gravity driven viscoelastic film-flow at low to moderate reynolds number,” Journal of Non-Newtonian Fluid Mechanics, vol. 31, no. 1, pp. 87–113, 1989. View at Google Scholar · View at Scopus
  48. R. V. Birikh, V. A. Briskman, M. G. Velarde, and J. C. Legros, Liquid Interfacial Systems: Oscillations and Instability, Marcel Dekker, New York, NY, USA, 2003.
  49. R. B. Bird, R. C. Armstrong, and O. Hassager, “Dynamics of polymeric liquids,” in Fluid Mechanics, vol. 1, John Wiley & Sons, New York, NY, USA, 1987. View at Google Scholar
  50. N. Aksel, “A brief note from the editor on the ‘second-order fluid’,” Acta Mechanica, vol. 157, no. 1–4, pp. 235–236, 2002. View at Publisher · View at Google Scholar · View at Scopus
  51. J. H. Spurk and N. Aksel, Fluid Mechanics, Springer, Berlin, Germany, 2nd edition, 2008.
  52. B. S. Dandapat and A. Samanta, “Bifurcation analysis of first and second order Benney equations for viscoelastic fluid flowing down a vertical plane,” Journal of Physics, vol. 41, no. 9, Article ID 095501, 2008. View at Publisher · View at Google Scholar · View at Scopus
  53. B. D. Coleman and W. Noll, “An approximation theorem for functionals with applications in continuum mechanics,” Archive for Rational Mechanics and Analysis, vol. 6, no. 1, pp. 355–370, 1960. View at Publisher · View at Google Scholar · View at Scopus
  54. K. C. Porteous and M. M. Denn, “Linear stability of plane poiseuille flow of viscoelastic liquids,” Trans Soc Rheol, vol. 16, pp. 295–308, 1972. View at Google Scholar · View at Scopus
  55. K. Walters, “The solution flow problems in the case of materials with memory,” Journal of Fluid Mechanics, vol. 1, p. 474, 1962. View at Google Scholar
  56. P. L. Bhatnagar, “Comparative study of some constitutive equations characterising non-Newtonian fluids,” Proceedings of the Indian Academy of Sciences, vol. 66, no. 6, pp. 342–352, 1967. View at Publisher · View at Google Scholar · View at Scopus
  57. N. Datta and R. N. Jana, “Flow and heat transfer in an elastico-viscous liquid over an oscillating plate in a rotating frame,” İstanbul Üniversitesi Fen Fakültesi, vol. 43, p. 121, 1978. View at Google Scholar
  58. C. W. Macosko, Rheology: Principles, Measurements and Applications, John Wiley & Sons, New York, NY, USA, 1994.
  59. B. Uma and R. Usha, “Nonlinear stability of thin viscoelastic liquid film down a vertical wall with interfacial phase change,” in Proceedings of the ASME Joint U.S.-European Fluids Engineering Conference, vol. 1, Fora, parts A and B, paper no. FEDSM2002-31035, pp. 1303–1310, Quebec, Canada, July 2002. View at Scopus
  60. P. Kumar and V. Kumar, “On thermosolutal-convective instability in Walters B heterogeneous viscoelastic fluid layer through porous medium,” American Journal of Fluid Dynamics, vol. 3, no. 1, 2013. View at Google Scholar
  61. D. W. Beard and K. Walters, “Elastico-viscous boundary-layer flow I. Two-dimensional flow over a stagnation point,” Proceedings of the Cambridge Philosophical Society, vol. 60, p. 667, 1964. View at Google Scholar
