Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-12-01T02:26:21.509Z Has data issue: false hasContentIssue false
  • English
  • Français

Viscous fingering of a miscible reactive A + B → C interface: a linear stability analysis

Published online by Cambridge University Press:  15 April 2010

S. H. HEJAZI ,
P. M. J. TREVELYAN ,
J. AZAIEZ  and
A. DE WIT
S. H. HEJAZI
Affiliation:
Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, CanadaT2N 1N4
P. M. J. TREVELYAN
Affiliation:
Nonlinear Physical Chemistry Unit, Faculté des Sciences, Université Libre de Bruxelles (ULB), CP 231, 1050 Brussels, Belgium
J. AZAIEZ
Affiliation:
Department of Chemical and Petroleum Engineering, University of Calgary, Calgary, Alberta, CanadaT2N 1N4
A. DE WIT*
Affiliation:
Nonlinear Physical Chemistry Unit, Faculté des Sciences, Université Libre de Bruxelles (ULB), CP 231, 1050 Brussels, Belgium
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

When one solution of reactant A is displacing another miscible solution of reactant B, a miscible product C can be generated in the contact zone if a simple A + B → C chemical reaction takes place. Depending on the relative effect of A, B and C on the viscosity, different viscous fingering (VF) instabilities can be observed. In this context, a linear stability analysis of this reaction–diffusion–convection problem under the quasi-steady-state approximation is performed to classify the various possible instability scenarios. To do so, we determine the criteria for an instability, obtain dispersion curves both at initial contact time using an analytical implicit solution and at later times via numerical stability analysis. Along with recovering known results for non-reactive systems where the displacement of a more viscous fluid by a less viscous one leads to a VF instability, it is found that in the presence of a chemical reaction, injecting a more viscous fluid into a less viscous fluid can also lead to instabilities. This occurs when the chemical reaction leads to the build up of non-monotonic viscosity profiles. Various instability scenarios are classified in a parameter plane spanned by Rb and Rc representing the log-mobility ratios of the viscosities of the B and C solution respectively with respect to that of the injected solution of A. A parametric study of the influence on stability of the Damköhler number and of the time elapsed after contact of the two reactive solutions is also conducted.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 652 , 10 June 2010 , pp. 501 - 528
Copyright
Copyright © Cambridge University Press 2010

