Uniform states

To reiterate, the development of these relations, (2.1)-(2.3), expresses conservation of mass, momentum, and energy across a planar shock discontinuity between an initial and a final uniform state. They are frequently called the jump conditions" because the initial values jump to the final values as the idealized shock wave passes by. It should be pointed out that the assumption of a discontinuity was not required to derive them. They are equally valid for any steady compression wave, connecting two uniform states, whose profile does not change with time. It is important to note that the initial and final states achieved through the shock transition must be states of mechanical equilibrium for these relations to be valid. The time required to reach such equilibrium is arbitrary, providing the transition wave is steady. For a more rigorous discussion of steady compression waves, see Courant and Friedrichs (1948). 

Note that in arriving at (2.58) it was assumed that there was no change in entropy along the rarefaction. This assumption is equivalent to stating that the rarefaction must be propagating into a uniform state. The Riemann Invarient has been defined in terms of the Riemann function... 

Riemann Invarient A constant defined by (2.61), which is independent of position on a rarefaction wave that propagates into a uniform state. 

Steady wave A propagating transition region that connects two uniform states of a material. The wave velocities of all parts of the disturbance are the same, so the profile does not change with time, and the assumptions that go into the jump conditions are valid. 

Boltzmann s H-Theorem. —One of the most striking features of transport theory is seen from the result that, although collisions are completely reversible phenomena (since they are based upon the reversible laws of mechanics), the solutions of the Boltzmann equation depict irreversible phenomena. This effect is most clearly seen from a consideration of Boltzmann s IZ-function, which will be discussed here for a gas in a uniform state (no dependence of the distribution function on position and no external forces) for simplicity. 

Recently [7] we constructed an example showing that interfacial flexibility can cause instability of the uniform state. Two elastic capacitors, C and C2, were connected in parallel. The total charge was fixed, but it was allowed to redistribute between C and C2. It was shown that if the interface was absolutely soft , i.e., contraction of the two gaps was not coupled, the uniform distribution became unstable at precisely the point where the dimensionless charge density s reached the critical value, = (2/3). In other words, the uniform distribution became unstable at the point where, under a control,... 

In summary, we may thus conclude that PGLa and GS do not form stable, NMR-observable pores in native membrane as readily as they do in model bilayers. The corresponding tilted and/or inserted states of our two representative MAPs could only be comprehensively characterized in DMPC-based samples, where the peptides could be trapped in a uniform state. In living cells, on the other hand, these states would seem to be only of a transient nature, i.e. at the very moment when the antimicrobial peptide attacks the membrane and passes through the lipid barrier along its concentration gradient towards the cytosol. 

The precise relationship is, however, very dependent on the conditions under which the powder bed was formed. In all cases the powder bed was brought to a uniform state of consolidation before determining the shear strength at a reduced load. 

The second issue is simply that the models in this section have never been fully unified. As discussed above, the model of Nelson and Powers considers thermal fluctuations in the curvature and the tilt direction about a uniform state but neglects the possibility of systematic modulations. By comparison, the models of Selinger et al. and Komura and Ou-Yang consider systematic... 

The fundamental difficulty is that polymeric substances cannot be obtained in a structurally and molecularly uniform state, unlike low-molecular-weight compounds. Thus, macromolecular materials of the same analytical composition may differ not only in their structure and configuration (see Sect. 1.2) but also in molecular size and molecular weight distribution they are polydisperse, i.e., they consist of mixtures of molecules of different size. Hence, it is understandable that the expression identical is not, in practice, applicable to macromolecules. Up to the present time, there is no possibility of preparing macromolecules of absolutely uniform structure and size. It follows, therefore, that physical measurements on polymers can only yield average values. The afore-... 

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. 

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. 

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. 

As a simple example, we might impose a perturbation with a cosine distribution as illustrated in Fig. 10.4. If the uniform state is stable to such a perturbation, the amplitude will decay to zero if the uniform state is unstable, the amplitude will grow. We could ask this question of stability with respect to any specific spatial pattern, but non-uniform solutions will also have to satisfy the boundary conditions. This latter requirement means that we should concentrate on perturbations composed of cosine terms, with different numbers of half-wavelengths between x = 0 and x = 1. 

Each component of the perturbations has been separated into two terms a time-dependent amplitude An and Tm, and a time-dependent spatial term cos (nnx). If the uniform state is stable, all the time-dependent coefficients will tend in time to zero. If the uniform state is temporally unstable even in the well-stirred case, but stable to spatial patterning, then the coefficients A0 and T0 will grow but the other amplitudes Ax-Ax and 7 1-7 0O will again tend to zero. If the uniform state becomes unstable to pattern formation, at least some of the higher coefficients will grow. This may all sound rather technical but is really only a generalization of the local stability analysis of chapter 3. 

One possible result of our perturbation is that all the modes associated with every possible integer value of n in the above equations will decay. Then A - 0, Tn - 0 for all n as t - oo. In this case, the uniform state will be stable to all 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... 

If the mass diffusion coefficient is sufficiently large compared with the thermal diffusivity, so P > / c, the range between n2- and n+ will be non-zero. There is another consideration n can only have discrete integer values, the lowest of which for a non-uniform state is n = 1. Thus for observable patterns we must make sure that at least n exceeds unity. Equation (10.49) shows that this last requirement puts a lower bound on the size parameter y we need y > yc, where... 

One way of presenting the various results above graphically is to take particular values for the chemical parameters p and k such that the uniform state is stable. The locus det(J) = 0 can then be plotted out in the p-y plane there will be different loci for different choices of the wave number n. 

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.

An alternative way of portraying the pattern formation behaviour in systems of the sort under consideration here is to delineate the regions in chemical parameter space (the h k plane) over which the uniform state is unstable to non-uniform perturbations. We have already seen in chapter 4, and in Fig. 10.3, that we can locate the boundary of Hopf instability (where the uniform state is unstable to a uniform perturbation and at which spatially uniform time-dependent oscillations set in). We can use the equations derived in 10.3.2 to draw similar loci for instability to spatial pattern formation. For this, we can choose a value for the ratio of the diffusivities / and then find the conditions where eqn (10.48), regarded as a quadratic in either y or n, has two real positive solutions. The latter requires that... 

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.

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. 

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