Summary
Secondary three-dimensional instabilities of nearly sinusoidal waves on vertically falling and nonuniformly heated films are studied by using a long-wave evolution equation. Two-dimensional waves are unstable with respect to transverse modulations with sufficiently long spanwise wavelength. Two distinct three-dimensional modes of instability are examined: a synchronous mode which does not alter the wave number of the basic two-dimensional waves and a subharmonic mode with one-half of the streamwise wave number. According to a Floquet analysis, the subharmonic instability is most likely to be dominant for streamwise wavenumbers close to the neutral curve. The three-dimensional instability mechanism depends on film heating. The secondary growth rate increases (decreases) with increasing (decreasing) film heating downstream, but the contribution of thermocapillarity to synchronous and subharmonic instabilities is different.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Chang, H.-C.: Wave evolution on a falling film. Ann. Rev. Fluid Mech.26, 103–136 (1994).
Alekseenko, S. V., Nakoryakov, V. Ye., Pokusaev, B. G.: Wave flow of liquid films. New York: Begell House 1994.
Joo, S. W., Davis, S. H.: Instabilities of three-dimensional viscous falling films. J. Fluid Mech.242, 529–547 (1992).
Joo, S. W., Davis, S. H.: Irregular waves on viscous falling films. Chem. Eng. Comm.118, 111–123 (1992).
Chang, H.-C., Demekhin, E. A., Kopelevich, D. I.: Nonlinear evolution of waves on a vertically falling film. J. Fluid Mech.250, 433–480 (1993).
Liu, J., Paul, J. D., Gollub, J. P.: Measurements of the primary instabilities of film flows. J. Fluid Mech.250, 69–101 (1993).
Liu, J., Schneider, J. B., Gollub, J. P.: Three dimensional instabilities of film flows. Phys. FluidsA7, 55–67 (1995).
Joo, S. W., Davis, S. H., Bankoff, S. G.: Long-wave instabilities of heated falling films: two-dimensional theory of uniform layers. J. Fluid Mech.230, 117–146 (1991).
Joo, S. W., Davis, S. H., Bankoff, S. G.: A mechanism for rivulet formation in heated falling films. J. Fluid Mech.321, 279–298 (1996).
Oron, A., Davis, S. H., Bankoff, S. G.: Long-scale evolution of thin liquid films. Rev. Modern Phys.69, 931–980 (1997).
Slavtchev, S., Miladinova, S., Lebon, G., Legros, J.-C.: Marangoni effect on the instability of nonuniformly heated falling liquid films. In: Proc. 1st Intern. Symp. on Microgravity Research & Appl. in Phys. Sciences & Biotechnology, Sorrento, September 2000, (SP-454),1, 33–39 (2001).
Kalitzova-Kurteva, P., Slavtchev, S., Kurtev, I.: Linear instability in liquid layers on an inclined, non-uniformly heated wall. Theor. Appl. Mech.30, 12–23 (2000).
Miladinova, S., Slavtchev, S., Lebon, G., Legros, J.-C.: Long wave instabilities of non-uniformly heated falling films. (to appear).
Benney, D. J.: Long waves on liquid films. J. Math. Phys.45, 150–155 (1966).
Kabov, O. A., Marchuk, I. V., Chupin, V. M.: Thermal imaging study of the liquid film flowing on a vertical surface with local heat source. Russ. J. Eng. Thermophys.6, 104–138 (1996).
Scheid, B., Kabov, O., Minetti, C., Colinet, P., Legros, J.-C.: Measurement of free surface deformation by reflectance-Schlieren method. In: Proc 3rd European Thermal-Science Conf., Heidelberg,2, 651–657 (2000).
Herbert, Th.: Secondary instability of boundary layers. Ann. Rev. Fluid Mech.20, 487–526 (1988).
Smith, B. T.: Matrix eigensystem routines: Eispack Guide. Springer 1976.
Kabov, O. A.: Formation of regular structures in a falling liquid film upon local heating. Thermophysics and Aeromechanics5, 547–551 (1998).
Kabov, O. A., Marchuk, I. V., Muzykantov, A. V., Legros, J.-C., Istasse, E., Dewandel, J. L.: Regular structures in locally heated falling liquid films. In: Proc. 2nd Int. Symp. on Two-Phase Flow Modelling and Experimentation (Celata, G. P., Di Marco, P., Shah, R. K., eds.)5, 1225–1233, Pisa 1999.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Miladinova, S., Staykova, D., Lebon, G. et al. Effect of nonuniform wall heating on the three-dimensional secondary instability of falling films. Acta Mechanica 156, 79–91 (2002). https://doi.org/10.1007/BF01188743
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01188743