### Abstract

The steady, spherically symmetric solutions to the reaction-diffusion equations based on a simple autocatalytic reaction followed by the decay of the autocatalyst are considered. Three parameters—the orders with respect to the autocatalyst in the autocatalysis p and in the decay q and the rate of decay of the autocatalyst relative to its autocatalytic production [Formula presented]—determine the steady concentration profiles. Numerical integrations for a fixed value of the order of the autocatalyst show that the concentration profiles have different forms depending on whether [Formula presented] or [Formula presented] In the former case, there is a critical decay rate [Formula presented] for solutions to exist, with multiple solutions for [Formula presented] In the latter case, there is a single solution for each value of K. This difference in the nature of the solution is confirmed by an analysis for p large. The temporal stability of the isothermal flame balls is examined, with temporally stable solutions being possible, provided that the ratio of the diffusion coefficient of the autocatalyst to that of the reactant is sufficiently small. The change in stability appears only when there are multiple solutions and is through a subcritical Hopf bifurcation.

Original language | English |
---|---|

Number of pages | 1 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 68 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 1 2003 |

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### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics

### Cite this

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*,

*68*(3). https://doi.org/10.1103/PhysRevE.68.036210

**Isothermal flame balls : Effect of autocatalyst decay.** / Jakab, Éva; Horváth, D.; Merkin, John H.; Scott, Stephen K.; Simon, Péter L.; Tóth, A.

Research output: Contribution to journal › Article

*Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics*, vol. 68, no. 3. https://doi.org/10.1103/PhysRevE.68.036210

}

TY - JOUR

T1 - Isothermal flame balls

T2 - Effect of autocatalyst decay

AU - Jakab, Éva

AU - Horváth, D.

AU - Merkin, John H.

AU - Scott, Stephen K.

AU - Simon, Péter L.

AU - Tóth, A.

PY - 2003/1/1

Y1 - 2003/1/1

N2 - The steady, spherically symmetric solutions to the reaction-diffusion equations based on a simple autocatalytic reaction followed by the decay of the autocatalyst are considered. Three parameters—the orders with respect to the autocatalyst in the autocatalysis p and in the decay q and the rate of decay of the autocatalyst relative to its autocatalytic production [Formula presented]—determine the steady concentration profiles. Numerical integrations for a fixed value of the order of the autocatalyst show that the concentration profiles have different forms depending on whether [Formula presented] or [Formula presented] In the former case, there is a critical decay rate [Formula presented] for solutions to exist, with multiple solutions for [Formula presented] In the latter case, there is a single solution for each value of K. This difference in the nature of the solution is confirmed by an analysis for p large. The temporal stability of the isothermal flame balls is examined, with temporally stable solutions being possible, provided that the ratio of the diffusion coefficient of the autocatalyst to that of the reactant is sufficiently small. The change in stability appears only when there are multiple solutions and is through a subcritical Hopf bifurcation.

AB - The steady, spherically symmetric solutions to the reaction-diffusion equations based on a simple autocatalytic reaction followed by the decay of the autocatalyst are considered. Three parameters—the orders with respect to the autocatalyst in the autocatalysis p and in the decay q and the rate of decay of the autocatalyst relative to its autocatalytic production [Formula presented]—determine the steady concentration profiles. Numerical integrations for a fixed value of the order of the autocatalyst show that the concentration profiles have different forms depending on whether [Formula presented] or [Formula presented] In the former case, there is a critical decay rate [Formula presented] for solutions to exist, with multiple solutions for [Formula presented] In the latter case, there is a single solution for each value of K. This difference in the nature of the solution is confirmed by an analysis for p large. The temporal stability of the isothermal flame balls is examined, with temporally stable solutions being possible, provided that the ratio of the diffusion coefficient of the autocatalyst to that of the reactant is sufficiently small. The change in stability appears only when there are multiple solutions and is through a subcritical Hopf bifurcation.

UR - http://www.scopus.com/inward/record.url?scp=85035301154&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85035301154&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.68.036210

DO - 10.1103/PhysRevE.68.036210

M3 - Article

AN - SCOPUS:85035301154

VL - 68

JO - Physical review. E

JF - Physical review. E

SN - 2470-0045

IS - 3

ER -