Reaction-diffusion manifold

In the following, reaction flow analysis, sensitivity analysis and the directed relation graph method will be presented as static and dynamic reduction procedures. Thereafter will the main features of ILDM (including extensions such as flamelet generated manifolds (FGM) and reaction-diffusion manifolds (REDIM)), CSP and the LOI be discussed, including the fundamentals of the quasi steady state elimination procedure and the rate-controlled constrained equilibria (RCCE) approach. 

The Reaction-Diffusion Manifolds (REDIM) approach represent an extension to the formulation of the standard ILDM. Where the ILDM is in fact a relaxation of a set of ordinary differential equations (ODE s) describing a homogenous system, the REDIM formulation generalizes for a set of partially differential equation (PDE s) where also the coupling between the reaction and diffusion processes are accounted for. Bykov and Maas (Bykov Maas, 2007) have performed the full derivation of this generalized system in the framework of ILDM and an optimized tabulation procedure of generalized coordinates. They present the method in... 

Bykov, V. Maas, U. (2007). The extension of the ILDM concept to reaction-diffusion manifolds. Combustion Theory and Modelling Vol. 11, No. 6, pp 839-862... 

Bykov, V. Maas, U. (2009). Problem adapted reduced models based on Reaction-Diffusion Manifolds (REDIMs), Proceedings of the Combustion Institute Vol. 32, Issue 1, pp 561-568... 

Among many techniques used to obtain skeletons of larger mechanisms, in what follows we discuss the Direct Relation Graph (DRG) technique, coupled with Depth First Search (DFS) technique, the Sensitivity Analysis of the Jacobian matrix for the chemical system, the Intrinsic Low-Dimensional Manifold (ILDM) technique, the Reaction Diffusion Manifolds (REDIM technique) and the flamelet technique. Among these, the flamelet technique is preferred for writing the simplified chemical system for premixed flames and diffusion flames presented in the following sections. 

Bykov V, Maas U. Problem adapted reduced models based on Reaction-Diffusion Manifolds... 

Maas, U., Bykov, V. The extension of the reaction/diffusion manifold concept to systems with detailed transport models. Proc. Combust. Inst. 33, 1253-1259 (2011)... 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

N. Jangle. Center manifolds in reaction-diffusion systems on unbounded domains (Zentrumsmannigfaltigkeiten in Reaktions-Diffusions-Systemen auf unbeschrankten Gebieten). Diploma Thesis, Free University of Berlin, 2003. 

Fig. 5. A perspective plot of the spatio-temporal variation of the variable u x t) as computed with the reaction-diffusion system (3), with Dirichlet boundary conditions (4), the slow manifold (6) and the model parameters iii = -1.5, e = 0.01. (a) Stationary single-front pattern (u() = 1.1,D = 0.1,q = 0.01) (b) periodically oscillating single-front pattern (uo = 1.1, D = 0.045, a = 0.01) (c) periodic alternation of a single-front and a three-front pattern (u[) = 1.1, D = 0.01, a = 0.01) (d) stationary three-front pattern (uo = 0.5, D = 0.08, a = 0.2) (e) periodically oscillating three-front pattern (uq = 0.5, D = 0.06, a = 0.2) (f) periodic alternation of a single-front and a three-front pattern (uo = 0.5, D = 0.02, a = 0.2).

The whole zoology of patterns reported in Sections 4.1.1 and 4.1.2 are robust , in the sense that they can generically be observed in any reaction-diffusion system with a S-shaped slow manifold [62]. In particular, we have reproduced the patterns reported in Figures 5,6 and 7 with the following slow manifolds ... 

Fig. 8. Diffusion-induced chaos obtained when integrating the reaction-diffusion system (3) with Dirichlet boundary conditions (4) and the slow manifold (7) 6 = 10 ). The model parameters are e = 0.01, D = 0.05, a = A, uo = 2.5, u = -3.1. (a) Spatio-temporal variation of the variable u x t) coded as in Figure 6 (b) phase portrait (c) Poincar map (d) ID map.

Fig. 9. Spatio-temporal pattern forming phenomena in the reaction-diffusion system (3) with symmetric Dirichlet boundary conditions and the slow manifold (8). The model parameters are e = 0.01, a = 0.01, uo = ui = —2. (a) D = 0.0322560 (7,7 oscillating pattern confined to the lower branch (b) D = 0.0322550 Cr crisis-induced intermittent bursting (c) D = 0.0322307 homoclinic intermittent bursting (d) D = 0.0322400 PJ periodic bursting,...

