Continuum mathematics

This technique becomes problematic when the particles touch—for example, for the constituents of atomic nuclei. Already, spin forced us to consider quantization without potentials. Many other strange quantum numbers have been posited, with no help from continuum mathematics. Perturbation expansions become funny, since the interaction is no longer smaller than some overriding field. Nucleon-nucleon potentials are discussed in terms of pion exchange, and may also be discussed in terms of quark-gluon interactions. 

Such a simplification is possible through the introduction of a continuum mathematical description of the gas-solid flow processes where this continuum description is based upon spatial averaging techniques. With this methodology, point variables, describing thermohydrodynamic processes on the scale of the particle size, are replaced by averaged variables which describe these processes on a scale large compared to the particle size but small compared to the size of the reactor. There is an extensive literature of such derivations of continuum equations for multiphase systems (17, 18, 19). In the present study, we have developed (17, ) a system of equations for... 

Multiscale models can be based only on continuum mathematical descriptions, e.g., as in the zeolite-based chemical reactor illustrated in Fig. 5, where the conservation equation describing the reactant transport along the reactor has a source term being calculated from the boundary condition of a macropore diffusion model with a source term calculated in turn from the boundary condition of a micropore diffusion model with a reaction kinetics source term [44]. 

Khludnev A. M. (1992) Contact viscoelastoplastic problem for a beam. In Free boundary problems in continuum mechanics. S.N.Antontsev, K.-H. Hoffmann, A.M.Khludnev (Eds.). Int. Series of Numerical Mathematics 106, Birkhauser Verlag, Basel, 159-166. 

The continuity equation is a mathematical formulation of the law of conservation of mass of a gas that is a continuum. The law of conservation of mass states that the mass of a volume moving with the fluid remains unchanged... 

From this kind of continuum mechanics one can move further towards the domain of almost pure mathematics until one reaches the field of rational mechanics, which harks back to Joseph Lagrange s (1736-1813) mechanics of rigid bodies and to earlier mathematicians such as Leonhard Euler (1707-1783) and later ones such as Augustin Cauchy (1789-1857), who developed the mechanics of deformable bodies. The preeminent exponent of this kind of continuum mechanics was probably Clifford Truesdell in Baltimore. An example of his extensive writings is A First Course in... 

A mathematical description of continuum fluid dynamics can proceed from two fundamentally different points of view (1) A macroscopic, or top-down ([hass87], [hass88]), approach, in which the equations of motions are derived as the most... 

On the continuum level of gas flow, the Navier-Stokes equation forms the basic mathematical model, in which dependent variables are macroscopic properties such as the velocity, density, pressure, and temperature in spatial and time spaces instead of nf in the multi-dimensional phase space formed by the combination of physical space and velocity space in the microscopic model. As long as there are a sufficient number of gas molecules within the smallest significant volume of a flow, the macroscopic properties are equivalent to the average values of the appropriate molecular quantities at any location in a flow, and the Navier-Stokes equation is valid. However, when gradients of the macroscopic properties become so steep that their scale length is of the same order as the mean free path of gas molecules,, the Navier-Stokes model fails because conservation equations do not form a closed set in such situations. 

Omitting all discussion of the mathematical properties and the subtleties with regard to the continuum of final state energies, we will hop to the perturbation expansion in three short equations. The initial state describing the unperturbed reactant (DA) system describes the solution to the zeroth order Schrodinger equation ... 

TNC.22. P. Glansdorff and I. Prigogine, On the general theory of stability of thermodynamic equilibrium, in Problems of Hydrodynamics and Continuum Mechanics, Society for Industrial and Applied Mathematics, Philadelphia, 1969. 

The basic tenet of continuum fracture mechanics is,- therefore, that the strength of most real solids is governed by the presence of flaws and, since the various theories enable the manner in which the flaws propagate under stress to be analysed mathematically, the application of fracture mechanics to crack growth in polymers has received considerable attention. Two main, inter-relatable, conditions for fracture are proposed. 

In principle, solvent trapping is also included in the vibrational overlap intergral in equation (25). As noted above, the solvent dipole reorientational motions associated with solvent trapping can be treated as a series of collective vibrations of the medium using approaches devised for treating the collective vibrations of ions or atoms that occur in a crystalline lattice.32-33 However, the problem is mathematically intractable because of the many solvent molecules involved which lead to many normal modes and the existence of a near continuum of energy levels. In addition,... 

As early as 1829, the observation of grain boundaries was reported. But it was more than one hundred years later that the structure of dislocations in crystals was understood. Early ideas on strain-figures that move in elastic bodies date back to the turn of this century. Although the mathematical theory of dislocations in an elastic continuum was summarized by [V. Volterra (1907)], it did not really influence the theory of crystal plasticity. X-ray intensity measurements [C.G. Darwin (1914)] with single crystals indicated their mosaic structure (j.e., subgrain boundaries) formed by dislocation arrays. Prandtl, Masing, and Polanyi, and in particular [U. Dehlinger (1929)] came close to the modern concept of line imperfections, which can move in a crystal lattice and induce plastic deformation. 

The abstract conception of a continuum and the mathematics required to describe it and its variations are discussed below. 

Here c(x, t)dx is the concentration of material with index in the slice (x, x + dx) whose rate constant is k(x) K(x, z) describes the interaction of the species. The authors obtain some striking results for uniform systems, as they call those for which K is independent of x (Astarita and Ocone, 1988 Astarita, 1989). Their second-order reaction would imply that each slice reacted with every other, K being a stoichiometric coefficient function. Only if K = S(z -x) would we have a continuum of independent parallel second-order reactions. In spite of the physical objections, the mathematical challenge of setting this up properly remains. Ho and Aris (1987) have shown how not to do it. Astarita and Ocone have shown how to do something a little different and probably more sensible physically. We shall see that it can be done quite generally by having a double-indexed mixture with parallel first-order reactions. The first-order kinetics ensures the individuality of the reactions and the distribution... 

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