Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-19T05:50:15.543Z Has data issue: false hasContentIssue false
  • English
  • Français

Dynamics of a Reactive Thin Film

Published online by Cambridge University Press:  09 July 2012

P.M.J. Trevelyan ,
A. Pereira  and
S. Kalliadasis
P.M.J. Trevelyan
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom Division of Mathematics & Statistics, University of Glamorgan, Pontypridd, CF37 1DL, Wales
A. Pereira
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom
S. Kalliadasis*
Affiliation:
Department of Chemical Engineering, Imperial College London, London SW7 2AZ, United Kingdom
*
∗ Corresponding author. E-mail: [email protected]
Get access

Abstract

Consider the dynamics of a thin film flowing down an inclined plane under the action of gravity and in the presence of a first-order exothermic chemical reaction. The heat released by the reaction induces a thermocapillary Marangoni instability on the film surface while the film evolution affects the reaction by influencing heat/mass transport through convection. The main parameter characterizing the reaction-diffusion process is the Damköhler number. We investigate the complete range of Damköhler numbers. We analyze the steady state, its linear stability and nonlinear regime. In the latter case, long-wave models are compared with integral-boundary-layer ones and bifurcation diagrams for permanent solitary wave solutions of the different models are constructed. Time-dependent computations with the integral-boundary-layer models show that the system approaches a train of coherent structures that resemble the solitary pulses obtained in the bifurcation diagrams.

Keywords

thin filmsMarangoni effectchemical reactions
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 7 , Issue 4: Modelling phenomena on micro- and nanoscale , 2012 , pp. 99 - 145
Copyright
© EDP Sciences, 2012

