Solution upper-branch

For the adiabatic condition in which RHL is suppressed, the flame response exhibits the conventional upper and middle branches of the characteristic ignition-extinction curve, with the upper branch representing the physically realistic solutions. It can be noted that the effective Le of this lean methane/air mixture is sub-unity. It can be seen from Figure 6.3.1 that, with increasing stretch rate, first increases owing to the nonequidiffusion effects (S > 0), and then decreases as the extinction state is approached, owing to incomplete reaction. Furthermore, is also expected to degenerate to the adiabatic flame temperature, when v = 0. 

We can see from eqn (8.42) that k2 must be small if there are to be real solutions for the Hopf point. In fact the condition k212 < i is exactly the same as that for the existence of isolas (k2 < iV)- Thus all isolas have a point of Hopf bifurcation along their upper branch. 

We now know that if a system on the upper branch of the isola, just below the Hopf bifurcation point, is given a small perturbation which remains within the unstable limit cycle, it will decay back to the upper solution. If, however, the perturbation is larger, so we move to a point outside the cycle, we will not be able to get back to the upper solution the system must move to the other stable state, with no reactant consumption. 

If a saturated solution is represented by a point of the lower branch of the solubility curve, this solution becomes non-saturated when the concentration is diminished and supersaturated when the concentration is increased if, on the contrary, a saturated solution is represented by a point on the upper branch C, of the solubility curve, this solution becomes non-saturated when the concentration is increased and supersaturated when the concentration is diminished. 

Otherwise expressed, the unsaturated solutions are represented by the points of the plane TOs (Fig. 56) which are situated bdow the lower branch or above the upper branch Cj of the solubility curve the supersaturated solutions are represented by the points situated between the two branches. 

Pickering recognized for the sulphuric hydrates S0,-5H30 and SO,-2HjO the two branches Cj, Cj of the solubility curve, and he could follow each of these branches over a considerable temperature range for the hydrate S0, H,0 he found an indication of the existence of the upper branch for solutions more concentrated than the hydrate. 

Eutectic point between ferric chloride hydrates. Investigations of Bakhuis RoozboonL—Of all the solid hydrates which a salt solution may form, ice is always the most hydrated and the anhydrous salt the least hydrated always less rich in water than the first, the solution is always richer in water than the second the solubility curve of the first is reduced to its upper branch, the solubility curve of the other to its lower branch when these two branches meet their point of intersection is necessarily a eutectic point. 

Thus, in the galaxies observed, oxygen could be as abundant as in the interstellar medium near the Sun or as low as 1/10 solar. When the R23 method is applied to cB58, a similar ambiguity obtains (Teplitz et al. 2000). The results from the analysis of the interstellar absorption lines described above ( 4.2) resolve the issue by showing that the upper branch solution is favoured (we have no reason to suspect that the neutral and ionised ISM have widely different abundances). It remains to be established whether this is also the case for other LBGs. 

Type II (Solid-Fluid) System. In type II systems (when the solid and the SCF component are very dissimilar in molecular size, structure, and polarity), the S-L-V line is no longer continuous, and the critical (L = V) mixture curve also is not continuous. The branch of the three-phase S-L-V line starting with the triple point of the solid solute does not bend as much toward lower temperature with increasing pressure as it does in the case of type I system. This is because the SCF component is not very soluble in the heavy molten solute. The S-L-V line rises sharply with pressure and intersects the upper branch of the critical mixture (L = V) curve at the upper critical end point (LfCEP), and the lower temperature branch of the S-L-V line intersects the critical mixture curve at the lower critical end point (LCEP). Between the two branches of the S-L-V line there exists S-V equilibrium only (13). 

By addition of heat and increase in the amount of chlorine, the iodine monochloride disappears, and the system passes along the curve DE, which represents the composition of the solutions in equilibrium with solid iodine trichloride. The concentration of chlorine in the solution increases as the temperature is raised, until at the point E, where the solution has the same composition as the solid, the maximum temperature is reached the iodine trichloride melts. On increasing still further the concentration of chlorine in the solution, the temperature of equilibrium falls, and a continuous curve, similar to that for the monochloride, is obtained. The upper branch of this curve has been followed down to a temperature of 30 , the solution at this point containing 99 6 per cent, of chlorine. The very rounded form of the curve is due to the trichloride being largely dissociated in the liquid state. 

