Abstract
The stability of a front of chemoconvective finger structures being spontaneously formed in a two-layer system of fluids filling a vertical Hele-Shaw cell, each of them containing a reactant of an exothermic A + B → S reaction is examined. If the configuration consists of more dense acid (or salt) on top of less dense base in the presence of gravity, the development of the Rayleigh- Taylor instability leads to the standard scenario of density fingering. Despite the widespread perception of fingering as an irregular process, we show that, at least in some cases, the exact balance between the instabilities involved can result quasi-regular fingering pattern formation. In the case of immiscible fluids, we demonstrate that the Rayleigh-Bénard mechanism associated with intensive heat release during the reaction performs fine-tuning of the envelope of salt fingers. The mathematical model we develop consists in a set of reaction-diffusion-convection equations governing the evolution of concentrations and temperature coupled to Navier-Stokes and energy equations, written in a Hele-Shaw approximation. The results of linear stability analysis and direct numerical simulations of the fully nonlinear system are presented.
Similar content being viewed by others
References
Nepomnyashchy, A., Simanovskii, I., Legros, J.C.: Interfacial Convection in Multilayer Systems, p 498. Springer, New York (2012)
Quincke, G.: Ueber Periodische Ausbreitung an Flussigkeitsoberflachen und Dadurch Hervorgerufene Bewegungserscheinungen. Ann. Phys. (Leipzig) 35, 580–642 (1888)
Sherwood, T.S., Wei, J.C.: Interfacial Phenomena in Liquid Extraction. Ind. Eng. Chem. 49, 1030–1034 (1957)
Dupeyrat, M., Nakache, E.: Direct Conversion of Chemical Energy Into Mechanical Energy at an Oil Water Interface. Bioelectrochem Bioenerg. 5, 134–141 (1978)
Berg, J.C., Morig, C.R.: Density Effects in Interfacial Convection. Chem. Eng. Sci. 24 (6), 937–946 (1969)
Avnir, D., Kagan, M.L: The Evolution of Chemical Patterns in Reactive Liquids Driven by Hydrodynamic Instabilities. Chaos 5, 589601 (1995)
Pons, A.J., Sagues, F., Bees, M.A., Sorensen, P.G.: Pattern Formation in the Methylene-Blue-Glucose System. J. Phys. Chem. B 104, 22512259 (2000)
Eckert, K., Grahn, A.: Plume and Finger Regimes Driven by a Exothermic Interfacial Reaction. Phys. Rev. Lett. 82, 4436–4439 (1999)
Trevelyan, P.M.J., Almarcha, C., De Wit, A.: Buoyancy -Driven Instabilities of Miscible Two-Layer Stratifications in Porous Media and Hele-Shaw Cells. J. Fluid Mech. 670, 3865 (2011)
Bratsun, D.A., De Wit, A.: Buoyancy-Driven Pattern Formation in Reactive Immiscible Two-Layer Systems. Chem. Eng. Sci. 66 (22), 5723–5734 (2011)
Eckert, K., Acker, M., Shi, Y.: Chemical Pattern Formation Driven by a Neutralization Reaction. Part I: Mechanism and Basic Features. Phys. Fluids 16, 385–399 (2004)
De Wit, A.: Fingering of Chemical Fronts in Porous Media. Phys. Rev. Lett. 054502, 87 (2001)
Almarcha, C., Trevelyan, P.M.J., Grosfils, P., De Wit, A.: Chemically Driven Hydrodynamic Instabilities. Phys. Rev. Lett 044501, 104 (2010)
Almarcha, C., R’Honi, Y., De Decker, Y., Trevelyan, P.M.J., Eckert, K., De Wit, A.: Convective Mixing Induced by Acid-Base Reactions. J. Phys. Chem. B 115, 9739–9744 (2011)
Bratsun, D.A., De Wit, A: On Marangoni Convective Patterns Driven by an Exothermic Chemical Reaction in Two-Layer Systems. Phys. Fluids 16 (4), 1082–1096 (2004)
Bratsun, D.A., De Wit, A.: Control of Chemoconvective Structures in a Slab Reactor. Techn. Phys. 53, 146–153 (2008)
Shi, Y., Eckert, K.: Orientation-Dependent Hydrodynamic Instabilities from Chemo-Marangoni Cells to Large Scale Interfacial Deformations. Chin. J. Chem. Eng. 15, 748–753 (2007)
Tan, C.T., Homsy, G.M.: Stability of Miscible Displacements in Porous Media: Rectilinear Ow. Phys. Fluids 29, 3549–3557 (1986)
Simanovskii, I.B., Nepomnyashchy, A.A.: Convective Instabilities in Systems with Interface, pp. 280, Gordon and Breach Science, Amsterdam (1993)
Bratsun, D.A., Shi, Y., Eckert, K., De Wit, A.: Control of chemo-Hydrodynamic Pattern Formation by External Localized Cooling. Europhys. Lett. 69 (5), 746–752 (2005)
Acknowledgements
We wish to thank K. Eckert and A. De Wit for stimulating discussions. The research has been carried out with the financial support from the Ministry of Education of Perm Region (grant C-26/244), Program of Strategic Development of Perm State Humanitarian Pedagogical University (project 031-F) and grants of Russian Fund for Basic Research (13-01-00508a, 14-01-96021r_ural_a)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bratsun, D.A. On Rayleigh-Bénard Mechanism of Alignment of Salt Fingers in Reactive Immiscible Two-Layer Systems. Microgravity Sci. Technol. 26, 293–303 (2014). https://doi.org/10.1007/s12217-014-9374-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12217-014-9374-0