  62. H. A. Barnes, J. F. Hutton, and K. Walters, An Introduction to Rheology, Elsevier, Amsterdam, The Netherlands, 1989.
  63. A. S. Gupta, “Stability of a non-Newtonian liquid film flowing down an inclined plane,” Physics of Fluids, vol. 28, p. 17, 1967. View at Google Scholar · View at Scopus
  64. B. S. Dandapat and A. S. Gupta, “Solitary waves on the surface of a viscoelastic fluid running down an inclined plane,” Rheologica Acta, vol. 36, no. 2, pp. 135–143, 1997. View at Google Scholar · View at Scopus
  65. W. Nusselt, “Die Oberfläächenkondensation des Wasserdampfes,” Zeitschrift des Vereines Deutscher Ingenieure, vol. 60, p. 541, 1916. View at Google Scholar
  66. M. Scholle and N. Aksel, “An exact solution of visco-capillary flow in an inclined channel,” Zeitschrift für Angewandte Mathematik und Physik, vol. 52, no. 5, pp. 749–769, 2001. View at Google Scholar · View at Scopus
  67. I. M. R. Sadiq and R. Usha, “Thin Newtonian film flow down a porous inclined plane: stability analysis,” Physics of Fluids, vol. 20, no. 2, Article ID 022105, 2008. View at Publisher · View at Google Scholar · View at Scopus
  68. K. A. Smith, “On convective instability induced by surface gradients,” Journal of Fluid Mechanics, vol. 24, p. 401, 1966. View at Google Scholar
  69. D. A. Goussis and R. E. Kelly, “On the thermocapillary instabilities in a liquid layer heated from below,” International Journal of Heat and Mass Transfer, vol. 33, p. 2237, 1990. View at Google Scholar
  70. S. J. Vanhook, M. F. Schatz, J. B. Swift, W. D. McCormick, and H. L. Swinney, “Long wave-length surface-tension-driven Bénard convection: experiment and theory,” Journal of Fluid Mechanics, vol. 345, pp. 45–78, 1997. View at Google Scholar · View at Scopus
  71. E. Kamke, Differentialgleichungen: Losungsmethoden Und Losungen, Band 1: Gewohnliche Differentialgleichungen, Akademische Verlagsgesellschaft Geest & Portig, Leipzig, Germany, 1961.
  72. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, New York, NY, USA, 1988.
  73. L. N. Trefthen, Spectral Methods in MATLAB, SIAM, Philadelphia, Pa, USA, 2000.
  74. A. Pumir, P. Manneville, and Y. Pomeau, “On solitary waves running down an inclined plane,” Journal of Fluid Mechanics, vol. 135, pp. 27–50, 1983. View at Google Scholar · View at Scopus
  75. S. V. Alekseenko, V. Y. Nakoryakov, and B. G. Pokusaev, “Wave formation on a vertical falling liquid film,” AIChE, vol. 31, no. 9, pp. 1446–1460, 1985. View at Google Scholar · View at Scopus
  76. C. Heining and N. Aksel, “Bottom reconstruction in thin-film flow over topography: steady solution and linear stability,” Physics of Fluids, vol. 21, no. 8, Article ID 083605, 2009. View at Publisher · View at Google Scholar · View at Scopus
  77. A. Mazouchi and G. M. Homsy, “Free surfaces stokes flow over topography,” Physics of Fluids, vol. 13, no. 10, pp. 2751–2761, 2001. View at Publisher · View at Google Scholar · View at Scopus
  78. S. Nadeem, N. S. Akbar, T. Hayat, and A. A. Hendi, “Peristaltic flow of Walter's B fluid in endoscope,” Applied Mathematics and Mechanics, vol. 32, no. 6, pp. 689–700, 2011. View at Publisher · View at Google Scholar · View at Scopus
  79. B. Scheid, S. Kalliadasis, C. Ruyer-Quil, and P. Colinet, “Interaction of three-dimensional hydrodynamic and thermocapillary instabilities in film flows,” Physical Review, vol. 78, no. 6, Article ID 066311, 2008. View at Publisher · View at Google Scholar · View at Scopus
  80. N. Amatousse, H. A. Abderrahmane, and N. Mehidi, “Traveling waves on a falling weakly viscoelastic fluid film,” International Journal of Engineering Science, vol. 54, pp. 27–41, 2012. View at Publisher · View at Google Scholar · View at Scopus