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

Oron, A., Davis, S. H., Bankoff, S. G.. Long-scale evolution of thin liquid films. Rev. Mod. Phys., 69 (1997), 931–980. CrossRefGoogle Scholar
S. Kalliadasis, U. Thiele (Ed) Thin Films of Soft Matter. Springer-Wien, New York, 2007.
Craster, R. V., Matar, O. K.. Dynamics and stability of thin liquid films. Rev. Mod. Phys., 81 (2009), 1131–1198. CrossRefGoogle Scholar
Kapitza, P. L.. Wave flow of thin layers of viscous fluid : I. Free flow. Zh. Eksp. Tear. Fiz., 18 (1948), 3–18. Google Scholar
Kapitza, P. L.. Wave flow of thin layers of a viscous fluid : II. Fluid flow in the presence of continuous gas flow and heat transfer. Zh. Eksp. Teor. Fiz., 18 (1948), 19–28. Google Scholar
Kapitza, P. L., Kapitza, S. P.. Wave flow of thin layers of a viscous fluid : III. Experimental study of undulatory flow conditions. Zh. Eksp. Teor. Fiz., 19 (1949), 105–120. Google Scholar
Chang, H.-C.. Wave evolution on a falling film. Ann. Rev. Fluid Mech., 26 (1994), 103–136. CrossRefGoogle Scholar
H.-C. Chang, E. A. Demekhin. Complex Wave Dynamics on Thin Films. Elsevier, 2002.
Dagan, Z., Pismen, L. M.. Marangoni waves induced by a multistable chemical reaction on thin liquid films. J. Colloid Interface Sci., 99 (1984), 215–225. CrossRefGoogle Scholar
Pereira, A., Trevelyan, P. M. J., Thiele, U., Kalliadasis, S.. Interfacial hydrodynamic waves driven by chemical reactions. J. Eng. Math., 59 (2007), 207–220. CrossRefGoogle Scholar
Pereira, A., Trevelyan, P. M. J., Thiele, U., Kalliadasis, S.. Dynamics of a horizontal thin liquid film in the presence of reactive surfactants. Phys. Fluids, 19 (2007), 112102. CrossRefGoogle Scholar
Pereira, A., Trevelyan, P. M. J., Thiele, U., Kalliadasis, S.. Interfacial instabilities driven by chemical reactions. Eur. Phys. J. Special Topics, 166 (2009), 121–125. CrossRefGoogle Scholar
Rednikov, A. Y., Ryazantsev, Y. S., Velarde, M. G.. Drop motion with surfactant transfer in a homogeneous surrounding. Phys. Fluids, 6 (1994), 451–468. CrossRefGoogle Scholar
Pismen, L. M.. Chemocapillary instabilities of a contact line. Phys. Rev. E, 81 (2010), 026307. CrossRefGoogle ScholarPubMed
Gilliland, E. R., Baddour, R. F., Brian, P. L. T.. Gas Absorption Accompanied by a Liquid-phase Chemical Reaction. Am. Inst. Chem. Eng. J., 4 (1958), 223. CrossRefGoogle Scholar
Trevelyan, P. M. J., Kalliadasis, S.. Dynamics of a reactive falling film at large Péclet numbers. I. Long-wave approximation. Phys. Fluids, 16 (2004), 3191-3208. CrossRefGoogle Scholar
Trevelyan, P. M. J., Kalliadasis, S.. Dynamics of a reactive falling film at large Péclet numbers. II. Nonlinear waves far from criticality : Integral-boundary-layer approximation. Phys. Fluids, 16 (2004), 3209-3226. CrossRefGoogle Scholar
A. A. Nepomnyashchy, M. G. Velarde, P. Colinet. Interfacial Phenomena and Convection. Chapman & Hall, London, 2002.
R. C. Weast, M. J. Astle. Handbook of Chemistry and Physics. CRC Press, Boca Raton, FL, 1979.
Wylock, C. E., Colinet, P., Cartage, T., Haut, B.. Coupling between mass transfer and chemical reactions during the absorption of CO2 in a NaHCO3-Na2HCO3 brine : Experimental and theoretical study. Int. J. Chem. React. Engng., 6 (2008), A4. Google Scholar
Kalliadasis, S., Demekhin, E. A., Ruyer-Quil, C., Velarde, M. G.. Thermocapillary instability and wave formation on a film falling down a uniformly heated plane. J. Fluid Mech., 492 (2003), 303–338. CrossRefGoogle Scholar
Trevelyan, P., Kalliadasis, S.. Wave dynamics on a thin-liquid film falling down a heated wall. J. Eng. Math., 50 (2004), 177–208. CrossRefGoogle Scholar
Ruyer-Quil, C., Scheid, B., Kalliadasis, S., Velarde, M. G., Zeytounian, R. Kh.. Thermocapillary long waves in a liquid film flow. Part 1. Low-dimensional formulation. J. Fluid Mech., 538 (2005), 199–222. CrossRefGoogle Scholar
Trevelyan, P. M. J., Scheid, B., Ruyer-Quil, C., Kalliadasis, S.. Heated falling films. J. Fluid Mech., 592 (2007), 295–334. CrossRefGoogle Scholar
Benjamin, T. B.. Wave formation in laminar flow down an inclined plane. J. Fluid Mech., 2 (1957), 554–574. CrossRefGoogle Scholar
