Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-04T19:33:44.721Z Has data issue: false hasContentIssue false
  • English
  • Français

Generation of nonlinear Marangoni waves in a two-layer film by heating modulation

Published online by Cambridge University Press:  17 April 2015

Alexander Nepomnyashchy  and
Ilya Simanovskii
Alexander Nepomnyashchy
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, 32000, Haifa, Israel Minerva Center for Nonlinear Physics of Complex Systems, Technion – Israel Institute of Technology, 32000, Haifa, Israel
Ilya Simanovskii*
Affiliation:
Department of Mathematics, Technion – Israel Institute of Technology, 32000, Haifa, Israel
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Longwave Marangoni convection in two-layer films under the action of heating modulation is considered. The analysis is carried out in the lubrication approximation. The capillary forces are assumed to be sufficiently strong, and they are taken into account. Periodic or symmetric boundary conditions are applied on the boundaries of the computational region. Numerical simulations are performed by means of a finite-difference method. Two regions of parametric instabilities have been found. In the first region, one observes the competition or coexistence of standing waves parallel to the boundaries of the computational region. The multistability of the flow regimes is revealed. In the second region, the regimes found in the case of periodic boundary conditions are more diverse than in the case of symmetric boundary conditions.

JFM classification

Instability: Instability Interfacial Flows (free surface): Interfacial Flows (free surface) Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 771 , 25 May 2015 , pp. 159 - 192
Check for updates
Copyright
© 2015 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Birikh, R. V. 1966 Thermocapillary convection in horizontal layer of liquid. J. Appl. Mech. Tech. Phys. 7 (3), 43.CrossRefGoogle Scholar
Castillo, J. L. & Velarde, M. G. 1978 Thermal diffusion and the Marangoni–Bénard instability of a two-component fluid layer heated from below. Phys. Lett. A 66, 489.Google Scholar
Castillo, J. L. & Velarde, M. G. 1980 Microgravity and the thermoconvective stability of a binary liquid layer open to the ambient air. J. Non-Equilib. Thermodyn. 5, 111.Google Scholar
Colinet, P., Legros, J. C. & Velarde, M. G. 2001 Nonlinear Dynamics of Surface-Tension Driven Instabilities. Wiley.Google Scholar
Cross, M. C., Daniels, P. G., Hohenberg, P. C. & Siggia, E. D. 1980 Effect of distant sidewalls on wave-number selection in Rayleigh–Bénard convection. Phys. Rev. Lett. 45, 898.Google Scholar
Czechowsky, L. & Floryan, J. M. 2001 Marangoni instability in a finite container – transition between short and long wavelengths modes. Trans. ASME J. Heat Transfer 123, 96.CrossRefGoogle Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19, 403.Google Scholar
Fayzrakhmanova, I. S., Shklyaev, S. & Nepomnyashchy, A. A. 2013a Influence of low-frequency vibration on thermocapillary instability in a binary mixture with the Soret effect: long-wave versus short-wave perturbations. J. Fluid Mech. 714, 190.Google Scholar
Fayzrakhmanova, I. S., Shklyaev, S. & Nepomnyashchy, A. A. 2013b Influence of heat flux modulation on thermocapilary instability in a binary mixture with the Soret effect. In Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics (ed. Rubio, R. G. et al. ), pp. 133–144. Springer.CrossRefGoogle Scholar
Fayzrakhmanova, I. S., Shklyaev, S. & Nepomnyashchy, A. A. 2014 Longwave convection in a layer of binary mixture with modulated heat flux: weakly nonlinear analysis. Fluid Dyn. Res. 46, 041406.CrossRefGoogle Scholar
Fisher, L. S. & Golovin, A. A. 2005 Nonlinear stability analysis of a two-layer thin liquid film: dewetting and autophobic behavior. J. Colloid Interface Sci. 291, 515.Google Scholar
Floryan, J. M. & Chen, C. 1994 Thermocapillary convection and existence of continuous liquid layers in the absence of gravity. J. Fluid Mech. 277, 303.Google Scholar
Freund, G., Pesch, W. & Zimmerman, W. 2011 Rayleigh–Bénard convection in the presence of spatial temperature modulations. J. Fluid Mech. 673, 318.Google Scholar
Géoris, Ph., Hennenberg, M., Lebon, G. & Legros, J. C. 1999 Investigation of thermocapillary convection in a three-liquid-layer system. J. Fluid Mech. 389, 209.CrossRefGoogle Scholar
Gershuni, G. Z. & Lyubimov, D. V. 1998 Thermal Vibrational Convection. Wiley.Google Scholar
Gershuni, G. Z., Nepomnyashchy, A. A., Smorodin, B. L. & Velarde, M. G. 1994 On parametric excitation of thermocapillary and thermogravitational convective instability. Microgravity Q. 4, 215.Google Scholar
Gershuni, G. Z., Nepomnyashchy, A. A., Smorodin, B. L. & Velarde, M. G. 1996 On parametric excitation of Marangoni instability in a liquid layer with free deformable surface. Microgravity Q. 6, 203.Google Scholar
