Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-23T09:17:21.845Z Has data issue: false hasContentIssue false
  • English
  • Français

Conjugated liquid layers driven by the short-wavelength Bénard–Marangoni instability: experiment and numerical simulation

Published online by Cambridge University Press:  13 October 2015

Iman Nejati ,
Mathias Dietzel  and
Steffen Hardt
Iman Nejati
Affiliation:
Institute for Nano- and Microfluidics, Center of Smart Interfaces, TU Darmstadt, Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany
Mathias Dietzel*
Affiliation:
Institute for Nano- and Microfluidics, Center of Smart Interfaces, TU Darmstadt, Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany
Steffen Hardt
Affiliation:
Institute for Nano- and Microfluidics, Center of Smart Interfaces, TU Darmstadt, Alarich-Weiss-Strasse 10, 64287 Darmstadt, Germany
*
†Email address for correspondence: dietzel@csi.tu-darmstadt.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The coupled dynamics of two conjugated liquid layers of disparate thicknesses, which coat a solid substrate and are subjected to a transverse temperature gradient, is investigated. The upper liquid layer evolves under the short-wavelength Bénard–Marangoni instability, whereas the lower, much thinner film undergoes a shear-driven long-wavelength deformation. Although the lubricating film should reduce the viscous stresses acting on the up to one hundred times thicker upper layer by only 10 %, it is found that the critical Marangoni number of marginal stability may be as low as if a stress-free boundary condition were applied at the bottom of the upper layer, i.e. much lower than the classical value of 79.6 known for a single film. Furthermore, it is experimentally verified that the deformation of the liquid–liquid interface, albeit small, has a non-negligible effect on the temperature distribution along the liquid–gas interface of the upper layer. This stabilizes the hexagonal pattern symmetry towards external disturbances and indicates a two-way coupling of the different layers. The experiments also demonstrate how convection patterns formed in a liquid film can be used to pattern a second conjugated film. The experimental findings are verified by a numerical model of the coupled layers.

JFM classification

Convection: Marangoni convection Nonlinear Dynamical Systems: Pattern formation Interfacial Flows (free surface): Thin films
Type
Papers
Information
Journal of Fluid Mechanics , Volume 783 , 25 November 2015 , pp. 46 - 71
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