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

REFERENCES

Azaiez, J. & Singh, B. 2002 Stability of miscible displacements of shear thinning fluids in a Hele-Shaw cell. Phys. Fluids 14, 1557.CrossRefGoogle Scholar
Bánsági, T. Jr., Horváth, D., Tóth, A., Yang, J., Kalliadasis, S. & De Wit, A. 2003 Density fingering of an exothermic autocatalytic reaction. Phys. Rev. E 68, 55301.CrossRefGoogle ScholarPubMed
Chadam, J., Hoff, D., Merino, E., Ortoleva, P. & Sen, A. 1986 Reactive infiltration instabilities. IMA J. Appl. Math. 36, 207.CrossRefGoogle Scholar
Cross, M. & Hohenberg, P. C. 1993 Pattern-formation outside of equilibrium. Rev. Mod. Phys. 65, 851.CrossRefGoogle Scholar
De Wit, A. 2001 Fingering of chemical fronts in porous media. Phys. Rev. Lett. 87, 054502.CrossRefGoogle ScholarPubMed
De Wit, A., Bertho, Y. & Martin, M. 2005 Viscous fingering of miscible slices. Phys. Fluids 17, 054114.CrossRefGoogle Scholar
De Wit, A., De Kepper, P., Benyaich, K., Dewel, G. & Borckmans, P. 2003 Hydrodynamical instability of spatially extended bistable chemical systems. Chem. Engng Sci. 58, 4823.CrossRefGoogle Scholar
De Wit, A. & Homsy, G. M. 1999 a Viscous fingering in reaction–diffusion systems. J. Chem. Phys. 110, 8663.CrossRefGoogle Scholar
De Wit, A. & Homsy, G. M. 1999 b Nonlinear interactions of chemical reactions and viscous fingering in porous media. Phys. Fluids 11, 949.CrossRefGoogle Scholar
Fernandez, J. & Homsy, G. M. 2003 Viscous fingering with chemical reaction: effect of in situ production of surfactants. J. Fluid Mech. 480, 267.CrossRefGoogle Scholar
Gálfi, L. & Rácz, Z. 1988 Properties of the reaction front in an A + B → C type reaction–diffusion process. Phys. Rev. A 38, 3151.CrossRefGoogle Scholar
Gérard, T. & De Wit, A. 2009 Miscible viscous fingering induced by a simple A + B → C chemical reaction. Phys. Rev. E 79, 016308.CrossRefGoogle Scholar
Ghesmat, K. & Azaiez, J. 2009 Miscible displacements of reactive and anisotropic dispersive flows in porous media. Transport Porous Media 77, 489.CrossRefGoogle Scholar
Hejazi, S. H. & Azaiez, J. 2010 Non-linear interactions of dynamic reactive interfaces in porous media. Chem. Engng Sci. 65, 938.CrossRefGoogle Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271.CrossRefGoogle Scholar
Jahoda, M. & Hornof, V. 2000 Concentration profiles of reactant in a viscous finger formed during the interfacially reactive immiscible displacements in porous media. Powder Technol. 110, 253.CrossRefGoogle Scholar
Jiang, Z. & Ebner, C. 1990 Simulation study of reaction fronts. Phys. Rev. A 42, 7483.CrossRefGoogle ScholarPubMed
Loggia, D., Rakotomalala, N., Salin, D. & Yortsos, Y. C. 1995 Evidence of new instability thresholds in miscible displacements in porous media. Europhys. Lett. 32, 633.CrossRefGoogle Scholar
Loggia, D., Salin, D. & Yortsos, Y. C. 1998 The effect of dispersion on the stability of non-monotonic mobility profiles in porous media. Phys. Fluids 10, 747.CrossRefGoogle Scholar
Manickam, O. & Homsy, G. M. 1993 Stability of miscible displacements in porous media with nonmonotonic viscosity profiles. Phys. Fluids A 5, 1356.CrossRefGoogle Scholar
Mishra, M., Martin, M. & De Wit, A. 2007 Miscible viscous fingering with linear adsorption on the porous matrix. Phys. Fluids 19, 073101.CrossRefGoogle Scholar
Mishra, M., Martin, M. & De Wit, A. 2008 Differences in miscible viscous fingering of finite width slices with positive or negative log-mobility ratio. Phys. Rev. E 78, 066306.CrossRefGoogle ScholarPubMed
Nagatsu, Y., Bae, S. K., Kato, Y. & Tada, Y. 2008 Miscible viscous fingering with a chemical reaction involving precipitation. Phys. Rev. E 77, 067302.CrossRefGoogle ScholarPubMed
Nagatsu, Y., Kondo, Y., Kato, Y. & Tada, Y. 2009 Effects of moderate Damköhler number on miscible viscous fingering involving viscosity decrease due to a chemical reaction. J. Fluid Mech. 625, 97.CrossRefGoogle Scholar
Nagatsu, Y., Matsuda, K., Kato, Y. & Tada, Y. 2007 Experimental study on miscible viscous fingering involving viscosity changes induced by variations in chemical species concentrations due to chemical reactions. J. Fluid Mech. 571, 475.CrossRefGoogle Scholar
Pankiewitz, C. & Meiburg, E. 1999 Miscible porous media displacements in the quarter five-spot configuration. Part 3. Non-monotonic viscosity profiles. J. Fluid Mech. 388, 171.CrossRefGoogle Scholar
Podgorski, T., Sostarecz, M. C., Zorman, S. & Belmonte, A. 2007 Fingering instabilities of a reactive micellar interface. Phys. Rev. E 76, 016202.CrossRefGoogle ScholarPubMed
Rongy, L., Trevelyan, P. M. J. & De Wit, A. 2008 Dynamics of A + B → C reaction fronts in the presence of buoyancy-driven convection. Phys. Rev. Lett. 101, 084503.CrossRefGoogle Scholar
Schafroth, D., Goyal, N. & Meiburg, E. 2007 Miscible displacements in Hele-Shaw cells: nonmonotonic viscosity profiles. Eur. J. Mech. B 26, 444.CrossRefGoogle Scholar
Swernath, S. & Pushpavanam, S. 2007 Viscous fingering in a horizontal flow through a porous medium induced by chemical reactions under isothermal and adiabatic conditions. J. Chem. Phys. 127, 204701.CrossRefGoogle Scholar
Swernath, S. & Pushpavanam, S. 2008 Instability of a vertical chemical front: effect of viscosity and density varying with concentration. Phys. Fluids 20, 012101.CrossRefGoogle Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements in porous media: rectilinear flow. Phys. Fluids 29, 3549.CrossRefGoogle Scholar
Wei, C. & Ortoleva, P. 1990 Reaction front fingering in carbonate-cemented sandstones. Earth Sci. Rev. 29, 183.CrossRefGoogle Scholar