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Transient Rayleigh–Bénard–Marangoni convection due to evaporation: a linear non-normal stability analysis

Published online by Cambridge University Press:  07 April 2010

F. DOUMENC ,
T. BOECK ,
B. GUERRIER  and
M. ROSSI
F. DOUMENC*
Affiliation:
UPMC Université Paris 06, Université Paris-Sud, CNRS, UMR 7608, Lab FAST, Bat 502, Campus Universitaire, Orsay F-91405, France
T. BOECK
Affiliation:
Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau, Postfach 100565, 98684 Ilmenau, Germany
B. GUERRIER
Affiliation:
UPMC Université Paris 06, Université Paris-Sud, CNRS, UMR 7608, Lab FAST, Bat 502, Campus Universitaire, Orsay F-91405, France
M. ROSSI
Affiliation:
UPMC Université Paris 06, CNRS, UMR 7190, IJLRA, 4 place Jussieu, Paris F-75005, France
*
†Email address for correspondence: frederic.doumenc@upmc.fr
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Abstract

The convective instability in a plane liquid layer with time-dependent temperature profile is investigated by means of a general method suitable for linear stability analysis of an unsteady basic flow. The method is based on a non-normal approach, and predicts the onset of instability, critical wavenumber and time. The method is applied to transient Rayleigh–Bénard–Marangoni convection due to cooling by evaporation. Numerical results as well as theoretical scalings for the critical parameters as function of the Biot number are presented for the limiting cases of purely buoyancy-driven and purely surface-tension-driven convection. Critical parameters from calculations are in good agreement with those from experiments on drying polymer solutions, where the surface cooling is induced by solvent evaporation.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 648 , 10 April 2010 , pp. 521 - 539
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Andersson, P., Berggren, M. & Henningson, D. S. 1999 Optimal disturbances and bypass transition in boundary layers. Phys. Fluids 11, 134–150.CrossRefGoogle Scholar
Berg, J. C., Boudart, M. & Acrivos, A. 1966 Natural convection in pools of evaporating liquids. J. Fluid Mech. 24, 721–735.CrossRefGoogle Scholar
Bhadauria, B. S. & Bathia, P. K. 2002 Time-periodic heating of Rayleigh–Benard convection. Phys. Scr. 66, 59–65.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Colinet, P., Joannes, L., Iorio, C. S., Haute, B., Bestehorn, M., Lebon, G. & Legros, J. C. 2003 Interfacial turbulence in evaporating liquids: theory and preliminary results of the itel-master 9 sounding rocket experiment. Adv. Space Res. 32 (2), 119–127.Google Scholar
Colinet, P., Legros, J. C. & Velarde, M. G. 2001 Nonlinear Dynamics of Surface-Tension-Driven Instabilities. Wiley-VCH.Google Scholar
Currie, I. G. 1967 The effect of heating rate on the stability of stationary fluids. J. Fluid Mech. 29, 337–347.CrossRefGoogle Scholar
Davis, S. H. 1987 Thermocapillary instabilities. Annu. Rev. Fluid Mech. 19, 403–435.CrossRefGoogle Scholar
Eckert, K., Acker, M. & Shi, Y. 2004 Chemical pattern formation driven by a neutralization reaction. I. Mechanism and basic features. Phys. Fluids 16 (2), 385–399.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part I. Autonomous operators. J. Atmos. Sci. 53, 2025–2040.2.0.CO;2>CrossRefGoogle Scholar
Foster, T. D. 1965 Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids 8, 1249–1257.CrossRefGoogle Scholar
Goussis, D. A. & Kelly, R. E. 1990 On the thermocapillary instabilities in a liquid layer heated from below. Intl J. Heat Mass Transfer 33, 2237–2245.Google Scholar
Homsy, G. M. 1973 Global stability of time-dependent flows: impulsively heated or cooled fluid layers. J. Fluid Mech. 60, 129–139.CrossRefGoogle Scholar
Kang, K. H. & Choi, C. K. 1997 A theoretical analysis of the onset of surface-tension-driven convection in a horizontal liquid layer cooled suddenly from above. Phys. Fluids 9, 7–15.Google Scholar
Lick, W. 1965 The instability of a fluid layer with time-dependent heating. J. Fluid Mech. 21, 565–576.Google Scholar
Luchini, P. 2000 Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations. J. Fluid Mech. 404, 289–309.CrossRefGoogle Scholar
Mancini, H. & Maza, D. 2004 Pattern formation without heating in an evaporative convection experiment. Europhys. Lett. 66, 812–818.Google Scholar
Merkt, D. & Bestehorn, M. 2003 Bénard–Marangoni convection in a strongly evaporating fluid. Physica D 185, 196–208.Google Scholar
Moussy, C., Lebon, G. & Margerit, J. 2004 Influence of evaporation on Bénard–Marangoni instability in a liquid–gas bilayer with a deformable interface. Eur. Phys. J. B 40, 327–335.Google Scholar
Nepomnyashchy, A., Simanovskii, I. & Legros, J. C. 2006 Interfacial Convection in Multilayer Systems. Springer.Google Scholar
Ozen, O. & Narayanan, R. 2004 The physics of evaporative and convective instabilities in bilayer systems: linear theory. Phys. Fluids 16, 4644–4652.Google Scholar
Pearson, J. R. A. 1958 On convection cells induced by surface tension. J. Fluid Mech. 4, 489–500.CrossRefGoogle Scholar
Rapaka, S., Chen, S., Pawar, R. J., Stauffer, P. H. & Zhang, D. 2008 Non-modal growth of perturbations in density-driven convection in porous media. J. Fluid Mech. 609, 285–303.CrossRefGoogle Scholar
Rayleigh, L. 1916 On convection currents in a horizontal layer of fluid when the higher temperature is on the under side. Phil. Mag. 32, 529–546.CrossRefGoogle Scholar
Reddy, S. C., Schmid, P. J., Bagget, P. & Henningson, D. S. 1998 On the stability of streamwise streaks and transition thresholds in plane channel flow. J. Fluid Mech. 365, 269–303.Google Scholar
Reichenbach, J. & Linde, H. 1981 Linear perturbation analysis of surface-tension-driven convection at a plane interface (Marangoni instability). J. Colloid Interface Sci. 84, 433–443.Google Scholar
Rosenblat, S. & Tanaka, G. A. 1971 Modulation of thermal convection instability. Phys. Fluids 14, 1319–1322.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.CrossRefGoogle ScholarPubMed
Schmid, P. J. & Henningson, D. S 2001 Stability and Transition in Shear Flows. Springer.CrossRefGoogle Scholar
Scriven, L. E. & Sternling, C. V. 1964 On cellular convection driven by surface-tension gradients: effects of mean surface tension and surface viscosity. J. Fluid Mech. 19, 321–340.Google Scholar
Sparrow, E. M., Goldstein, R. J. & Jonsson, V. K. 1963 Thermal instability in a horizontal fluid layer: effect of boundary conditions and nonlinear temperature profile. J. Fluid Mech. 18, 513–528.CrossRefGoogle Scholar
Touazi, O., Chénier, E., Doumenc, F. & Guerrier, B. 2010 Simulation of transient Rayleigh–Bénard–Marangoni convection induced by evaporation. Intl J. Heat Mass Transfer. 53, 656–664.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.Google Scholar
Vidal, A. & Acrivos, A. 1968 Effect of nonlinear temperature profiles on onset of convection driven by surface tension gradients. Ind. Engng Chem. Fundam. 7, 53–58.CrossRefGoogle Scholar
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