Gershuni, G. Z., Nepomnyashchy, A. A. & Velarde, M. G. 1992 On dynamics excitation of Marangoni instability. Phys. Fluids A 4, 2394.Google Scholar
Gershuni, G. Z. & Zhukhovitsky, E. M. 1963 On parametric excitation of convective instability. Z. Angew. Math. Mech. 27, 1197.CrossRefGoogle Scholar
Gershuni, G. Z., Zhukhovitsky, E. M. & Yurkov, Yu. S. 1970 On convective stability in the presence of periodically varying parameter. Z. Angew. Math. Mech. 34, 442.CrossRefGoogle Scholar
Hamed, M. & Floryan, J. M. 2000 Marangoni convection. Part 1. A cavity with differentially heated sidewalls. J. Fluid Mech. 405, 79.Google Scholar
Hossain, M. Z. & Floryan, J. M. 2013 Instabilities of natural convection in a periodically heated layer. J. Fluid Mech. 733, 33.Google Scholar
Kapitza, P. L. 1951 Dynamic stability of a pendulum when its point of suspension vibrates. Sov. Phys. JETP 21, 588.Google Scholar
Nepomnyashchy, A. A. & Simanovskii, I. B. 2007 Marangoni instability in ultrathin two-layer films. Phys. Fluids 19, 122103.Google Scholar
Nepomnyashchy, A. & Simanovskii, I. 2012 Nonlinear Marangoni waves in a two-layer film in the presence of gravity. Phys. Fluids 24, 032101.CrossRefGoogle Scholar
Nepomnyashchy, A. & Simanovskii, I. 2013 The influence of vibration on Marangoni waves in two-layer films. J. Fluid Mech. 726, 476.Google Scholar
Nepomnyashchy, A., Simanovskii, I. & Legros, J. C. 2012 Interfacial Convection in Multilayer Systems, 2nd edn. Springer.Google Scholar
Or, A. C. & Kelly, R. E. 1999 Time-modulated convection with zero-mean temperature gradient. Phys. Rev. E 60, 1741.Google Scholar
Or, A. C. & Kelly, R. E. 2002 The effects of thermal modulation upon the onset of Marangoni–Bénard convection. J. Fluid Mech. 456, 161.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931.CrossRefGoogle Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489.CrossRefGoogle Scholar
Prakash, A. & Koster, J. N. 1994 Convection in multiple layers of immiscible liquids in a shallow cavity – I. Steady natural convection. Intl J. Multiphase Flow 20, 383.Google Scholar
Pshenichnikov, A. & Tokmenina, G. A. 1983 Deformation of the free surface of a liquid by thermocapillary motion. Fluid Dyn. 18, 463.CrossRefGoogle Scholar
Rosenblat, S. & Tanaka, G. A. 1971 Modulation of thermal convection instability. Phys. Fluids 14, 1319.Google Scholar
Scriven, L. E. & Sternling, C. V. 1964 On cellular convection driven by surface-tension gradients: effect of mean surface tension and surface viscosity. J. Fluid Mech. 19, 321.Google Scholar
Sen, A. K. & Davis, S. H. 1982 Steady thermocapillary flows in two-dimensional slots. J. Fluid Mech. 121, 163.Google Scholar
Simanovskii, I. B. & Nepomnyashchy, A. A. 1993 Convective Instabilities in Systems with Interface. Gordon and Breach.Google Scholar
Smith, M. K. & Davis, S. H. 1983a Instabilities of dynamic thermocapillary liquid layers. Part 1. Convective instabilities. J. Fluid Mech. 132, 119.CrossRefGoogle Scholar
Smith, M. K. & Davis, S. H. 1983b Instabilities of dynamic thermocapillary liquid layers. Part 2. Surface-wave instabilities. J. Fluid Mech. 132, 145.Google Scholar
Smorodin, B. L. & Lüecke, M. 2009 Convection in binary fluid mixtures with modulated heating. Phys. Rev. E 79, 026315.Google Scholar
Smorodin, B. L. & Lüecke, M. 2010 Binary fluid mixture convection with low frequency modulated heating. Phys. Rev. E 82, 016310.Google Scholar
Smorodin, B. L., Mikishev, A. B., Nepomnyashchy, A. A. & Myznikova, B. I. 2009 Thermocapillary instability of a liquid layer under heat flux modulation. Phys. Fluids 21, 062102.Google Scholar
Swift, J. B. & Hohenberg, P. C. 1987 Modulated convection at high frequencies and large modulation amplitudes. Phys. Rev. A 36, 4870.Google Scholar
Van Hook, S. J., Schatz, M. F., McCormick, W. D., Swift, J. B. & Swinney, H. L. 1995 Long-wavelength instability in surface-tension-driven Bénard convection. Phys. Rev. Lett. 75, 4397.Google Scholar
Van Hook, S. J., Schatz, M. F., Swift, J. B., McCormick, W. D. & Swinney, H. L. 1997 Long-wavelength instability in surface-tension-driven Bénard convection: experiment and theory. J. Fluid Mech. 345, 45.Google Scholar
Venezian, G. 1969 Effect of modulation on the onset of thermal convection. J. Fluid Mech. 35, 243.Google Scholar
Yih, C. S. & Li, C. H. 1972 Instability of unsteady flows or configurations. Part 2. Convective instability. J. Fluid Mech. 54, 143.Google Scholar
Zaleski, S. 1984 Cellular patterns with boundary forcing. J. Fluid Mech. 149, 101.Google Scholar
Zhou, B., Liu, Q. & Tang, Z. 2004 Rayleigh–Marangoni–Bénard instability in two-layer fluid system. Acta Mechanica Sin. 20, 366.Google Scholar
Zuev, A. L. & Pshenichnikov, A. F. 1987 Deformation and breakup of a liquid film under the action of thermocapillary convection. J. Appl. Mech. Tech. Phys. 28, 399.Google Scholar