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Marangoni instability in a viscoelastic binary film with cross-diffusive effect

Published online by Cambridge University Press:  15 January 2021

Rajkumar Sarma Open the ORCID record for Rajkumar Sarma [Opens in a new window]  and
Pranab Kumar Mondal Open the ORCID record for Pranab Kumar Mondal [Opens in a new window]
Rajkumar Sarma
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam781039, India
Pranab Kumar Mondal*
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam781039, India
*
† Email address for correspondence: mail2pranab@gmail.com
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Abstract

Viscoelastic liquids are usually blends of a polymeric solute and a Newtonian solvent. In the presence of a temperature gradient, stratification of these solutes can take place via the Soret effect. Here, we investigate the classical Marangoni instability problem for a thin viscoelastic film considering this binary aspect of the liquid. The film, bounded above by a deformable free surface, is subjected to heating from below by a solid substrate. Linear stability analysis performed numerically for perturbations of finite wavelength (short-wave perturbations) reveals that both monotonic and oscillatory instabilities can emerge in this system. In the presence of Soret diffusion, the interaction between thermocapillary and solutocapillary forces is found to give rise to two different oscillatory instabilities, of which one mode was overlooked previously, even for the Newtonian binary mixtures. As a principal result of this work, we provide a complete picture of the susceptibility to different instability modes based on model parameter values. Finally, an approximate model is developed under the framework of long-wave analysis, which can qualitatively depict the stability behaviour of the system without numerically solving the problem.