Consider again the steady-state temperature versus the residence-time curve shown in Figure 6.9 We have labeled seven points on this cun e A, B and C on the lower branch D on the middle branch and E, F and G on the upper branch. The numerical values at these points are listed in Table 6.2. If we first substitute the solution of the mass balance, Equation 6,.36, for CA into the energy balance, Equation 6.37, we obtain... 

The curve yielding Jq as a function of a shows the existence of a phenomenon of bistability in the reduced system when the substrate concentration is held constant. Three distinct values of yo are obtained in fig. 6.10 in the range a linear stability analysis of eqns (6.5) reveals that the steady state on the lower branch of the hysteresis curve is always stable, while it is unstable on the median branch, between points Sj and S2. On the upper branch, the steady state is unstable in the domain ai < a < 2. Two families of periodic solutions, denoted Tj and Fj, appear through a Hopf bifurcation at the points Hj and H2 they disappear at the points H and H 2, of abscissae a l, a 2, when the amplitude of the limit cycle is such that the latter reaches the... 

When the limit point 2 is reached, an increase in a elicits the abrupt transition towards the upper branch of the hysteresis curve, given that the lower branch has now vanished. But the upper branch of the steady-state curve of the (pr, y) system is unstable, and the analysis of the reduced system predicts that oscillations belonging to the branch of periodic solution F2 should occur when a is close to the value a"2. These oscillations correspond to the active phase of bursting represented in fig. 6.9b. 

Equation 6 which is the general solution to a variety of exitation schemes, can be solved graphically for ( ij (L). This is shown here for two cases. The solutions for a medium exhibiting Sg — transition (Fig. 2) incorporated in a matched ring resonator. The results for a coherent field pumping are depicted in Fig. The results for are used to calculate the output intensity as a function of lin (Fig. 4). The bistable behavior of this system (lower branch due to almost linear absorption and upper branch due to saturated absorption) can be also demonstrated in many other cases. The solutions for reverse saturable absorber - a medium exhibiting Sg — S. Ay T- —Tjj ( -jn on transitions - in the same optical resonator... 

Equistability of a homogeneous stable stationary state on the upper branch of the hysteresis loop, labelled I in Fig. 11.1, with a homogeneous stable stationary state on the lower branch, labelled II, occurs at one value of the influx coefficient k within the loop. Say that point occms at the location of line A. The predictions of the stationary solution of the master stochastic master equation are (a) the minimmn of the bimodal stationary probability distribution is located on the separatrix, and (b) at equistability the probability of fluctuations P(c) obeys the condition... 

Different stabilities and characters are represented by sn, stable node sf, stable focus un, unstable node uf, unstable focus and sp, saddle point. Also shown are the envelopes of stable limit cycles (sic) surrounding unstable solutions on upper branch. 

We see that for short times we obtain a unique solution. But there exists a critical time, t, beyond which new branches of solution appear, reflecting the formation of a second peak of the probability distribution. Eventually the upper branch dissapears, and the system evolves to extinction as combustion is completed. 

The solution in the upper branch correspond to Eq. (1.171) and it is characterized by cop-. in this case the dielectric constant of the metal is positive and both p (a>) and k (co) are imaginary, thus describing a propagating electromagnetic wave in the z direction. 

We now see in Eq. [5] a self-consistent equation for the determination of the temperature dependence of < P2 >. The order parameter < P2 > appears on both the left and right hand sides of the equation. For every temperature T (or / ) we can use a computer to obtain the value (or values) of < P2 > that satisfies the self-consistency equation. This process has been accomplished and the results are depicted in Fig. 2. < P2 > = 0 is a solution at all temperatures this is the disordered phase, the normal isotropic liquid. For temperatures T below 0.22284 v/k, two other solutions to Eq. [5] appear. The upper branch tends... 

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