Well stirred system

Well-stirred systems are particularly convenient for the theoretician but are often less easy to realize in practice. Indeed, spatial inhomogeneities, with consequent molecular diffusion or thermal conduction processes, arise in many important situations—as varied as a single biological cell and a haystack. In the next three chapters we turn to unstirred systems, again seeking to determine bifurcation phenomena driven by non-linear kinetics. 

Of particular interest is the special case of a complex pair of principal eigenvalues whose real parts are passing through zero. This is the situation which we have seen corresponding to a Hopf bifurcation in the well-stirred systems examined previously. Hopf bifurcation points locate the conditions for the emergence of limit cycles. Using the CSTR behaviour as a guide it is relatively easy to find conditions for Hopf bifurcations, and then locally values of the diffusion coefficient for which a unique stationary state is unstable. Indeed the stationary-state profile shown in Fig. 9.5 is such a... 

So far almost all aspects of the stationary-state and even the time-dependent behaviour of this reaction-diffusion system differ only qualitatively from that found in the corresponding CSTR. In this section, however, we can consider a variation for which there can be no parallel in the well-stirred system—that of a reaction-diffusion cell set up with asymmetric boundary conditions. Thus we might consider our infinite slab with separate reservoirs on each side, with different concentrations of the autocatalyst in each reservoir. (For simplicity we will take the reactant concentration to be equal on each side.) Thus if we identify the reservoir concentration for p < — 1 as / L and on the other side (p + 1) as / R, the simple boundary conditions in eqn (9.11) are replaced by... 

First, we ask whether it is possible that the diffusion of the intermediate A and the conduction of heat along the box might destabilize a stable uniform state. An important condition for this is that the diffusion and conduction rates should proceed at different rates (i.e. be characterized by different timescales). Secondly, if the well-stirred system is unstable, can diffusion stabilize the system into a time-independent spatially non-uniform state Here we find a qualified yes , although the resulting steady patterns may be particularly fragile to some disturbances. 

We start with a well-stirred system, so the diffusion terms d2a./dx2 and d20/dx2 make no contribution. The stationary states of this spatially uniform case satisfy... 

These are very similar to the equations derived in chapter 3 for the decay or growth of small perturbations in well-stirred systems. Again we can expect exponential growth or decay, depending on the relative magnitudes of the four coefficients in these equations which, in turn, depend on y, fi, k, and / . 

We can recognize the first term as the trace of the matrix for the well-stirred system of chapter 4 (let us call this tr(U)) multiplied by the positive quantity y. We have specified that we are to consider here systems which have a stable stationary state when well stirred, i.e. for which tr(U) is negative. The additional term associated with diffusion in eqn (10.47) can only make tr(J) more negative, apparently enhancing the stability. There are no Hopf bifurcations (where tr(J) = 0) induced by choosing a spatial perturbation with non-zero n. 

Figure 10.7 shows this locus for a system with / = 10. Also shown, as a broken curve, is the Hopf locus for the well-stirred system. The latter is important, since we must remain outside this region for the uniform system to be stable in the absence of diffusion. Clearly, for this particular choice of / , there is a significant region in which the well-stirred system is stable (and hence the uniform state is stable to uniform perturbations) but unstable to pattern formation. 

We may also note, for the special case / = 1, that the locus described by eqns (10.58) and (10.59) is exactly that corresponding to the boundary between unstable focus and unstable node for the well-stirred system. This seems to be a general equivalence between the existence of unstable nodal solutions in the well-stirred system and the possibility of diffusion-driven pattern formation in the absence of stirring. We have seen in chapter 5 that unstable nodes are not found in the present model if the full Arrhenius rate law is used and the activation energy is low, i.e. iff <4 RTa. In that case we would also not expect spatial instability. 

Apart from the common factor y, these are the same as the equations governing growth or decay of small perturbations in the well-stirred system. The eigenvalues A1>2 are given by... 

Provided condition (10.70) is satisfied (i.e. provided the well-stirred system is unstable), this equation has a real solution for the wave number n for given values of /r and k. 

For these values of k there are no real conditions for which the determinant of J can change sign. This is equivalent to saying that the well-stirred system does not have unstable nodal states for k in this range. 

If we consider the well-stirred system, the stationary state has two Hopf bifurcation points at /r 2, where tr(U) = 0. In between these there are two values of the dimensionless reactant concentration /r 1>2 where the state changes from unstable focus to unstable node. In between these parameter values we can have (tr(U))2 — 4det(U) > 0, so there are real roots to eqn (10.76). 

When the dimensionless reaction rate constant lies in the range given by eqn (10.77), the well-stirred system has two Hopf bifurcation points /i 2. Over the range of reactant concentration... 

If the dimensionless rate constant satisfies inequality (10.78), the well-stirred system again has two Hopf bifurcation points fi and n. However, within the range of reactant concentrations between these, the uniform state also changes character from unstable focus to unstable node at n and n 2, as shown in Fig. 10.10. 

Kinetic coefficients fc, entering equations for the well-stirred systems like (8.1.4) to (8.1.6) are defined, in principle, by mutual diffusive approach of... 

The principal role of diffusion in these processes could be established considering rather simple examples [2]. If the kinetic equations for a well-stirred system are able to reproduce self-oscillations (the limit cycle), the extended system could be presented as a set of non-linear oscillators continuously distributed in space. Diffusion acts to conjunct these local oscillations and under certain conditions it can result in the synchronisation of oscillations. Thus, autowave solutions could be interpreted as a result of a weak coupling (conjunction) of local oscillators when they are not synchronised completely. The stationary spatial distributions in an initially homogeneous systems can also arise due to diffusion, which makes homogeneous solutions unstable. 

In this presentation we, therefore, investigate the kinetics of ion exchange in such mixtures for the case vdiere diffusion of the ions across a hydrostatic boundary layer (Nemst film) surrounding the particles is the rate controlling step (film diffusion). In well-stirred systems, liquid-phase mass transfer will usually be fevored by a low concentration of the external solution, a high ion-exchange capacity, and a small particle size [I]-... 

If we ignore the effect of p on k, which may be much smaller in well-stirred systems, the ignition condition can be written as... 

Figure 4 shows the experimental data taken for hydrogen-air combustion. The solid squares are the result of a complete kinetic calculation (not assuming partial equilibration) (26), treating the jet-stirred reactor as a well stirred system and inputing measured reactor temperature or heat loss rates. The reaction scheme included the following ... 

Fig. 4.5 Plot of measured (I ) versus inflow rate coefficient in the iodate-arsenous acid reaction run in an open, well-stirred system, a CSTR. The arrows indicate observed transitions from one branch of stable stationary states to the other stable branch, as the inflow rate coefficient is varied, and define the hysteresis loop. (Taken from [21] with permission.)...

Consider the steady state of the well-stirred system, [X]s, [Y]s, and small perturbations that move the system away from the steady state defined by x = [X] - [X]s and y = [Y] - [Y]s. These are substituted into Eqs. [56], and the resulting expressions are linearized by dropping nonlinear terms. As described earlier, this is formally carried out by writing Taylor series expansions for /"([X], [Y]) and g([X], [Y]) around the steady state concentrations [X]s, [Y]s and retaining only the linear terms. This procedure yields equations for the evolution of the perturbation in the linear regime of the steady state... 

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