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Linear temporal and spatio-temporal stability analysis of a binary liquid film flowing down an inclined uniformly heated plate

Published online by Cambridge University Press:  06 March 2008

JUN HU ,
HAMDA BEN HADID ,
DANIEL HENRY  and
ABDELKADER MOJTABI
JUN HU
Affiliation:
Institute of Applied Physics and Computational Mathematics, Beijing 100088, [email protected]
HAMDA BEN HADID
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, CNRS/Université de Lyon, Ecole Centrale de Lyon/Université Lyon 1/INSA de Lyon, ECL, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France
DANIEL HENRY
Affiliation:
Laboratoire de Mécanique des Fluides et d'Acoustique, CNRS/Université de Lyon, Ecole Centrale de Lyon/Université Lyon 1/INSA de Lyon, ECL, 36 avenue Guy de Collongue, 69134 Ecully Cedex, France
ABDELKADER MOJTABI
Affiliation:
IMFT, UMR CNRS/INP/UPS 5502, UFR MIG, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex, France
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Abstract

Temporal and spatio-temporal instabilities of binary liquid films flowing down an inclined uniformly heated plate with Soret effect are investigated by using the Chebyshev collocation method to solve the full system of linear stability equations. Seven dimensionless parameters, i.e. the Kapitza, Galileo, Prandtl, Lewis, Soret, Marangoni, and Biot numbers (Ka, G, Pr, L, χ, M, B), as well as the inclination angle (β) are used to control the flow system. In the case of pure spanwise perturbations, thermocapillary S- and P-modes are obtained. It is found that the most dangerous modes are stationary for positive Soret numbers (χ≥0), and oscillatory for χ<0. Moreover, the P-mode which is short-wave unstable for χ=0 remains so for χ<0, but becomes long-wave unstable for χ>0 and even merges with the long-wave S-mode. In the case of streamwise perturbations, a long-wave surface mode (H-mode) is also obtained. From the neutral curves, it is found that larger Soret numbers make the film flow more unstable as do larger Marangoni numbers. The increase of these parameters leads to the merging of the long-wave H- and S-modes, making the situation long-wave unstable for any Galileo number. It also strongly influences the short-wave P-mode which becomes the most critical for large enough Galileo numbers. Furthermore, from the boundary curves between absolute and convective instabilities (AI/CI) calculated for both the long-wave instability (S- and H-modes) and the short-wave instability (P-mode), it is shown that for small Galileo numbers the AI/CI boundary curves are determined by the long-wave instability, while for large Galileo numbers they are determined by the short-wave instability.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 599 , 25 March 2008 , pp. 269 - 298
Copyright
Copyright © Cambridge University Press 2008

