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Thermosolutal Marangoni instability in a viscoelastic liquid film: effect of heating from the free surface

Published online by Cambridge University Press:  22 December 2020

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, Assam781 039, India
Pranab Kumar Mondal*
Affiliation:
Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Assam781 039, India
*
†Email address for correspondence: mail2pranab@gmail.com
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Abstract

We investigate the Marangoni instability in a thin polymeric liquid film heated from the free surface. The polymeric solutions are usually a binary mixture of a Newtonian solvent with a polymeric solute, and exhibit viscoelastic behaviour. In the presence of a temperature gradient, stratification of these solutes can take place via the Soret effect, giving rise to the solutocapillary effect at the free surface. Considering this cross-diffusive effect and incorporating the effects of gravity, here we analyse the stability characteristics of this polymeric film when bounded between its deformable free surface and a poorly conductive rigid substrate. Linear stability analysis around the quiescent base state reveals that, under the combined influences of thermosolutocapillarity and the elasticity of the liquid, apart from the monotonic disturbances, two different oscillatory instabilities can emerge in this system. The characteristics of each instability mode are discussed, and a complete stability picture is perceived in terms of the phase diagrams, identifying the model parameter regimes for which a particular instability mode becomes dominant.

JFM classification

Convection: Marangoni convection Interfacial Flows (free surface): Thin films Non-Newtonian Flows: Viscoelasticity
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 909 , 25 February 2021 , A12
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

