Perturbation wave length

The classical linear stability theory for a planar interface was formulated in 1964 by Mullins and Sekerka. The theory predicts, under what growth conditions a binary alloy solidifying unidirectionally at constant velocity may become morphologically unstable. Its basic result is a dispersion relation for those perturbation wave lengths that are able to grow, rendering a planar interface unstable. Two approximations of the theory are of practical relevance for the present work. In the thermal steady state, which is approached at large ratios of thermal to solutal diffusivity, and for concentrations close to the onset of instability the characteristic equation of the problem... 

A weakness of these methods lies in the limited number of zeroth-order states that are used for an expansion of the first-order perturbed wave function. In particular, it has been demonstrated that probabilities of spin-forbidden radiative transitions converge slowly with the length of the perturbation expansion.92... 

A forbidden energy state, that is to say, an electron wave (defined as to direction and wave length) which is perturbed by the lattice signifies an electron wave which is reflected by the lattice on penetration from the outside. 

The analogous expansion may be developed for the nth-order perturbed wave function P(n) in Eq. (22b), but they must be expressed in terms of open diagrams, i.e., those in which all the lines are not closed into loops. Strictly speaking, such diagrams occur in the expansion of the wave operator, a concept studied at length by Lowdin,35 since their algebraic equivalents are second-quantized operators. The nth-order wave... 

We further assume that the perturbation is small, while the wave length is very large compared with the tube radius. Thus, the nonlinear inertia variables are negligible and the linearized conservation equation of mass and momentum become, respectively. 

Landau and Darrieus pioneering works on hydrodynamic instability of a flat laminar flame are well known [1]. According to Darrieus-Landau theory, small perturbations, independent of the wave length, make a flat flame unstable. 

Numerical solution of Chazelviel s equations is hampered by the enormous variation in characteristic lengths, from the cell size (about one cm) to the charge region (100 pm in the binary solution experiments with cell potentials of several volts), to the double layer (100 mn). Bazant treated the full dynamic problem, rather than a static concentration profile, and found a wave solution for transport in the bulk solution [42], The ion-transport equations are taken together with Poisson s equation. The result is a singular perturbative problem with the small parameter A. 

For the purpose of illustration, in this paper we use a viscosity-capillarity model (Truskinovsky, 1982 Slemrod, 1983) as an artificial "micromodel",and investigate how the information about the behavior of solutions at the microscale can be used to narrow the nonuniqueness at the macroscale. The viscosity-capillarity model contains a parameter -Je with a scale of length, and the nonlinear wave equation is viewed as a limit of this "micromodel" obtained when this parameter tends to zero. As we show, the localized perturbations of the form x /-4I) can influence the choice of attractor for this type of perturbation, support (but not amplitude) vanishes as the small parameter goes to zero. Another manifestation of this effect is the essential dependence of the limiting solution on the... 

In non-relativistic perturbative atomic Z-expansion theory, as recently sum-marked [11], a new scaled length, p = Zr, and a scaled energy, e= T2 E, are introduced in the many-electron wave equation. That is, the units of length and energy are changed to 1/Z and Z2 a.u., respectively. The Hamiltonian then takes the form... 

Before recombination, the radiation pressure is so great that the Jeans length is greater than the horizon size, and so no perturbations within the horizon can grow they can only oscillate as sound waves. Conversely, after recombination, the pressure drops precipitously and all of a sudden, perturbations within the horizon can grow. 

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