JFM classification

Convection: Marangoni convection Interfacial Flows (free surface): Thin films Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 910 , 10 March 2021 , A30
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Arafune, K., Yamamoto, K. & Hirata, A. 2001 Interactive thermal and solutal Marangoni convection during compound semiconductor growth in a rectangular open boat. Intl J. Heat Mass Transfer 44 (13), 2405–2411.CrossRefGoogle Scholar
Bassou, N. & Rharbi, Y. 2009 Role of Bénard-Marangoni instabilities during solvent evaporation in polymer surface corrugations. Langmuir 25, 624–632.CrossRefGoogle ScholarPubMed
Batson, W., Cummings, L. J., Shirokoff, D. & Kondic, L. 2019 Oscillatory thermocapillary instability of a film heated by a thick substrate. J. Fluid Mech. 872, 928–962.CrossRefGoogle Scholar
Bénard, H. 1901 Les tourbillons cellulaires dans une nappe de liquide transportant de la chaleur par convection en régime permanent. Ann. Chim. Phys. 23, 62–144.Google Scholar
Bestehorn, M. & Borcia, I. D. 2010 Thin film lubrication dynamics of a binary mixture: Example of an oscillatory instability. Phys. Fluids 22 (10), 104102.CrossRefGoogle Scholar
Bhattacharjee, J. K. 1994 Marangoni convection in binary liquids. Phys. Rev. E 50 (2), 1198–1205.CrossRefGoogle ScholarPubMed
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of Polymeric Liquids. Vol. 1: Fluid Mechanics. Wiley.Google Scholar
Bormashenko, E., Balter, S., Pogreb, R., Bormashenko, Y., Gendelman, O. & Aurbach, D. 2010 On the mechanism of patterning in rapidly evaporated polymer solutions: Is temperature-gradient-driven Marangoni instability responsible for the large-scale patterning? J. Colloid Interface Sci. 343, 602–607.CrossRefGoogle ScholarPubMed
Chen, Z. W., Li, Y. & Zhan, J. M. 2010 Double-diffusive Marangoni convection in a rectangular cavity: onset of convection. Phys. Fluids 22 (3), 034106.CrossRefGoogle Scholar
Colinet, P., Legros, J. C. & Velarde, M. G. 2001 Nonlinear Dynamics of Surface-Tension-Driven Instabilities. Wiley.CrossRefGoogle Scholar
Craster, R. V. & Matar, O. K. 2009 Dynamics and stability of thin liquid films. Rev. Mod. Phys. 81 (3), 1131–1198.CrossRefGoogle Scholar
D'Alessio, S. J. D., Pascal, J. P., Ellaban, E. & Ruyer-Quil, C. 2020 Marangoni instabilities associated with heated surfactant-laden falling films. J. Fluid Mech. 887, A20.CrossRefGoogle Scholar
Dauby, P. C., Parmentier, P., Lebon, G. & Grmela, M. 1993 Coupled buoyancy and thermocapillary convection in a viscoelastic Maxwell fluid. J. Phys: Condens. Matter 5 (26), 4343–4352.Google Scholar
Doumenc, F., Boeck, T., Guerrier, B. & Rossi, M. 2010 Transient Rayleigh–Bénard–Marangoni convection due to evaporation: a linear non-normal stability analysis. J. Fluid Mech. 648, 521–539.CrossRefGoogle Scholar
Doumenc, F., Chénier, E., Trouette, B., Boeck, T., Delcarte, C., Guerrier, B. & Rossi, M. 2013 Free convection in drying binary mixtures: solutal versus thermal instabilities. Intl J. Heat Mass Transfer 63, 336–350.CrossRefGoogle Scholar
Fayzrakhmanova, I. S., Shklyaev, S. & Nepomnyashchy, A. A. 2013 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, 190–212.CrossRefGoogle Scholar
de Gans, B. J., Kita, R., Wiegand, S. & Luettmer-Strathmann, J. 2003 Unusual thermal diffusion in polymer solutions. Phys. Rev. Lett. 91 (24), 245501.CrossRefGoogle ScholarPubMed
Getachew, D. & Rosenblat, S. 1985 Thermocapillary instability of a viscoelastic liquid layer. Acta Mech. 55 (1–2), 137–149.CrossRefGoogle Scholar
de Groot, S. R. & Mazur, P. 2011 Non-Equilibrium Thermodynamics. Dover Publications.Google Scholar
Heriot, S. Y. & Jones, R. A. L. 2005 An interfacial instability in a transient wetting layer leads to lateral phase separation in thin spin-cast polymer-blend films. Nat. Mater. 4, 782.CrossRefGoogle Scholar
Hirata, S. C., De B. Alves, L. S., Delenda, N. & Ouarzazi, M. N. 2015 Convective and absolute instabilities in Rayleigh-Bénard-Poiseuille mixed convection for viscoelastic fluids. J. Fluid Mech. 765, 167–210.CrossRefGoogle Scholar
Ho, K. L. & Chang, H. C. 1988 On nonlinear doubly-diffusive marangoni instability. AIChE J. 34 (5), 705–722.CrossRefGoogle Scholar