Yih, C. S.. Stability of liquid flow down an inclined plane. Phys. Fluids, 6 (1963), 321–334. CrossRefGoogle Scholar
Benney, D. J.. Long waves on liquid films. J. Math. Phys., 45 (1966), 150–155. CrossRefGoogle Scholar
Scheid, B., Ruyer-Quil, C., Kalliadasis, S., Velarde, M. G., Zeytounian, R. Kh.. Thermocapillary long waves in a liquid film flow. Part 2. Linear stability and nonlinear waves. J. Fluid Mech., 538 (2005), 223–244. CrossRefGoogle Scholar
Ruyer-Quil, C., Trevelyan, P., Giorgiutti-Dauphiné, F., Duprat, C., Kalliadasis, S.. Modelling film flows down a fibre. J. Fluid Mech., 603 (2008), 431–462. CrossRefGoogle Scholar
Pumir, A., Manneville, P., Pomeau, Y.. On solitary waves running down an inclined plane. J. Fluid Mech., 135 (1983), 27–50. CrossRefGoogle Scholar
Nakaya, C.. Waves on a viscous fluid film down a vertical wall. Phys. Fluids, 1 (1989), 1143–1154. CrossRefGoogle Scholar
Oron, A., Gottlieb, O.. Nonlinear dynamics of temporally excited falling liquid films. Phys. Fluids, 14 (2002), 2622–2636. CrossRefGoogle Scholar
Scheid, B., Ruyer-Quil, C., Thiele, U., Kabov, O.A., Legros, J.C., Colinet, P.. Validity domain of the Benney equation including Marangoni effect for closed and open flows. J. Fluid Mech., 527 (2004), 303–335. CrossRefGoogle Scholar
Ya. Shkadov, V.. Wave modes in the flow of thin layer of a viscous liquid under the action of gravity. Izv. Akad. Nauk SSSR, Mekh. Zhidk Gaza, 1 (1967), 43–50. Google Scholar
Ruyer-Quil, C., Manneville, P.. Modeling film flows down inclined planes. Eur. Phys. J. B, 6 (1998), 277–292. CrossRefGoogle Scholar
Ruyer-Quil, C., Manneville, P.. Improved Modeling of flows down inclined planes. Eur. Phys. J. B, 15 (2000), 357-369. CrossRefGoogle Scholar
Ruyer-Quil, C., Manneville, P.. Further accuracy and convergence results on the modeling of flows down inclined planes by weighted-residual approximations. Phys. Fluids, 14 (2002), 170–183. CrossRefGoogle Scholar
Kalliadasis, S., Kiyashko, A., Demekhin, E. A.. Marangoni instability of a thin liquid film heated from below by a local heat source. J. Fluid Mech., 475 (2003), 377–408. CrossRefGoogle Scholar
Kliakhandler, I. L., Davis, S. H., Bankoff, S. G.. Viscous beads on vertical fibre. J. Fluid Mech., 429 (2001), 381–390. CrossRefGoogle Scholar
Duprat, C., Ruyer-Quil, C., Kalliadasis, S., Giorgiutti-Dauphiné, F.. Absolute and convective instabilities of a viscous film flowing down a vertical fiber. Phys. Rev. Lett., 98 (2007), 244502. CrossRefGoogle Scholar
Malamataris, N. A., Vlachogiannis, M., Bontozoglou, V.. Solitary waves on inclined films : Flow structure and binary interactions. Phys. Fluids, 14 (2002), 1082–1094. CrossRefGoogle Scholar
Scheid, B., Ruyer-Quil, C., Manneville, P.. Wave patterns in film flows : modelling and three-dimensional waves. J. Fluid Mech., 562 (2006), 183–222. CrossRefGoogle Scholar
E. Doedel, A. Champneys, T. Fairfrieve, Y. Kuznetsov, B. Sandstede, X. Wang. AUTO97 : Continuation and bifurcation software for ordinary differential equations. Concordia University, Montreal, 1997.
Huerre, P., Monkewitz, P. A.. Local and global instabilities in spatially developing flows. Annu. Rev. Fluid. Mech., 22 (1990), 473–537. CrossRefGoogle Scholar
P. Huerre, M. Rossi. Hydrodynamic and Nonlinear Instabilities. In : Hydrodynamic Instabilities in Open Flows (Ed. C. Godréche, P. Manneville), Cambridge University Press, 1998, 81-294.
Chang, H.-C., Demekhin, E. A., Kopelevich, D. I.. Stability of a solitary pulse against wave packet disturbances in an active medium. Phys. Rev. Lett., 75 (1995), 1747–1750. CrossRefGoogle Scholar
Demekhin, E. A., Kalaidin, E. N., Kalliadasis, S., Yu. Vlaskin, S.. Three-dimensional localized coherent structures of surface turbulence. II. Λ solitons., Phys. Fluids, 19 (2007), 114104. CrossRefGoogle Scholar
Golovin, A. A., Nepomnyashchy, A. A., Pismen, L. M.. Interaction between short-scale Marangoni convection and long-scale deformation instability. Phys. Fluids, 6 (1994), 34–48. CrossRefGoogle Scholar
Thiele, U., Knobloch, E.. Thin liquid films on a slightly inclined heated plate. Physica D, 190 (2004), 213–248. CrossRefGoogle Scholar