Autocatalytic reactions in the isothermal, continuous stirred tank reactor

Gray, Peter; Scott, Stephen K.

doi:10.1016/0009-2509(84)87017-7

Cited by 552 publications

(238 citation statements)

References 19 publications

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“…The GS system, introduced for continuously stirred systems in [14], models an irreversible reaction involving two reactants in a gel reactor, where the reactor is maintained in contact with a reservoir of one of the chemical species. The dimensionless GS model is (cf.…”

Section: Introductionmentioning

confidence: 99%

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

Wei

2006

Interfaces Free Bound.

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Slow translational instabilities of symmetric k-spike equilibria for the one-dimensional singularly perturbed two-component Gray-Scott (GS) model are analyzed. These symmetric spike patterns are characterized by a common value of the spike amplitude. The GS model is studied on a finite interval in the semi-strong spike-interaction regime, where the diffusion coefficient of only one of the two chemical species is asymptotically small. Two distinguished limits for the GS model are considered: the low feed-rate regime and the intermediate regime. In the low feed-rate regime it is shown analytically that k−1 small eigenvalues, governing the translational stability of a symmetric kspike pattern, simultaneously cross through zero at precisely the same parameter value at which k −1 different asymmetric k-spike equilibria bifurcate off of the symmetric k-spike equilibrium branch. These asymmetric equilibria have the general form SBB . . . BS (neglecting the positioning of the B and S spikes in the overall spike sequence). For a one-spike equilibrium solution in the intermediate regime it is shown that a translational, or drift, instability can occur from a Hopf bifurcation in the spike-layer location when a reaction-time parameter τ is asymptotically large as ε → 0. Locally, this instability leads to small-scale oscillations in the spike-layer location. For a certain parameter range within the intermediate regime such a drift instability for the GS model is shown to be the dominant instability mechanism. Numerical experiments are performed to validate the asymptotic theory.

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Section: Introductionmentioning

confidence: 99%

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Kolokolnikov

Ward

Wei

2006

Interfaces Free Bound.

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“…One of the simplest models of biochemical relevance leading to spatial and temporal patterns when diffusion is included is that due to Gray and Scott (GS) [4]. Numerical simulations of the deterministic GS system have revealed a rich set of strikingly complex and irregular patterns [5].…”

Section: Introductionmentioning

confidence: 99%

Spatiotemporal patterns driven by autocatalytic internal reaction noise

Hochberg

Zorzano

Morán

2005

The Journal of Chemical Physics

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The influence that intrinsic local density fluctuations can have on solutions of mean-field reactiondiffusion models is investigated numerically by means of the spatial patterns arising from two species that react and diffuse in the presence of strong internal reaction noise. The dynamics of the GrayScott (GS) model with constant external source is first cast in terms of a continuum field theory representing the corresponding master equation. We then derive a Langevin description of the field theory and use these stochastic differential equations in our simulations. The nature of the multiplicative noise is specified exactly without recourse to assumptions and turns out to be of the same order as the reaction itself, and thus cannot be treated as a small perturbation. Many of the complex patterns obtained in the absence of noise for the GS model are completely obliterated by these strong internal fluctuations, but we find novel spatial patterns induced by this reaction noise in regions of parameter space that otherwise correspond to homogeneous solutions when fluctuations are not included.

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“…Bifurcation theory, dynamical systems and limit cycles are discussed in many texts, for example see [19] or [20]. For the cubic autocatalytic system ((1) when p = 0), both the CSTR and the reaction-diffusion cell exhibit Hopf bifurcations, see [4] and [9]. Here the stability of the semi-analytical model of the mixed quadratic-cubic system is considered by finding the Hopf degenerate points and mapping these in parameter space.…”

Section: Local Stability and Oscillatory Solutionsmentioning

confidence: 99%

Mixed quadratic-cubic autocatalytic reaction–diffusion equations: Semi-analytical solutions

Alharthi

Marchant

Nelson

2014

Applied Mathematical Modelling

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Semi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a onedimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction-diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semi-analytical results are show to be highly accurate, in comparison to numerical solutions of the governing partial differential equations. AbstractSemi-analytical solutions for autocatalytic reactions with mixed quadratic and cubic terms are considered. The kinetic model is combined with diffusion and considered in a one-dimensional reactor. The spatial structure of the reactant and autocatalyst concentrations are approximated by trial functions and averaging is used to obtain a lower-order ordinary differential equation model, as an approximation to the governing partial differential equations. This allows semi-analytical results to be obtained for the reaction-diffusion cell, using theoretical methods developed for ordinary differential equations. Singularity theory is used to investigate the static multiplicity of the system and obtain a parameter map, in which the different types of steady-state bifurcation diagrams occur. Hopf bifurcations are also found by a local stability analysis of the semi-analytical model. The transitions in the number and types of bifurcation diagrams and the changes to the parameter regions, in which Hopf bifurcations occur, as the relative importance of the cubic and quadratic terms vary, is explored in great detail. A key outcome of the study is that the static and dynamic stability of the mixed system exhibits more complexity than either the cubic or quadratic autocatalytic systems alone. In addition it is found that varying the diffusivity ratio, of the reactant and autocatalyst, causes dramatic changes to the dynamic stability. The semianalytical ...

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Autocatalytic reactions in the isothermal, continuous stirred tank reactor

Cited by 552 publications

References 19 publications

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Slow translational instabilities of spike patterns in the one-dimensional Gray-Scott model

Spatiotemporal patterns driven by autocatalytic internal reaction noise

Mixed quadratic-cubic autocatalytic reaction–diffusion equations: Semi-analytical solutions

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