Reaction-diffusion problems in the physics of hot plasmas
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July 1, Gas Turbines Power. July ; 3 : — The object of this study is overlay coatings of MCrAlY alloy sprayed by a vacuum plasma spray VPS process for the protection against high-temperature corrosion and oxidation in the field of gas turbine components. Reaction diffusion behaviors at the interface between the MCrAlY coatings and the substrate, which have an important effect on coating degradation, have not always been clarified. The experimental results showed that the reaction diffusion layers consisted of aluminum compound layer and aluminum depleted layer, excepting that the aluminum depleted layer could not be observed in the case of CoNiCrAlY and NiCoCrAlY coatings.
It also indicated that the diffusion thickness could be observed to follow a parabolic time dependence. A convenient computer-aided system was developed for analyzing the reaction diffusion behaviors at the interface between coating and substrate. It was also clear that the estimated results of long time diffusion behaviors by simulation analysis was in good agreement with experiments.
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However, except for a very special choice of initial conditions, one amplitude will, after some time, become negative while the other starts growing. One way to limit the explosive growth would be to make a time dependent by introducing, after a certain time, a continued decrease of a. In the explosive type equation, the constant a accounts for an average general trend encompassing an extremely complicated internal structure in many cases of application.
One can draw parallels of the description of the overall situation with the behaviour of hot plasmas, where the parameters e. The reason is that the ultraviolet UV radiation from the sun, which usually ionizes the atoms of the ionosphere, experiences the shadowing effect by the moon.
Accordingly, the balance between the ionization and recombination, which is quadratic in the free electron density diffusion and attachment of free electrons to neutral atoms can be neglected , will be perturbed. The variation in density can easily be recorded by a ground-based electromagnetic radiation backscatter device.
The result is strictly valid for non-relativistic oscillations of a cold plasma if the ion motion is neglected [1. These relations illustrate an interesting interplay between the effects of nonlinearity and temperature effects and serves as simple examples of the forthcoming discussions. References [1. SSSR 80 [1. SSSR [1. Scripta 10 10 [1. Fluids 4 [1. Scripta 5 5 [1. Quantum Chem.
A continuum model is intrinsically non-linear and the structure of this nonlinear behaviour can be understood as follows. A plasma macroscopically can be considered a globally electrically neutral gas of charged particles electrons and ions interacting through long-range Coulomb electrostatic forces. Appropriate references on the full theoretical foundations of plasma kinetic theory and magnetized plasma transport theory are provided and should be consulted for a rigorous description, and to frame correctly the picture of linear and non-linear plasma response in special problems presented in this text.
When dimensional quantities are used, we use Gaussian cgs units, as is customary in plasma physics theory. Furthermore, the long-range Coulomb interaction among particles dominates the short-range interactions that can be described as a sudden variation of the velocity vector and are similar to collisions between neutral particles.
The solution of the kinetic problem 2. The reader should consult [2. For convenience, we report here the essential formal steps. Integrating equation 2.
Multiplication of equation 2. Although neo-classical theory is considered the rock-bottom theory of transport, providing the lower limit to transport effects, real experiments show largely anomalous values, attributed to different kinds of turbulence and instability regimes. In practice, the problems discussed will stem from appropriate variants of the continuity equation 2.
To be clear, we reconsider the energy balance equation 2. The most recent scaling obtained from databases of all the relevant tokamaks [2. The reader is therefore encouraged to consult the appropriate references for in-depth study of the formal aspects, which, for conciseness cannot be dealt with in this context. These partial differential equations belong to the class of second-order parabolic equations and have attracted considerable attention because of their ubiquitous appearance in science and their interesting mathematical properties.
General approximation methods of treating the problems of this type of non-linear equation must resort to preliminary, physically based ordering of time and space scales to allow the use of a variety of approximate methods. However, many interesting results escape systematic approximation procedures and can be found more conveniently by ad-hoc transformations of both the independent and the dependent variables, which lead to directly integrable equations or allow synthetic understanding of the properties of the solution in appropriate representation spaces.
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These techniques are well founded, but often not systematic, except for the topological methods of phase space analysis. The most useful approach is therefore the presentation of a collection of instructive examples that may become the paradigms of treatment of many new problems. References [2. Fluids 17 [2. I p [2. Pure Appl.
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Fluids 10 [2. Scripta 43 [2. Plasma Phys. Furthermore, the type of the evolution differential equations is still formidable as they are non-linear and partial, PDE. A strong motivation is that in the real world, all the concrete information about the state of a system is always available even in principle! The procedure is applicable also in the general and most important case of non-linear evolution equations [3. In this way, a synthetic view of the dynamics of the systems is often possible, achieving great insight even with a limited amount of explicit calculations.
However, this motivates a rational review and orderly collection of interdisciplinary applications, which on one hand produce new, sometimes unexpected results that guide full-blown numerical simulations, and on the other can help progress in achieving a higher standard of rigour. We shall consider examples in burning plasma physics transport problems, in non-linear electromagnetic waves and in the description of some non-linear resistive MHD perturbations. The main broad distinction is, however, between the two classes of elastic and inelastic processes among the interacting species, which have different properties with respect to conservation of number of particles, momentum and energy.
An important channel of electron energy loss in plasmas is due to inelastic collisions of the electrons with the residual neutral atoms present in the plasma, and their subsequent excitation to higher energy levels. The minimum energy for excitation of electron energy levels is of the order of 10 eV.
The information on the excitation cross-section is available from experiments for many atoms. The cross-section is a fast growing function of energy, just below the excitation threshold, with a maximum close to the threshold, followed by a decaying behaviour, due to the decrease of the interaction time of the electron with the atom. The recombination processes are complex, and require a three-body interaction e.
Therefore, radiative recombination may occur with emission of a quantum of radiation associated with the formation of a neutral atom and this is the inverse of a process of photoionization. This process becomes more important at lower densities and higher temperatures. Also a form of dissociative recombination exists where neutralization of a molecular ion is accompanied by dissociation of the molecule into two neutral atoms. These are examples of non-linear source or loss terms that may appear in particle balance equations describing complex processes. Inelastic collisions of ions with neutral atoms and with other ions are similar but more complex processes producing non-linear rates of accretion of charged species.
3(2,1)-D Solutions of Generalized Reaction Diffusion Equations for Dense Plasmas
As anticipated, the relevance of these processes depends on the temperature regime of the plasma considered. The interest here is primarily that of studying the formal properties of these solutions. In hot plasma oriented to fusion, the other non-linear processes that are relevant in determining a power balance are radiative losses and auxiliary heating input. In order to frame a problem correctly, it is useful to have a rather detailed knowledge of the properties of formal solutions of this type of equations, and this is the scope of this chapter.
The essential physical processes of diffusion and reaction occurring simultaneously lead to special solutions, which do not exist if only one kind of process contributes to the dynamics of the system [3. Exact solutions are rare in these conditions and systematic analytic methods of approximation are also cumbersome.
Interesting results concern exact particular solutions of equation 3. Here, we present the results for a wider range of combination of parameters and make comparisons with other exact particular solutions. We point out that equation 3. As shall be discussed further, the solutions of the type 3. In the model described by equation 3.