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Influence of Periodic and Quasi-periodic Gravitational Modulation on Convective Instability of Reaction Fronts in Porous Media

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Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

This work gives an overview on the effect of vertical periodic and QP gravitational modulations on the convective instability of reaction fronts in porous media. The model consists of the heat equation, the equation for the depth of conversion and the equations of motion under the Darcy law. Attention is focused on two cases. The case of a periodic gravitational vibration with a modulated amplitude, and the case of quasi-periodic vibration having two incommensurate frequencies. In both cases the heating is acted from below such that the sense of reaction is opposite to the gravity sense. The convective instability threshold is obtained by reducing the original reaction-diffusion problem to a singular perturbation one using the matched asymptotic expansion. The obtained reduced problem is then solved numerically after performing the linear stability analysis of the steady-state solution for the interface. It is shown that in the case of the modulation of the periodic vibration amplitude, a destabilizing effect of reaction fronts can be gained for a frequency modulation equal to half the frequency of the vibration, whereas a stabilizing effect is observed when the frequency of the modulation is twice that of the vibration. In the case of a quasi-periodic gravitational vibration it is indicated that for appropriate values of amplitudes and frequencies ratio of the quasi-periodic excitation, a stabilizing effect of reaction fronts can be successfully achieved.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University Hassan II-Mohammedia, 146, FST-Mohammadia, Morocco

    K. Allali

  2. Department of Mechanics, University Hassan II-Casablanca, 5366, Maârif, Casablanca, Morocco

    M. Belhaq

Authors
  1. K. Allali
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  2. M. Belhaq
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Corresponding author

Correspondence to M. Belhaq .

Editor information

Editors and Affiliations

  1. , Dpt. Quimica-Fisica-I, UCM, Madrid, 28040, Spain

    Ramon G. Rubio

  2. Instituto Pluridisciplinar, UCM, Paseo Juan XXIII 1, Madrid, 28040, Spain

    Yuri S. Ryazantsev

  3. , Dpt. of Chemical Engineering, Loughborough University, Loughborough, LE11 3TU, United Kingdom

    Victor M Starov

  4. , Physics Dpt., East China Normal University, Zhongshan Northern Road 3663, Shanghai, 200062, China, People's Republic

    Guo-Xiang Huang

  5. , Faculty of Physics, Saratov State University, Astrakhanskaya Str. 83, Saratov, 410012, Russia

    Alexander P Chetverikov

  6. , Dpt. di Ingegneria Elettrica Elettronica, Università di Catania, V. le A. Doria 6, Catania, 95125, Italy

    Paolo Arena

  7. , Dept. of Mathematics, Technion, Haifa, 32000, Israel

    Alex A. Nepomnyashchy

  8. , Instituto Cajal, CSIC, Av. Dr. Arce 37, Madrid, 28002, Spain

    Alberto Ferrus

  9. Shirshov Institute of Oceanology, Nakhimovsky prospekt 36, Moscow, 117851, Russia

    Eugene G. Morozov

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Allali, K., Belhaq, M. (2013). Influence of Periodic and Quasi-periodic Gravitational Modulation on Convective Instability of Reaction Fronts in Porous Media. In: Rubio, R., et al. Without Bounds: A Scientific Canvas of Nonlinearity and Complex Dynamics. Understanding Complex Systems. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34070-3_14

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  • DOI: https://doi.org/10.1007/978-3-642-34070-3_14

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  • Print ISBN: 978-3-642-34069-7

  • Online ISBN: 978-3-642-34070-3

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