Zeldovich number

RT /E(T - T0) the Zeldovich number and give this number the symbol I in their book. Thus... 

If the matter is more important than the recognition, it was also true, for Zeldovich in particular, that the manner was as significant as the matter. He always proceeded by a direct intuitive physical approach to problems. Even in areas where his ultimate accomplishment was a mathematical formulation adopted by others such as the Zeldovich number in combustion theory or the Zeldovich spectrum and the Zeldovich approximation to linear... 

B Zeldovich number (or dimensionless adiabatic temperature rise) Bi Biot number... 

This term specifles the ratio Si /S and has been determined explicitly by Linan and Williams [13] by the procedure they call activation energy asymptotics. Essentially, this is the technique used by Zeldovich, Frank-Kamenetskii, and Semenov [see Eq. (59)]. The analytical development of the asymptotic approach is not given here. For a discussion of the use of asymptotics, one should refer to the excellent books by Williams [ 12], Linan and Williams [13], and Zeldovich et al. [10]. Linan and Williams have called the term RT /E(Tf—To) the Zeldovich number and give this number the symbol j3 in their book. Thus... 

Thus, the nondimensional adiabatic and initial temperatures are equal to zero and negative one, respectively. The nondimensional concentration varies from zero to one. Next, Z is the so-called Zeldovich number, which can be thought of as a nondimensional activation energy of the reaction and is 10 in both combustion and FP problems. It is the large parameter in our asymptotic studies below. Finally, the parameter 5 is very close to unity in combustion problems because T To, while in FP problem, it is about 0.5. 

The neutral stabihty curve in the (s, z)-plane has a minimum at s = 0.5. The neutral stability curve is shown in Figure 8 (the lowest curve). The uniformly propagating wave is unstable in the region above the curve, i.e., for z > Zc, and it is stable below the curve. It can be easily checked that the parameter z is nothing else but the Zeldovich number Z and can be also written in the form... 

Numerical simulations were performed on the planar problem in [83], which showed that the transition to the self-oscillatory combustion occurs when a parameter related to the Zeldovich number is increased. Numerical studies [5, 6, 79] of the one-dimensional problem found transitions to relaxation oscillations and period doublings, and these studies demonstrated two routes to chaotic dynamics as the bifurcation parameter related to the Zeldovich number was increased. Numerical studies [36 1,1,7,4] of the two- and three-dimensional model found spinning modes of propagation as well as standing modes, which describe multiple point propagation, and quasi-periodic modes of propagation. 

The objective of this paper is to examine the roles of the Zeldovich number Z and the nondimensional sample size R on the different modes of propagation possible in SHS. In particular, we expand on results in [6] for surface modes on a cylinder of radius R as well as chaotic modes occurring in planar SHS combustion. The characteristics of the resulting combustion wave depend significantly on the mode of propagation and impact on the nature of the product synthesized and indeed the ability of the combustion wave to propagate in large samples. 

The stability of this solution depends on the parameters of the problem. We first note that the Zeldovich number Z can be interpreted as the ratio of the diffusion time scale to and the reaction time scale U. Thus, for Z sufficiently small all the heat released in the reaction can be diffused away from the interface. However, for Z sufficiently large this is not the case. The heat released in the reaction which exceeds the amount of heat that can be carried away by diffusion necessarily raises the temperature at the interface. This, in turn, leads to an increase in the thermal gradient into the fresh fuel (leading to the interface speeding up), which lowers the interface temperature (leading to the interface slowing down), and the process repeats. This is the mechanism for the onset of oscillatory propagation as Z exceeds a critical value Zc. 

Traveling waves, 201, 227, 234, 237 Vortices, 284-285, 291-292 Wetting, 127-128 Young-Laplace condition, 159 Zeldovich number, 247, 250, 254 Zero mode, 130, 132, 154... 

The linear stability analysis of the longitudinally propagating fronts in the cylindrical adiabatic reactors with one overall reaction predicted that the expected frontal mode for the given reactive medium and diameter of reactor is governed by the Zeldovich number ... 

The single-head spin mode was studied in detail by Ilyashenko and Pojman (46), They were able to estimate the Zeldovich number using kinetic parameters for the initiator and the methacrylic acid. The value at room temperature was about 7, less than the critical value for spin modes. In fact, fronts at room temperature were planar and spin modes only appeared by lowering the initial temperature. However, spin modes could be observed by increasing the heat loss from the reactor by immersing the tube in water or oil. The simple analysis assumes an adiabatic system. 

Solovyov et al. performed a two-dimensional numerical study using a standard three-step free-radical mechanism (47), They calculated the Zeldovich number from the overall activation energy using the steady-state theory and determined the critical values for bifurcations to periodic modes and found that the complex kinetics stabilized the front. 

If the front propagates downwards, the convective motion decreases the heat loss from the high-temperature spot to the unreacted monomer. The heat is conserved near the reaction front, and the perturbation of the temperature has better conditions to increase. Hence the critical value of the Zeldovich number decreases. We note that the influence of convection on the spinning modes for polymerization fronts with a liquid polymer is different in comparison with the case of a solid polymer 31),... 

In stirred reactions, we saw that a steady state could lose its stability as a bifurcation parameter was varied, leading to oscillations. Propagating thermal fronts can show analogous behavior. The bifurcation parameter for a thermal front is the Zeldovich number (Zeldovich et al., 1985). Zeldovich assumed that the reaction occurs in an infinitely narrow region in a single step with activation energy eff, initial temperature Tq, and maximum temperature T ... 

The first analysis of the wrinkled flame structure was carried out by Barenblatt, Zeldovich and Istratov (1962) but in the diffusive-thermal model where the gas expansion effects i) and ii) are neglected. This model was extensively used these ten last years to culminate in the derivation by G. Sivashinsky (1977) of a non linear differential equation for the flame motion describing a self turbulizing behavior of the cellular structures (Michelson 6e Sivashinsky 1977). The main interest of this model is to provide us with a simple framework for studying systematically all the d3mamical effects that can possibely be produced by the diffusion of heat and mass. The asymptotic technique applied to solve this model in the limit of large values of the Zeldovich number is presented in the paper of Jou 1 in 6e Clavin (1979)... 

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