Semi-analytical solutions for cubic autocatalytic reaction-diffusion equations; the effect of a precursor chemical

Muteb R Alharthi, Tim Marchant, Mark Nelson

Abstract


Semi-analytical solutions for a cubic autocatalytic reaction, with linear decay and a precursor chemical, are considered. The model is coupled with diffusion and considered in a one-dimensional reactor. In this model the reactant is supplied by two mechanisms, diffusion via the cell boundaries and decay of an abundant precursor chemical present in the reactor. The Galerkin method is used to approximate the spatial structure of the reactant and autocatalyst concentrations in the reactor. Ordinary differential equations are then obtained as an approximation to the governing partial differential equations and analyzed to obtain semi-analytical results for the reaction-diffusion cell. Singularity theory and a local stability analysis are used to determine the regions of parameter space in which the different types of bifurcation diagrams and Hopf bifurcations occur. The effect of the precursor chemical concentration is examined in detail and some novel behaviours are identified.

References
  • P. Gray and S. K. Scott. Chemical oscillations and instabilities: nonlinear chemical kinetics. Oxford University Press, London, 1990.
  • P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal continuous, stirred tank reactor: isolas and other forms of multistability. Chem. Engng. Sci., 38:29--43, 1983. doi:10.1016/0009-2509(83)80132-8
  • P. Gray and S. K. Scott. Autocatalytic reactions in the isothermal continuous, stirred tank reactor: oscillations and instabilities in the system $a + 2b \to 3b$; $b \to c$. Chem. Engng. Sci., 39:1087--1097, 1984. doi:10.1016/0009-2509(84)87017-7
  • S. R. Kay, S. K. Scott, and P. G. Lignola. The application of singularity theory to isothermal autocatalytic reactions: the influence of uncatalysed reactions. Proc. Roy. Soc. Lond A, 409:433--448, 1987. doi:10.1098/rspa.1986.0015
  • S. K. Scott. Isolas, mushrooms and oscillations in isothermal, autocatalytic reaction-diffusion equations. Chem. Engng. Sci., 42:307--315, 1987. doi:10.1016/0009-2509(87)85060-1
  • S. R. Kay and S. K. Scott. Multiple stationary states, sustained oscillations and transient behavior in autocatalytic reaction-diffusion equations. Proc. Roy. Soc. Lond A, 418:345--364, 1988. doi:10.1098/rspa.1988.0088
  • J. H. Merkin, D. J. Needham, and S. K. Scott. On the creation, growth and extinction of oscillatory solutions for a simple pooled chemical reaction scheme. SIAM Journal on Applied Mathematics, 47:1040--1060, 1987. http://www.jstor.org/stable/2101706
  • T. R. Marchant. Cubic autocatalytic reaction-diffusion equations: semi-analytical solutions. Proc. Roy. Soc. Lond A, 458:873--888, 2002. doi:10.1098/rspa.2001.0899
  • V. Balakotaiah and D. Luss. Multiplicity features of reacting systems. Chem. Engng. Sci., 38:1709--1721, 1983. doi:10.1016/0009-2509(83)85028-3
  • J. Guckenheimer and P. Holmes. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag, New York, 1983.
  • M. Golubitsky and D. G. Schaeffer. Singularites and groups in bifurcation theory. Springer-Verlag, New York, 1985.

Keywords


reaction-diffusion equations; cubic autocatalytic reactions; singularity theory; Hopf bifurcations; semi-analytical solutions

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DOI: http://dx.doi.org/10.21914/anziamj.v53i0.5340



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