In some experiments performed with some variants of the chlorite-iodide reaction, the oscillating front patterns have been observed to invade one of the end CSTRs [32, 33]. Henceforth the two CSTRs cannot be considered to be in a steady state during the experimental run as before. In order to account for the interplay of the dynamics inside the Couette reactor and in the CSTRs, we have performed subsequent numerical simulations [59,64] of our reaction-diffusion model (3)Avith the CSTR boundary conditions defined in Equation (5). We give here a short description of the patterns observed when considering the slow-manifold (6). The following parameters are kept fixed e = 10 , a = 0.5, uq = 2,ui = —4, Vi = f ui), i = 0,1. D = hg is our control parameter. 

Fig. 13. Displacement of a stationary front when increasing the diffusion coefficient D in the reaction-diffusion model (3). The boundary conditions are of CSTR type (Equation (5)), the slow manifold is given by Equation (6) and the model parameters are e = 10 - a = 0.5, U() = 2, wi = -4.

Fig. 16. Comparison between the exact stationary single-front solution (n(a ),t (a )) (full line) and the external approximation of the same solution (U x),V x)) (dashed line) of the reaction-diffusion model (3) with Dirichlet boundary conditions the piecewise linear slow manifold is given by Equation (7) with 6 = 0. (a) The functions u x) and U x) note that u and U differ significantly in the inner region only (b) phase portrait in the (n, v) plane by definition, the points (U x),V[x)) are located on the slow manifold (dashed line) (c) comparison between the exact solution (full line) and fhe function U[x) + i(0 (dashed line) (d) comparison between the exact solution v x) (full line) and the external function V x) (dashed line).

Fig. 18. Periodically oscillating two-front solution (breathing pattern) of the reaction-diffusion system (3) with Dirichlet boundary conditions, the slow manifold (6) and the model parameters e — 0.01, D = 0.03, a = 0.2, uo = ui = —1.5. (aj Unstable stationary profile u(x) (b) spatio-temporal variation of u(x, t) using the same coding as in Figure 6 (c) real part and (d) imaginary part of the u component of the critical Hopf eigenmode.

Using center-manifold theorem and normal form techniques [65,66], we have explicitly reduced our reaction-diffusion system (3) to the Hopf normal form (64) of the single-front solution. Technically we have used the normal form reduction method proposed by Coullet and Spiegel [110] (see also [111]). We refer the reader to [62] and [104], where this lengthy calculation has been carried out step by step. For the sake of simplicity, we will skip this technical part here, and we will focus on the theoretical prediction so obtained (on the critical surface n = 0) for the coefficient k in Equation (64). Our purpose is actually to compare the prediction for the value of Re K, with the measurement of the same quantity from direct simulations of the reaction-diffusion system (3). The numerical estimate is easily obtained from the amplitude of oscillation of the single-front solution in the ( , ") direction (Figure 20). If we write 2 = p e, the real part of Equation (64) yields ... 

Davis, M.J. Low-dimensional manifolds in reaction—diffusion equations. 1. Fundamental aspects. J. Phys. Chem. A 110, 5235-5256 (2006a)... 

Fig. 4. Schematic vacuum system for metal atom reactions. X represents the stopcock or Teflon-in-glass valve. Satisfactory components (for a general discussion of vacuum line design see References 1 and 4) forepump, 25 L/min free air capacity diffusion pump, 2 L/sec main trap is removable and measures about 300 mm deep main manifold has a diameter of about 25 mm, stopcock or valve in manifold should be at least 10 mm substrate container is removable container with 1-2 mm Teflon-in-glass needle valve connected to bottom of container. Connection between this needle valve and the reactor may be 1/8 in. od. Teflon tubing is used. Alternatively, the substrate may be added as shown in Fig. 3.

The vacuum line used in the following preparations is similar to that described by Shriver.14 It consists of a pump station, a main reaction manifold with six reaction stations, a fractionation manifold with four U-traps and a reaction station at each end, a McLeod Guage, and a Topler pump. The pump station employs a two-stage mechanical forepump and a two-stage mercury diffusion pump. Operating vacuum is 1.0 X 10"S torr. Teflon valves are employed throughout. 

A further consequence of the upstream diffusion to the burner face could be heterogeneous reaction at the burner. Such reaction is likely on metal faces that may have catalytic activity. In this case the mass balance as stated in Eq. 16.99 must be altered by the incorporation of the surface reaction rate. In addition to the burner face in a flame configuration, an analogous situation is encountered in a stagnation-flow chemical-vapor-deposition reactor (as illustrated in Fig. 17.1). Here again, as flow rates are decreased or pressure is lowered, the enhanced diffusion tends to promote species to diffuse upstream toward the inlet manifold. 

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