Instabilities of Thin Viscous Liquid Film Flowing down a Uniformly Heated Inclined Plane

Document Type : Full Length Research Article

Authors

1 Vivekananda Mahavidyalaya, Burdwan, W.B., India.

2 South Asian University, Akbar Bhavan, Chanakyapuri, New Delhi-110021, India.

3 Vivekananda Mahavidyalaya, Burdwan,West Bengal, India

Abstract

Instabilities of a thin viscous film flowing down a uniformly heated plane are investigated in this study. The heating generates a surface tension gradient that induces thermocapillary stresses on the free surface. Thus, the film is not only influenced by gravity and mean surface tension but also the thermocapillary force is acting on the free surface. Moreover, the heat transfer at the free surface plays a crucial role in the evolution of the film. The main objective of this study is to scrutinize the impact of Biot number Bi which describes heat transfer at the free surface on instability mechanism. Using the long wave expansion method, a generalized non-linear evolution equation of Benney type, including the above mentioned effects, is derived for the development of the free surface. A normal mode approach and the method of multiple scales are used to obtain the linear and weakly nonlinear stability solution for the film flow. The linear stability analysis of the evolution equation shows that the Biot number plays a double role; for Bi < 1 it gives destabilizing effect but for Bi > 1 it produces stabilization. At Bi = 1, the instability is maximum. The weakly nonlinear study reveals that the impact of Marangoni number Mr is very strong on the bifurcation scenario even for its slight variation.

This behaviour of the Biot number is the consequence of the fact that the interfacial
temperature is held close to the plane temperature for Bi > 1, thus weakening the Marangoni effect. The weakly nonlinear
study reveals that the impact of Marangoni number Mr is very strong on the bifurcation scenario even for its slight variation.

Keywords

Main Subjects

References

References
[1] J. Marra, J. A. M. Huethorst, Physical principles of Marangoni drying, Langmuir, 7, 2748-2755 (1991).
[2]          S. B. G. M. O’ Brien, On Marangoni drying : nonlinear kinematic waves in a thin film, J.Fluid Mech., 254, 649-670 (1993).
[3]          A. Oron, Nonlinear dynamics of thin evaporating liquid films subject to internal heat generation, In Fluid Dynamics at Interfaces (ed. W. shyy and R. Narayanan), Cambridge University Press, (1999).
[4]          A. A. Nepomnyashchy, M. G. Velarde, P. Colinet, Interfacial phenomena and convection, Chapman and Hall, (2002).
[5]          P. Colinet, J. C. Legros, M. G. Velarde, Nonlinear dynamics of surface-tension driven instabilities, Wiley VCH, (2001).
[6]          M. G. Velarde, R. Kh. Zeytounian, Interfacial Phenomena and the Marangoni effect, Springer, (2002).
[7]          D. A. Goussis, R. E. Kelly, Surface wave and thermocapillary instabilities in a liquid film flow, J. Fluid Mech., 223, 25-45, (1991) and corrigendum J. Fluid. Mech. 226, 663- (1991).
[8] S. W. Joo, S. H. Davis, S. G. Bankoff, Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers, J. Fluid Mech., 230, 117-146 (1991).
[9]          D. J. Benny, Long waves on liquid films, J. Math. Phys., 45, 150-155 (1966).
[10]        W. Boos, A. Thess, Cascade of structures in long-wavelength Marangoni instability, Phys.Fluids, [1], 1484-1494 (1999).
[11]        S. Kalliadasis, E. A. Demekhin, C. Ruyer-Quil, M. G. Velarde, Thermocapillary instability and wave formation on a film falling down a uniformly heated plane, J. Fluid Mech., 492, 303-338 (2003).
[12]        P. M. J. Trevelyan, S. Kalliadasis, Wave dynamics on a thin-liquid film falling down a heated wall, J. Engineering Mathematics, 50, 177-208 (2004).
[13]        B. Scheid, C. Ruyer-Quil, U. Thiele, O. A. Kabov, J. Legros, P. Colinet, Validity domain of the Benney equation including Marangoni effect for closed and open flows, J. Fluid Mech., 527, 303-335 (2005).
[14]        C. Ruyer-Quil, B. Scheid, S. Kalliadasis, M. G. Velarde, R. Kh. Zeytounian, Thermocapil-lary long waves in a liquid film flow, Part 1 Low-dimensional formulation, J. Fluid Mech., 538, 199-222 (2005).
[15]        B. Scheid, C. Ruyer-Quil, S. Kalliadasis, M. G. Velarde, R. Kh. Zeytounian, Thermocapil-lary long waves in a liquid film flow, Part 2 Linear stability and nonlinear waves, J. Fluid Mech., 538, 223-244 (2005).
[16]        S. Miladinova, S. Slavtchev, G. Lebon, J. C. Legros, Long-wave instabilities of non-uniform heated falling films, J. Fluid Mech., 453, 153-175 (2002).
[17]        S. Miladinova, D. Staykova, G. Lebon, B. Scheid, Effect of non-uniform wall heating on the three-dimensional secondary instability of falling films, Acta Mechanica, 30, 1-13 (2002).
[18]        A. Mukhopadhyay, A. Mukhopadhyay, Nonlinear stability of viscous film flowing down an inclined plane with linear temperature variation, J. Phys. D: Appl. Phys., 40, 1-8 (2007).
[19]        Y. L. Yeo, V. R. Craster, O. K. Matar, Marangoni instability of a thin liquid film resting on a locally heated horizontal wall, Physical Review E, 67, 056315 1-14 (2003).
[20]        S. Saprykin, P. M. J. Trevelyan, R. J. Koopmans, S. Kalliadasis, Free surface thin film flows over uniformly heated topography, Physical Review E, 75, 026306 1-17 (2007).
[21]        H. C. Chang, Wave evolution on a falling film, Annu. Rev. Fluid Mech., 26, 103-136 (1994).
[22] .      L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Springer, (1997).
[23]        B. S. Dandapat, A. Samanta, Bifurcation analysis of first and second order Benney equa-tions for viscoelastic fluid flowing down a vertical plane, J. Phys. D: Appl. Phys., 41, 095501(9PP) (2008).

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History

  • Receive Date: 27 January 2015
  • Revise Date: 13 July 2015
  • Accept Date: 30 December 2015

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