Adam, M. & Delsanti, M. 1983 Viscosity and longest relaxation time of semi-dilute polymer solutions. I. Good solvent. J. Phys. 44 (10), 1185–1193.10.1051/jphys:0198300440100118500CrossRefGoogle Scholar
Arshad, T. A., Kim, C. B., Prisco, N. A., Katzenstein, J. M., Janes, D. W., Bonnecaze, R. T. & Ellison, C. J. 2014 Precision Marangoni-driven patterning. Soft Matt. 10 (40), 8043–8050.10.1039/C4SM01284DCrossRefGoogle ScholarPubMed
Bénard, H. 1901 Les tourbillons cellulaires dans une nappe de liquide transportant de la chaleur par convection en régime permanent. Ann. Chem. 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, 104102.10.1063/1.3489434CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C. & Hassager, O. 1987 Dynamics of polymeric liquids. Vol. 1: fluid mechanics. Wiley.Google Scholar
Boggon, T. J., Chayen, N. E., Snell, E. H., Dong, J., Lautenschlager, P., Potthast, P., Siddons, D. P., Stojanoff, V., Gordon, E., Thompson, A. W., et al. 1998 Protein crystal movements and fluid flows during microgravity growth. Phil. Trans. R. Soc. A Math. Phys. Engng Sci. 356, 1045–1061.10.1098/rsta.1998.0208CrossRefGoogle Scholar
Castillo, J. L. & Velarde, M. G. 1982 Buoyancy-thermocapillary instability: the role of interfacial deformation in one- and two-component fluid layers heated from below or above. J. Fluid Mech. 125, 463–474.10.1017/S0022112082003449CrossRefGoogle Scholar
Colinet, P., Legros, J. C. & Velarde, M. G. 2001 Nonlinear dynamics of surface-tension-driven instabilities, 1st edn. Wiley-VCH.10.1002/3527603115CrossRefGoogle 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
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
De Groot, S. R. & Mazur, P. 2011 Non-equilibrium thermodynamics. Dover Publications.Google 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
Ebagninin, K. W., Benchabane, A. & Bekkour, K. 2009 Rheological characterization of poly(ethylene oxide) solutions of different molecular weights. J. Colloid Interface Sci. 336, 360–367.CrossRefGoogle ScholarPubMed
Getachew, D. & Rosenblat, S. 1985 Thermocapillary instability of a viscoelastic liquid layer. Acta Mech. 55 (1–2), 137–149.CrossRefGoogle Scholar
Hartung, M., Rauch, J. & Köhler, W. 2006 Thermal diffusion of dilute polymer solutions: the role of solvent viscosity. J. Chem. Phys. 125, 214904.CrossRefGoogle ScholarPubMed
Hu, K. X., He, M. & Chen, Q. S. 2016 Instability of thermocapillary liquid layers for Oldroyd-B fluid. Phys. Fluids 28 (3), 033105.CrossRefGoogle Scholar
Joo, S. W. 1995 Marangoni instabilities in liquid mixtures with Soret effsects. J. Fluid Mech. 293, 127–145.10.1017/S0022112095001662CrossRefGoogle Scholar
Joseph, D. D. 1990 Fluid dynamics of viscoelastic liquids, 1st edn. Springer.10.1007/978-1-4612-4462-2CrossRefGoogle Scholar
Khattab, I. S., Bandarkar, F., Fakhree, M. A. A. & Jouyban, A. 2012 Density, viscosity, and surface tension of water + ethanol mixtures from 293 to 323 K. Korean J. Chem. Engng 29 (6), 812–817.CrossRefGoogle Scholar
Kita, R., Wiegand, S. & Luettmer-Strathmann, J. 2004 Sign change of the Soret coefficient of poly(ethylene oxide) in water/ethanol mixtures observed by thermal diffusion forced Rayleigh scattering. J. Chem. Phys. 121 (8), 3874–3885.CrossRefGoogle ScholarPubMed
Kumar, A. & Roy, S. 2009 Effect of three-dimensional melt pool convection on process characteristics during laser cladding. Comput. Mater. Sci. 46 (2), 495–506.CrossRefGoogle Scholar
Lappa, M. & Ferialdi, H. 2018 Multiple solutions, oscillons, and strange attractors in thermoviscoelastic Marangoni convection. Phys. Fluids 30 (10), 104104.CrossRefGoogle Scholar
Mark, J. E. 1999 Polymer data handbook. Oxford University Press.Google Scholar
Mills, K. C., Keene, B. J., Brooks, R. F. & Shirali, A. 1998 Marangoni effects in welding. Phil. Trans. R. Soc. A Math. Phys. Engng Sci. 356, 911-925.CrossRefGoogle Scholar
Morozov, M., Oron, A. & Nepomnyashchy, A. A. 2014 Long-wave Marangoni convection in a layer of surfactant solution. Phys. Fluids 26 (11), 112101.10.1063/1.4901950CrossRefGoogle Scholar
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 (5), 489–500.10.1017/S0022112058000616CrossRefGoogle 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), 104104.10.1063/1.2075287CrossRefGoogle 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
Schmid, P. J. & Henningson, D. S. 2001 Stability and transition in shear flows. Springer.10.1007/978-1-4613-0185-1CrossRefGoogle 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
Singh, M. 2007 Survismeter unit for surface tension, viscosity, and dipole moment determination for polystyrene interactions in benzene. J. Disper. Sci. Technol. 28, 1278–1286.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, 109–133.10.1017/S0022112098001220CrossRefGoogle 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 (17–18), 4228–4237.CrossRefGoogle 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 (1), 45–78.CrossRefGoogle Scholar
Wang, J. & Fiebig, M. 1995 Measurement of the thermal diffusivity of aqueous solutions of alcohols by a laser-induced thermal grating technique. Intl J. Thermophys. 16 (6), 1353–1361.CrossRefGoogle Scholar
Würger, A. 2007 Thermophoresis in colloidal suspensions driven by marangoni forces. Phys. Rev. Lett. 98 (13), 138301.10.1103/PhysRevLett.98.138301CrossRefGoogle ScholarPubMed
Yiantsios, S. G. & Higgins, B. G. 2006 Marangoni flows during drying of colloidal films. Phys. Fluids 18 (8), 082103.CrossRefGoogle Scholar
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, J., Behringer, R. P. & Oron, A. 2007 Marangoni convection in binary mixtures. Phys. Rev. E 76 (1), 016306.10.1103/PhysRevE.76.016306CrossRefGoogle ScholarPubMed
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