Andereck, C. D., Colovas, P. W., Degen, M. M. & Renardy, Y. Y. 1998 Instabilities in two layer Rayleigh–Bénard convection: overview and outlook. Intl J. Engng Sci. 36 (1214), 1451–1470.Google Scholar
Bénard, H. 1900 Étude expérimentale des courants de convection dans une nappe liquide. Régime permanent: tourbillons cellulaires. J. Phys. Theor. Appl. 9 (1), 513–524.Google Scholar
Bestehorn, M. & Busse, F. H. 2006 Hydrodynamik und Strukturbildung: Mit einer kurzen Einführung in die Kontinuumsmechanik. Physica.Google Scholar
Boeck, T. & Thess, A. 1997 Inertial Bénard–Marangoni convection. J. Fluid Mech. 350, 149–175.Google Scholar
Burgess, J. M., Juel, A., McCormick, W. D., Swift, J. B. & Swinney, H. L. 2001 Suppression of dripping from a ceiling. Phys. Rev. Lett. 86 (7), 1203–1206.Google Scholar
Chandrasekhar, S. 1970 Hydrodynamic and Hydromagnetic Stability. Clarendon Press.Google Scholar
Colinet, P. & Legros, J. C. 1994 On the Hopf bifurcation occurring in the two layer Rayleigh–Bénard convective instability. Phys. Fluids 6 (8), 2631–2639.Google Scholar
Colinet, P., Legros, J. C. & Velarde, M. G. 2005 Instability Modes in Bénard Layers. pp. 39–84. Wiley-VCH.Google Scholar
Colinet, P. & Nepomnyashchy, A. 2010 Pattern Formation at Interfaces. Springer.Google Scholar
Comsol 2014 Comsol Multiphysics® . COMSOL, Inc.Google Scholar
Cross, M. & Hohenberg, P. 1993 Pattern formation outside of equilibrium. Rev. Mod. Phys. 65, 851–1112.Google Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19 (1), 403–435.Google Scholar
Degen, M. M., Colovas, P. W. & Andereck, C. D. 1998 Time-dependent patterns in the two-layer Rayleigh–Bénard system. Phys. Rev. E 57, 6647–6659.Google Scholar
Dietzel, M. & Troian, S. M. 2010 Mechanism for spontaneous growth of nanopillar arrays in ultrathin films subject to a thermal gradient. J. Appl. Phys. 108 (7), 074308.Google Scholar
Dutton, T. W., Pate, L. R. & Hollingsworth, D. K. 2010 Imaging of surface-tension-driven convection using liquid crystal thermography. J. Heat Transfer 132 (12), 121601.Google Scholar
Eckert, K., Bestehorn, M. & Thess, A. 1998 Square cells in surface-tension-driven Bénard convection: experiment and theory. J. Fluid Mech. 356, 155–197.Google Scholar
Golovin, A. A., Nepomnyashchy, A. A. & Pismen, L. M. 1994 Interaction between short-scale Marangoni convection and long-scale deformational instability. Phys. Fluids 6 (1), 34–48.Google Scholar
Hossain, M. Z. & Floryan, J. M. 2014 Natural convection in a fluid layer periodically heated from above. Phys. Rev. E 90, 023015.Google Scholar
Hurle, D. T. J. 1981 Surface aspects of crystal growth from the melt. Adv. Colloid Interface 15 (2), 101–130.Google Scholar
Israelachvili, J. N. 2011 Intermolecular and Surface Forces: Revised Third Edition. Elsevier Science.Google Scholar
Kang, Q., Zhang, J. F., Hu, L. & Duan, L. 2003 Experimental study on Bénard–Marangoni convection by PIV and TCL. Proc. SPIE 5058, 155–161.Google Scholar
Koschmieder, E. L. & Biggerstaff, M. I. 1986 Onset of surface-tension-driven Bénard convection. J. Fluid Mech. 167, 49–64.Google Scholar
McLeod, E., Liu, Y. & Troian, S. M. 2011 Experimental verification of the formation mechanism for pillar arrays in nanofilms subject to large thermal gradients. Phys. Rev. Lett. 106 (17), 175501.Google Scholar
Merkt, D. & Bestehorn, M. 2012 Pattern formation in anticonvective systems. Fluid Dyn. Res. 44 (3), 031413.Google Scholar
Merkt, D., Pototsky, A., Bestehorn, M. & Thiele, U. 2005 Long-wave theory of bounded two-layer films with a free liquid–liquid interface: short- and long-time evolution. Phys. Fluids 17 (6), 064104.Google Scholar
Merzkirch, W. 1987 Flow Visualization. Academic.Google Scholar
Mills, K. C. & Keene, B. J. 1990 Factors affecting variable weld penetration. Intl. Mater. Rev. 35 (1), 185–216.Google Scholar
Nicolis, G. & Prigogine, I. 1977 Self-Organization in Nonequilibrium Systems: From Dissipative Structures to Order Through Fluctuations. Wiley.Google Scholar
Oprisan, A., Hegseth, J. J., Smith, G. M., Lecoutre, C., Garrabos, Y. & Beysens, D. A. 2011 Dynamics of a wetting layer and Marangoni convection in microgravity. Phys. Rev. E 84, 021202.Google Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69, 931–980.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.Google Scholar
Pototsky, A., Bestehorn, M., Merkt, D. & Thiele, U. 2005 Morphology changes in the evolution of liquid two-layer films. J. Chem. Phys. 122 (22), 224711.Google Scholar
Prakash, A., Yasuda, K., Otsubo, F., Kuwahara, K. & Doi, T. 1997 Flow coupling mechanisms in two-layer Rayleigh–Bénard convection. Exp. Fluids 23 (3), 252–261.Google Scholar
Rahal, S., Cerisier, P. & Azuma, H. 2007 Bénard–Marangoni convection in a small circular container: influence of the Biot and Prandtl numbers on pattern dynamics and free surface deformation. Exp. Fluids 43 (4), 547–554.Google Scholar
Rasenat, S., Busse, F. H. & Rehberg, I. 1989 A theoretical and experimental study of double-layer convection. J. Fluid Mech. 199, 519–540.Google Scholar
Schatz, M. F. & Neitzel, G. P. 2001 Experiments on thermocapillary instabilities. Annu. Rev. Fluid Mech. 33 (1), 93–127.Google Scholar
Schatz, M. F., VanHook, S. J., McCormick, W. D., Swift, J. B. & Swinney, H. L. 1995 Onset of surface-tension-driven Bénard convection. Phys. Rev. Lett. 75, 1938–1941.Google Scholar
Smith, K. A. 1966 On convective instability induced by surface-tension gradients. J. Fluid Mech. 24, 401–414.Google Scholar
Thess, A. & Bestehorn, M. 1995 Planform selection in Bénard–Marangoni convection: l hexagons versus $g$ hexagons. Phys. Rev. E 52, 6358–6367.Google Scholar
Trice, J., Favazza, C., Thomas, D., Garcia, H., Kalyanaraman, R. & Sureshkumar, R. 2008 Novel self-organization mechanism in ultrathin liquid films: theory and experiment. Phys. Rev. Lett. 101 (1), 017802.Google Scholar
VanHook, S. J., Schatz, M. F., Swift, J. B., McCormick, W. D. & Swinney, H. L. 1997 Long-wavelength surface-tension-driven Bénard convection: experiment and theory. J. Fluid Mech. 345, 45–78.CrossRefGoogle Scholar
Vécsei, M., Dietzel, M. & Hardt, S. 2014 Coupled self-organization: thermal interaction between two liquid films undergoing long-wavelength instabilities. Phys. Rev. E 89, 053018.Google Scholar
Welander, P. 1964 Convective instability in a two-layer fluid heated uniformly from above. Tellus 16 (3), 349–358.Google Scholar