Uniform states stationary

Chemical reactions with autocatalytic or thermal feedback can combine with the diffusive transport of molecules to create a striking set of spatial or temporal patterns. A reactor with permeable wall across which fresh reactants can diffuse in and products diffuse out is an open system and so can support multiple stationary states and sustained oscillations. The diffusion processes mean that the stationary-state concentrations will vary with position in the reactor, giving a profile , which may show distinct banding (Fig. 1.16). Similar patterns are also predicted in some circumstances in closed vessels if stirring ceases. Then the spatial dependence can develop spontaneously from an initially uniform state, but uniformity must always return eventually as the system approaches equilibrium. 

From the results of chapters 4 and 5, we can predict the behaviour of the system if it is well stirred. For some experimental conditions, represented by particular values for the dimensionless reactant concentration /t and the rate constant k, the system will have a uniform, stable stationary state (really only a pseudo-stationary state as /t is decreasing slowly because of the inevitable consumption of the reactant discussed previously). For other conditions, the stationary state loses its stability and stable uniform oscillations can be... 

However, these stationary-state equations are still satisfied by the uniform solutions (10.19) and (10.20) for which the spatial derivatives are zero and the reaction terms cancel. We denote this uniform state a, 6. 

Thus, when the stirring stops, the uniform state remains a stationary solution of the system. Diffusion does not affect the existence of the uniform state, but it may influence its stability. In particular we are interested in determining whether this state can become unstable to spatially non-uniform perturbations. 

If det(J) is positive for all n, then the amplitudes of all the components of any perturbation will decay back to the spatially uniform stationary state. As mentioned above, det(J) is positive for n = 0, and clearly will always be positive for sufficiently large n when the last term dominates. However, eqn (10.48) is a quadratic in n2 a completely stable uniform state arises if there are no real solutions to the condition det(J) = 0- We can write the... 

Fig. 10.5. Representation of the conditions for which the uniform stationary state is unstable to perturbations with a particular number-of half-wavelengths (a) n = 2 (b) the loci for n = 1,2,3, and 4, showing how these overlap to create regions in which more than one spatial perturbation may cause departure from the uniform state.

Fig. 10.7. Representation of conditions in the K-fi parameter plane for instability of the uniform stationary state with respect to spatial perturbations for a system with f = 10. Also shown (broken curve) is the Hopf bifurcation locus, within which the uniform state is itself unstable. The two loci cross at some point on their upper shores.

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

Considering these relations simultaneously with Eq. (6) leads to a number of consequences of practical importance. From Eq. (10) it follows that nonuniform stationary states can arise only from the destruction of those uniform states which correspond to the ascending branch of rj) = 0, since the derivative dGjdt] < 0 within the entire interval 0 < i/j < 1, and 50/50 > 0 only in the region of 0 values lying just on this branch. 

Let us emphasize that there are no other spatially uniform stationary solutions except (p = 0. Thus, when the latter solution is unstable, the system tends to a non-uniform state, i.e. pattern formation takes place. A direct simulation shows that stripe patterns are formed [36], with the stripes wavelength near 2tt fkc- Note that because of the rotational invariance of problem (14) the orientation of the stripes is arbitrary. Initially, a disordered system of stripes is developed from random initial conditions, and then some large domains are developed with a definite orientation of stripes inside each domain. The mean domain size grows with time, i.e. domain coarsening takes place for differently oriented stripe patterns rather than for different uniform phases. 

The stationary state (being not necessarily uniform) corresponds to the minimum of F at given N = X), Ci ... 

Rod bundle heat transfer analysis (Anklam, 1981a) A 64-rod bundle was used with an axially and radially uniform power profile. Bundle dimensions are typical of a 17 X 17 fuel assembly in a PWR. Experiments were carried out in a steady-state mode with the inlet flow equal to the steaming rate. Generally, about 20-30% of the heated bundle was uncovered. Data were taken during periods of time when the two-phase mixture level was stationary and with parameters in the following ranges ... 

Up until the mid-1940s, most physical electrochemistry was based around the dropping mercury electrode. However, in 1942, Levich showed that rotating a disc-shaped electrode in a liquid renders it uniformly accessible to diffusion, yet the hydrodynamics of the liquid flow are soluble and the kinetic equations relatively simple. In addition, in contrast to the case of a stationary planar electrode, the current at an RDE rapidly attains a steady-state value. 

In steady-state operation, each stage of a CSTR is in a stationary state (uniform CA, T, etc.), which is independent of time. 

Protection of quantum states from the influence of noise is important. It has been shown that the alternating transport of a EEC generated by the fast-forward driving field suppresses the influence of a fluctuating random potential on the EEC [47], The EEC is kept undisturbed for a longer time than is characteristic of the simple trapping with a stationary potential because the effective potential, which the quanmm state feels, becomes uniform when the transport velocity is sufficiently large. 

Fig. 10.2. A typical non-uniform stationary-state profile. Note the vanishing spatial derivative at the end walls (r = 0 and r = a0) appropriate to zero-flux boundary conditions.

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... 

The division of the parameter plane into regions of stability and instability is reproduced in Fig. 10.3. If pi and k lie within the closed region, the stationary state is unstable and spatially uniform oscillations exist. 

Fig. 10.3. The locus of Hopf bifurcation points indicating the conditions for loss of local stability for the spatially uniform stationary-state solution. Inside this region the system may show spatially uniform time-dependent oscillations.

Fig. 10.4. A typical spatially dependent perturbation about a uniform stationary-state solution.

Let us consider the effect of small perturbations about the uniform stationary state a, 0 given by eqns (10.26) and (10.27). For ease of manipulation we can represent the concentration and temperature rise as... 

The stationary-state solution a is constant in time and uniform in space, so da/dx = d2a/dx2 = 0. Our specification that the perturbations Aa and A0 should be small also allows us to linearize the reaction terms, giving... 

Substituting for a and ff, the first two terms in the brackets cancel (because that is the uniform stationary-state condition). Proceeding in a similar way for the temperature perturbations, we finally obtain the governing equations... 

If P, n, and k are chosen suitably, this equation may have two real and positive solutions, ni and n+ say. The uniform stationary state is then unstable to perturbations which have n half-wavelengths where n is an integer lying in the range... 

Recognizing that n and k can be related to the uniform stationary-state temperature excess 0SS = h/k, the condition for real positive solutions to eqn (10.48) to exist can be expressed as... 

We can think of the reactant concentration and some initial spatial distribution of the intermediate concentration and temperature profiles specifying a point on Fig. 10.9. If we choose a point above the neutral stability curve, then the first response of the system will be for spatial inhomogeneity to disappear. If the value of /r lies outside the range given by (10.79), then the system adjusts to a stable spatially uniform stationary state. If ji lies between H and n, we may find uniform oscillations. 

The analysis starts when a small quantity of sample in liquid or gaseous state is injected. The dual role of the injector is to vaporise the analytes and to mix them uniformly in the mobile phase. Once the sample is vaporised in the mobile phase, it is swept into the column, which is usually a tube coiled into a very small section with a length that can vary from 1 to over 100 m. The column containing the stationary phase is situated in a variable temperature oven. At the end of the column, the mobile phase passes through a detector before it exits to the atmosphere. Some gas chromatograph models have their own power supply, permitting them to be used in the field (see Fig. 2.16). 

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