The asymptotic analyses of non-linear waves in magneto-thermoelastic solids with thermal relaxation

Summary We investigate the magneto-thermoelastic model with one-dimensional deformation proposed in [1] when Fourier's law is replaced by Cattaneo's Eq. [4]. By means of an asymptotic analysis, we point out the main features of the non-linear wave processes related to different hypotheses based on the order of magnitude of physical parameters connected to thermal relaxation, heat and electrical conduction. In the various cases we obtain the evolution equations which govern the wave amplitude.

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Sommario In questa Nota si prende in esame il modello della magneto-thermoelasticità con deformazioni undimensionali proposto in [1], sostituendo peró alla legge di Fourier l'equazione di Cattaneo [4]. Facendo uso di metodi perturbativi, si mette in evidenza il carattere dei processi ondosi non-lineari connessi con different ipotesi sugli ordini di grandezza dei parametri fisici legati alla conduzione del calore, al rilassamento termico e alla conducibilità elettrica finita. Nei vari casi si deducono le equazioni di evoluzione che governano l'ampiezza d'onda.

     

The Thomas-Fermi theory of atoms,molecules and solids

We place the Thomas-Fermi model of the quantum theory of atoms, molecules, and solids on a firm mathematical footing. Our results include: (1) A proof of existence and uniqueness of solutions of the nonlinear Thomas-Fermi equations as well as the fact that these solutions minimize the Thomas-Fermi energy functional, (2) a proof that in a suitable large nuclear charge limit, the quantum mechanical energy is asymptotic to the Thomas-Fermi energy, and (3) control of the thermodynamic limit of the Thomas-Fermi theory on a lattice.     

The foundation of the grey matrix and the grey input–output analysis

The grey systems theory aims at the objects that their information is inadequate and this situation is general in reality. It has been urgent work to study the uncertain problems using the missing information. With the help of the simple introduction of grey systems theory, we further study the covered operation and get some calculation rules about grey number. The definition of grey matrix (GM) and its covered operation are proposed. Particularly, some results of the inverse grey matrix are obtained. Also with the help of the proposed grey matrix theory and the traditional input–output analysis, we propose the grey input–output analysis. The most important results are the computational formulas and their rigorous proofs of the matrix-covered set of the inverse grey Leontief coefficient’s matrix. It provides an effective tool to study an economic system by the input–output analysis under the uncertain situation. The modified case verifies the effectiveness of our methodology.     

A theory of heating of Voigt solids and fluids by external energy sources and flame theory

The purpose of this paper is to develop

1. a theory of laser stimulated vaporization of droplets,

2. a theory of internal heating resulting from vibration waves in linearly responding elastic material, and

3. flame theory.

There are applications to sending information through clouds on laser beams and to the control of temperature in ultrasonic welding, and improvement of the design of aircraft engines and the processes used for the destruction of toxic chemicals.

We develop a theory of thermal excursions resulting from ultrasonic welding in 3 and 7 dimensions, and interpret it as an elastic interaction with damping in a Voigt solid. It is hypothesized that with good control of temperature, one could achieve strong and uniform welds by this process and greatly reduce the cost of manufacturing aircraft, and other aluminum structures. We consider equations describing the conservation of mass, momentum, and energy coupled by an equation of state, and consider general mass, momentum, and energy transfer relationships in a compressible body subjected to external stimuli. For the Voigt solid theory, a linear elastic theory with damping forces, we show how some simple local time averaging gives us a dovetailed system consisting of the elastic wave equations whose solution provides the source term for an otherwise uncoupled heat equation. For the more general theory of droplet vaporization, we illustrate a general nonlinear energy equation which includes a radiation energy conductivity term. We get a class of exact solutions for a nonlinear flame front boundary value problem.     

The modeling mechanism,extension and optimization of grey GM (1, 1) model

The modeling mechanism of GM (1, 1) model is studied by using the thought of matrix analysis in this paper, the extension form GGM (1, 1) model based on the fractional order accumulated generating is put forward and its theoretical significance is analyzed. Furthermore, the influence of multiple transformation, translation transformation for the initial value and generating series on model parameters and predictive value are researched, then the quantitative relation among them is deduced and an optimization model and corresponding algorithm in practical modeling are presented.     

The oxidation process of silicon: Modelling and mathematical treatment

In this paper we develop a mathematical model describing the oxidation process of silicon. At first we introduce a model with a sharp reaction front between the silicon and the oxide layer. From this we turn over to a phase field model where the reaction front is replaced by an extended reaction zone. The silicon dioxide and the silicon are regarded as components of a reacting mixture, and the oxygen is assumed to be dissolved in it. We formulate a local existence result of that second model in three space dimensions assuming some simplifications concerning the boundary conditions and solve the complete free boundary value problem numerically. For this, it has been implemented into the process simulator DIOS which allows two-dimensional computations.     

Blow-up result and energy decay rates for binary mixtures of solids with nonlinear damping and source terms

In this paper we study the long-time behavior of binary mixture problem of solids, focusing on the interplay between nonlinear damping and source terms. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions, and uniqueness of weak solutions. Moreover, we prove that every weak solution to our system blows up in finite time, provided the initial energy is negative and the sources are more dominant than the damping in the system. Additional results are obtained via careful analysis involving the Nehari Manifold. Specifically, we prove the existence of a unique global weak solution with initial data coming from the “good” part of the potential well. For such a global solution, we prove that the total energy of the system decays exponentially or algebraically, depending on the behavior of the dissipation in the system near the origin.     

The transport of mass, energy and momentum in physiologic systems

A generic equation is developed to define the flux, Φ, of mass, energy and momentum in physiologic systems. This equation is written in terms of the concentration, ρ, of the quantity to be transported, the force driving it, a corresponding kinematic transport coefficient, , and the path and mobility associated with the transport process. Driving forces are expressed in terms of the gradient of some intensive property, ø, of the system, such that:

, where n=0 for energy, for momentum and n=1 for mass; β is a nondimensional coefficient of proportionality. Values of φ and are tabulated for various scenarios typical of transport processes occuring in physiologic systems.     

The Navier-Stokes equations: Existence theorems and energy equalities

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