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The instability of liquid films with temperature-dependent properties flowing down a heated incline

1 Department of Mathematics, Ryerson University, Toronto, Ontario, M5B 2K3, Canada
2 Centre for Education in Mathematics and Computing, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada

Special Issues: Applied and Industrial Mathematics in Canada and Worldwide

We investigate the stability of free-surface flow on a heated incline. We develop a complete mathematical model for the flow which captures the Marangoni effect and also accounts for changes in the properties of the fluid with temperature. We apply a linear stability analysis to determine the stability of the steady and uniform flow. The associated eigenvalue problem is solved numerically by means of a spectral collocation method. Comparisons with nonlinear simulations are also made and the agreement is good.
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1. P. L. Kapitza, Wave flow of thin layers of a viscous fluid: II. Fluid flow in the presence of continuous gas flow and heat transfer, Zh. Eksp. Teor. Fiz., 18 (1948), 19-28.

2. P. L. Kapitza and S. P. Kapitza, Wave flow of thin layers of a viscous fluid: III. Experimental study of undulatory flow conditions, Zh. Eksp. Teor. Fiz., 19 (1949), 105-120.

3. T. B. Benjamin, Wave formation in laminar flow down an inclined plane, J. Fluid Mech., 2 (1957), 554-574.    

4. C.-S. Yih, Stability of liquid flow down an inclined plane, Phys. Fluids, 6 (1963), 321-334.    

5. D. J. Benney, Long waves on liquid films, J. Math. and Phys., 45 (1966), 150-155.

6. B. Gjevik, Occurrence of Finite-Amplitude Surface Waves on Falling Liquid Films, Phys. Fluids, 13 (1970), 1918-1925.

7. S. Kalliadasis, E. A. Demekhin, C. Ruyer-Quil, et al. Falling liquid films, Springer-Verlag, London, 2012.

8. S. Kalliadasis, E. A. Demekhin, C. Ruyer-Quil, et al. Thermocapillary instability and wave formation on a film flowing down a uniformly heated plane, J. Fluid Mech., 492 (2003), 303-338.    

9. C. Ruyer-Quil, B. Scheid, S. Kalliadasis, et al. Thermocapillary long waves in a liquid film flow. part 1. low-dimensional formulation, J. Fluid Mech., 538 (2005), 199-222.

10. B. Scheid, C. Ruyer-Quil, S. Kalliadasis, et al. Thermocapillary long waves in a liquid film flow. part 2. linear stability and nonlinear waves, J. Fluid Mech., 538 (2005), 223-244.

11. P. Trevelyan, B. Scheid, C. Ruyer-Quil, et al. Heated falling films, J. Fluid Mech., 592 (2007), 295-334.

12. S. Lin, Stability of a liquid flow down a heated inclined plane, Lett. Heat Mass Transfer, 2 (1975), 361-369.    

13. Y. O. Kabova and V. V. Kuznetsov, Downward flow of a nonisothermal thin liquid film with variable viscosity, J. Appl. Mech. Tech. Phys., 43 (2002), 895-901.

14. D. A. Goussis and R. A. Kelly, Effects of viscosity on the stability of film flow down heated or cooled inclined surfaces: Long-wavelength analysis, Phys. Fluids, 28 (1985), 3207-3214.

15. C.-C. Hwang and C.-I. Weng, Non-linear stability analysis of film flow down a heated or cooled inclined plane with viscosity variation, Int. J. Heat Mass Transfer, 31 (1988), 1775-1784.

16. K. A. Ogden, S. J. D. D'Alessio, J.P. Pascal, Gravity-driven flow over heated, porous, wavy surfaces, Phys. Fluids, 23 (2011), 122102.

17. J. P. Pascal, N. Gonputh and S. J. D. D'Alessio, Long-wave instability of flow with temperature dependent fluid properties down a heated incline, Intl J. Eng. Sci., 70 (2013), 73-90.    

18. S. J. D. D'Alessio, C. J. M. P. Seth and J. P. Pascal, The effects of variable fluid properties on thin film stability, Phys. Fluids, 26 (2014), 122105.

19. M. Chhay, D. Dutykhb, M. Gisclonb, et al. New asymptotic heat transfer model in thin liquid films, Appl. Math. Model., 48 (2017), 844-859.

20. J. H. Spurk and N. Aksel, Fluid Mechanics, 2nd edition, Springer-Verlag, Berlin, 2008.

21. C. Ruyer-Quil and P. Manneville, Improved modeling of flows down inclined planes, Eur. Phys. J. B, 15 (2000), 357-369.

22. C. Ruyer-Quil and P. Manneville, Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations, Phys. Fluids, 14 (2002), 170-183.

23. N. Gonputh, J. P. Pascal and S. J. D. D'Alessio, Inclined flow of a heated fluid film with temperature dependent fluid properties, In: Proceedings of the 2012 International Conference on Scientific Computing, pp. 247-253, CSREA Press, 2012.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

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