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References

REFERENCES

Benjamin, T. B. 1957 Wave formulation in laminar flow down an inclined plane. J. Fluid Mech. 2, 554–574.CrossRefGoogle Scholar
Benney, D. J. 1966 Long waves on liquid films. J. Math. Phys. 45, 150–155.CrossRefGoogle Scholar
Bers, A. 1973 Theory of absolute and convective instabilities. In Survey Lectures, Proceedings of the International Congress on Waves and Instabilities in Plasmas (ed. Auer, G. & Cap, F.), pp. B1–B52. Institute for Theoretical Physics, Innsbruck, Austria.Google Scholar
Brevdo, L., Laure, P., Dias, F. & Bridges, T. J. 1999 Linear pulse structure and signalling in a film flow on an inclined plane. J. Fluid Mech. 396, 37–71.CrossRefGoogle Scholar
Briggs, R. J. 1964 Electron-Stream Interaction with Plasmas. MIT Press.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer-Verlag.CrossRefGoogle Scholar
Chang, H.-C. 1994 Wave evolution on a falling film. J. Fluid Mech. 26, 103–136.CrossRefGoogle Scholar
Deissler, R. J. 1987 Spatially growing waves, intermittency, and convective chaos in an open-flow system. Physica D 25, 233.Google Scholar
Gjevik, B. 1970 Occurrence of finite-amplitude surface waves on falling liquid films. Phys. Fluids 13, 1918.CrossRefGoogle Scholar
Goussis, D. A. & Kelly, R. E. 1990 On the thermocapillary instabilities in a liquid layer heated from below. Intl J. Heat Mass Transfer 33, 2237–2245.CrossRefGoogle Scholar
Goussis, D. A. & Kelly, R. E. 1991 Surface wave and thermocapillary instabilities in a liquid film flow. J. Fluid Mech. 223, 25–45.CrossRefGoogle Scholar
de Groot, S. R. & Mazur, P. 1969 Non-Equilibrium Thermodynamics. North-Holland.Google Scholar
Hu, J., Ben Hadid, H. & Henry, D. 2007 Linear stability analysis of Poiseuille-Rayleigh-Bénard flows in binary fluids with Soret effect. Phys. Fluids 19, 034101.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1985 Absolute and convective instabilities in free shear layers. J. Fluid Mech. 159, 151–168.CrossRefGoogle Scholar
Huerre, P. & Monkewitz, P. A. 1990 Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech. 22, 473.CrossRefGoogle Scholar
Joo, S. W. 1995 Marangoni instabilities in liquid mixtures with Soret effects. J. Fluid Mech. 293, 127–145.CrossRefGoogle Scholar
Kapitza, P. L. & Kapitza, S. P. 1949 Wave flow of thin layers of a viscous fluid. Zh. Eksper. Teor. Fiz. 19, 105–120, also in Collected Papers of P. L. Kapitza (ed. Haar, D. Ter), pp. 690–709. Pergamon, 1965.Google Scholar
Kelly, R. E., Davis, S. H. & Goussis, D. A. 1986 On the instability of heated film flow with variable surface tension. In Heat Transfer 1986, Proc. 9th Intl Heat Transfer Conference, San Francisco, vol. 4, pp. 1937–1942. Hemisphere.Google Scholar
Kelly, R. E., Goussis, D. A., Lin, S. P. & Hsu, F. K. 1989 The mechanism for surface wave instability in film flow down an inclined plane. Phys. Fluids A 1, 819–828.CrossRefGoogle Scholar
Lin, S. P. 1969 Finite amplitude stability of a parallel flow with a free surface. J. Fluid Mech. 36, 113–126.CrossRefGoogle Scholar
Liu, J., Paul, J. D. & Gollub, J. P. 1993 Measurements of the primary instabilities of film flows. J. Fluid Mech. 250, 69–101.CrossRefGoogle Scholar
Ooshida, T. 1999 Surface equation of falling film flows with moderate Reynolds number and large but finite Weber number. Phys. Fluids 11, 3247.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Mod Rev Phys 69, 931–980.CrossRefGoogle Scholar
Oron, A. & Gottlieb, O. 2002 Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids 14, 2622.CrossRefGoogle Scholar
Pearlstein, A. J. & Goussis, D. A. 1988 Efficient transformation of certain singular polynomial matrix eigenvalue problems. J. Comput. Phys. 78, 305–312.CrossRefGoogle Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.CrossRefGoogle Scholar
Podolny, A., Oron, A. & Nepomnyashchy, A. A. 2005 Long-wave Marangoni instability in a binary-liquid layer with deformable interface in the presence of Soret effect: Linear theory. Phys. Fluids 17, 104104.CrossRefGoogle Scholar
Podolny, A., Oron, A. & Nepomnyashchy, A. A. 2006 Linear and nonlinear theory of long-wave Marangoni instability with the Soret effect at finite Biot numbers. Phys. Fluids 18, 054104.CrossRefGoogle Scholar
Pumir, A., Manneville, P. & Pomeau, Y. 1983 On solitary waves running down an inclined plane. J. Fluid Mech. 135, 27–50.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2000 Improved modeling of flows down inclined planes. Eur. Phys. J. B 15, 357.CrossRefGoogle Scholar
Ruyer-Quil, C. & Manneville, P. 2002 Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids 14, 170.CrossRefGoogle Scholar
Scheid, B. Ruyer-Quil, C. Thiele, U. Kabov, O. A. Legros, J. C. & Colinet, P. 2005 Validity domain of the Benney equation including Marangoni effect for closed and open flows. J. Fluid Mech. 527, 303–335.CrossRefGoogle Scholar
Scriven, L. E. & Sternling, C. V. 1964 On cellular convection driven by surface-tension gradients: Effects of mean surface tension and surface viscosity. J. Fluid Mech. 19, 321–340.CrossRefGoogle Scholar
Shkadov, V. 1967 Wave flow regimes of a thin layer of viscous fluid subject to gravity. Isv. Akad. Nauk SSSR 2, 43.Google Scholar
Smith, K. A. 1966 On convective instability induced by surface-tension gradients. J. Fluid Mech. 24, 401–414.CrossRefGoogle Scholar
Sreenivasan, S. & Lin, S. P. 1978 Surface tension driven instability of a liquid film down a heated incline. Intl J. Heat Mass Transfer 21, 1517.CrossRefGoogle Scholar
Takashima, M. 1979 Surface tension driven instabillity in a horizontal layer of binary liquid mixture in the presence of the Soret effect. J. Phys. Soc. Japan 47, 1321–1326.CrossRefGoogle Scholar
Thiele, U. & Knobloch, E. 2004 Thin liquid films on a slightly inclined heated plate. Physica D 190, 213.Google Scholar
Yih, C. S. 1963 Stability of liquid flow down an inclined plane. Phys. Fluids 6, 321.CrossRefGoogle Scholar
Yin, X.-Y. Sun, D.-J. Wei, M.-J. & Wu, J.-Z. 2000 Absolute and convective instability character of slender viscous vortices. Phys. Fluids 12, 1062.CrossRefGoogle Scholar