Hu, J., Hadid, H. B., Henry, D. & Mojtabi, A. 2008 Linear temporal and spatio-temporal stability analysis of a binary liquid film flowing down an inclined uniformly heated plate. J. Fluid Mech. 599, 269–298.CrossRefGoogle Scholar
Joo, S. W. 1995 Marangoni instabilities in liquid mixtures with Soret effects. J. Fluid Mech. 293, 127–145.CrossRefGoogle Scholar
Keller, H. B. 2018 Numerical Methods for Two-Point Boundary-Value Problems. Dover Publications.Google Scholar
Lebon, G., Parmentier, P., Teller, O. & Dauby, P. C. 1994 Bénard-Marangoni instability in a viscoelastic Jeffreys’ fluid layer. Rheol. Acta 33 (4), 257–266.CrossRefGoogle Scholar
Maxwell, J. C. 1867 On the dynamical theory of gases. Phil. Trans. R. Soc. Lond. 157 (235), 49–88.Google Scholar
McFadden, G. B., Coriell, S. R. & Lott, P. A. 2010 Onset of convection in two layers of a binary liquid. J. Fluid Mech. 647, 105–124.CrossRefGoogle Scholar
McTaggart, C. L. 1983 Convection driven by concentration- and temperature-dependent surface tension. J. Fluid Mech. 134 (1), 301–310.CrossRefGoogle Scholar
Nepomnyashchy, A. A. 2001 Interfacial Phenomena and Convection. Chapman and Hall/CRC.CrossRefGoogle Scholar
Oron, A., Davis, S. H. & Bankoff, S. G. 1997 Long-scale evolution of thin liquid films. Rev. Mod. Phys. 69 (3), 931–980.CrossRefGoogle Scholar
Oron, A. & Nepomnyashchy, A. A. 2004 Long-wavelength thermocapillary instability with the Soret effect. Phys. Rev. E 69 (1), 016313.CrossRefGoogle ScholarPubMed
Parmentier, P., Lebon, G. & Regnier, V. 2000 Weakly nonlinear analysis of Bénard–Marangoni instability in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 89 (1–2), 63–95.CrossRefGoogle Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.CrossRefGoogle Scholar
Pillai, D. S. & Narayanan, R. 2018 Rayleigh-Taylor stability in an evaporating binary mixture. J. Fluid Mech. 848, R1.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 (10), 104104.CrossRefGoogle Scholar
Ramkissoon, H., Ramdath, G., Comissiong, D. & Rahaman, K. 2006 On thermal instabilities in a viscoelastic fluid. Intl J. Non-Linear. Mech. 41 (1), 18–25.CrossRefGoogle Scholar
Sarma, R. & Mondal, P. K. 2018 Marangoni instability in a thin film heated from below: effect of nonmonotonic dependence of surface tension on temperature. Phys. Rev. E 97 (4), 043105.CrossRefGoogle Scholar
Sarma, R. & Mondal, P. K. 2019 Marangoni instability in a heated viscoelastic liquid film: long-wave versus short-wave perturbations. Phys. Rev. E 100 (1), 013103.CrossRefGoogle Scholar
Schatz, M. F. & Neitzel, G. P. 2001 Experiments on thermocapillary instabilities. Annu. Rev. Fluid Mech. 33 (1), 93–127.CrossRefGoogle Scholar
Shklyaev, S., Alabuzhev, A. A. & Khenner, M. 2012 Long-wave Marangoni convection in a thin film heated from below. Phys. Rev. E 85 (1), 016328.CrossRefGoogle Scholar
Shklyaev, S. & Nepomnyashchy, A. A. 2013 Longwave Marangoni convection in a surfactant solution between poorly conducting boundaries. J. Fluid Mech. 718, 428–456.CrossRefGoogle Scholar
Shklyaev, S. & Nepomnyashchy, A. A. 2017 Longwave Instabilities and Patterns in Fluids. Springer.CrossRefGoogle Scholar
Shklyaev, S., Nepomnyashchy, A. A. & Oron, A. 2009 Marangoni convection in a binary liquid layer with Soret effect at small Lewis number: linear stability analysis. Phys. Fluids 21 (5), 054101.CrossRefGoogle Scholar
Skarda, J. R. L., Jacqmin, D. & McCaughan, F. E. 1998 Exact and approximate solutions to the double-diffusive Marangoni–Bénard problem with cross-diffusive terms. J. Fluid Mech. 366, 109–133.CrossRefGoogle Scholar
Toussaint, G., Bodiguel, H., Doumenc, F., Guerrier, B. & Allain, C. 2008 Experimental characterization of buoyancy- and surface tension-driven convection during the drying of a polymer solution. Intl J. Heat Mass Transfer 51, 4228–4237.CrossRefGoogle Scholar
Würger, A. 2007 Thermophoresis in colloidal suspensions driven by marangoni forces. Phys. Rev. Lett. 98 (13), 138301.CrossRefGoogle ScholarPubMed
Yiantsios, S. G., Serpetsi, S. K., Doumenc, F. & Guerrier, B. 2015 Surface deformation and film corrugation during drying of polymer solutions induced by Marangoni phenomena. Intl J. Heat Mass Transfer 89, 1083–1094.CrossRefGoogle Scholar
Zhang, M. & Müller-Plathe, F. 2006 The Soret effect in dilute polymer solutions: influence of chain length, chain stiffness, and solvent quality. J. Chem. Phys. 125 (12), 124903.CrossRefGoogle ScholarPubMed