Exponentially unstable

Quite naturally, the next question which arises is whether strong, exponential unstable chaos, being sufficient, is also necessary. 

The capillary flow with distinct evaporative meniscus is described in the frame of the quasi-dimensional model. The effect of heat flux and capillary pressure oscillations on the stability of laminar flow at small and moderate Peclet number is estimated. It is shown that the stable stationary flow with fixed meniscus position occurs at low wall heat fluxes (Pe -Cl), whereas at high wall heat fluxes Pe > 1, the exponential increase of small disturbances takes place. The latter leads to the transition from stable stationary to an unstable regime of flow with oscillating meniscus. 

This equation continues to conserve mass but is no longer stable. The original upset grows exponentially in magnitude and oscillates in sign. This marching-ahead scheme is clearly unstable in the presence of small blunders or round-off errors. 

The important phenomenon of exponential decay is the prototype first-order reaction and provides an informative introduction to first-order kinetic principles. Consider an important example from nuclear physics the decay of the radioactive isotope of carbon, carbon-14 (or C). This form of carbon is unstable and decays over time to form nitrogen-14 ( N) plus an electron (e ) the reaction can be written as... 

OS 39] ]R 25] ]P 27] A 24 h run of a pilot-scale micro reactor for azo pigment production was performed using a diazo suspension [55], At the end of this period, the pressure loss of the micro reactor increased exponentially. Special means were developed to prevent clogging and unstable operation. By partial removal of the deposits, the pressure loss was brought back to normal. 

In fact, an important advance in the phosphorescence theory was realized by Wiedemann in 1889, stating that a phosphor exists in two forms, a stable one, A, and an unstable one, B. Light absorption brings along conversion of form A to B, which then returns to A emitting light. This hypothesis was in agreement with the exponential decay law as postulated years before by Becquerel, but who did not provide any information about the nature of both forms [5],... 

For < 0 (unstable system). If the damping coefficient is negative, the exponential term increases without bound as time becomes large. Thus the system is unstable. 

The stability of the system is dictated by the values of the real parts of the roots. The system is stable if the real parts of all roots are negative, since the exponential terms go to zero as time goes to infinity. If the real part of any one of the roots is positive, the system is unstable. 

The solution to these kinetics is clearly unstable because it predicts unlimited exponential growth. Therefore, we need to assume that there is some sort of a death reaction. One form of this might be... 

Under these conditions there is a theoretical equilibrium vapor pressure of monomer above solid polymer which varies with temperature. This pressure is substantial at ambient temperature for poly (olefin sulfone)s and will increase exponentially with temperature. Thus, there will be a critical temperature above which the depropagation equilibrium vapor pressure will be higher than the condensation pressure and the polymer will be thermodynamically completely unstable. 

Again 2, and k2 are complex eigenvalues of the form Re(x) + ilm(A), but now they have positive real parts. Thus the exponential terms in (3.47) and (3.48) grow in time. The perturbation grows away from the unstable stationary state in a divergent oscillatory or unstable focal manner. 

Now ij and k2 are real and both positive. The exponential terms in (3.40) and (3.41) all increase monotonically in time. The perturbations grow exponentially and the system moves directly away from the unstable nodal state. 

Fig. 4.3. The ft-K parameter plane showing loci of changes in local stability or character for the model with the exponential approximation (a) full plane (b) enlargement of region near origin showing, in particular, the locus of Hopf bifurcation (change from stable to unstable focus) and the locus corresponding to the maximum in the ass(/r) curves (broken line).

This rather cluttered equation at least reduces very simply to the correct form for the exponential approximation (eqn (5.43)) in the limit as y->0. The numerator is a quadratic in 6. We will be interested to see if there are any conditions under which p2 becomes zero. If P2 is negative, the emerging limit cycle is stable as before. If P2 becomes positive, the emerging limit cycle will lose its stability and become unstable. 

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