One spatial dimension

For a set of Gaussians, it is rather difficult to establish the analytic behavior of Eqs. f55), or of (50), in the t plane. However, with a single Gaussian (in one spatial dimension) and a harmonic potential surface one classically has... 

The reader is encouraged to use a two-phase, one spatial dimension, and time-dependent mathematical model to study this phenomenon. The UCKRON test problem can be used for general introduction before the particular model for the system of interest is investigated. The success of the simulation will depend strongly on the quality of physical parameters and estimated transfer coefficients for the system. 

The simplest kind of gridpoint model is one where only one spatial dimension is considered, most often the vertical. Such one-dimensional models are particularly useful when the conditions are horizontally homogeneous and the main transport occurs in the vertical direction. Examples of such situations are the vertical distribution of CO2 within the ocean (except for the downwelling regions in high latitudes, Sie-genthaler, 1983) and the vertical distribution of... 

This equation has been derived as a model amplitude equation in several contexts, from the flow of thin fluid films down an inclined plane to the development of instabilities on flame fronts and pattern formation in reaction-diffusion systems we will not discuss here the validity of the K-S as a model of the above physicochemical processes (see (5) and references therein). Extensive theoretical and numerical work on several versions of the K-S has been performed by many researchers (2). One of the main reasons is the rich patterns of dynamic behavior and transitions that this model exhibits even in one spatial dimension. This makes it a testing ground for methods and algorithms for the study and analysis of complex dynamics. Another reason is the recent theory of Inertial Manifolds, through which it can be shown that the K-S is strictly equivalent to a low dimensional dynamical system (a set of Ordinary Differentia Equations) (6). The dimension of this set of course varies as the parameter a varies. This implies that the various bifurcations of the solutions of the K-S as well as the chaotic dynamics associated with them can be predicted by low-dimensional sets of ODEs. It is interesting that the Inertial Manifold Theory provides an algorithmic approach for the construction of this set of ODEs. 

In one spatial dimension (usually denoted as the slice-select direction), a selective RF pulse with a certain... 

Fick s first law States that a net diffusive flux occurs spontaneously, moving solutes from regions of high concentration to regions of low concentration, with a magnitude that is proportional to the concentration gradient ([dC]/dz). In one spatial dimension (z), Flux = -D ([dQ/dz). 

We can easily recover the mass- and energy-balance equations used in previous chapters from these equations. First we assume variation in only one spatial dimension, the direction of flow z, to obtain... 

The mass- and energy-balance equations for our new system, allowing for diffusion and conduction along one spatial dimension r, can be written as... 

In order to fully appreciate the consequences of the rather simple mathematical rules which describe the random walk, we move one step further and combine Fick s first law with the principle of mass balance which we used in Section 12.4 when deriving the one-box model. For simplicity, here we just consider diffusion along one spatial dimension (e.g., along the x-axis.)... 

In this section we consider problems in which there is convective and diffusive transport in one spatial dimension, as well as elementary chemical reaction. The computational solution of such problems requires attention to discretization on a mesh network and solution algorithms. For steady-state situations the computational problem is one of solving a boundary-value problem. In chemically reacting flow problems it is not uncommon to have steep reaction fronts, such as in a flame. In such a case it is important to provide adequate mesh resolution within the front. Adaptive mesh schemes are used to accomplish this objective. 

The vector notation is dropped when there is only one spatial dimension. Unless indicated otherwise, the diffusion is assumed to be isotropic in this chapter. Anisotropic diffusion is treated in Section 4.5. 

Optically excited semiconductor nanostructures show effects of QC if at least one spatial dimension of the material becomes comparable to, or smaller than the characteristic length scale (the classical Bohr radius) of an e-h pair. Different regimes of QC have been defined which depend on the semiconductor nanocrystallite size R relative to that of the Bohr radius of the exciton an, the electron ae or the hole ah ... 

For the description of phase separation we choose again the generic Cahn-Hilliard model in one spatial dimension [124, 125]... 

Consider solutions of the Poisson equation in one spatial dimension for a system with a uniform charge density /Cm the interval — /2 general solution for this domain, and discuss the possibilities for making this general solution periodic with period L. 

This can be shown with a current-density generalization [24] of the so-called Harriman construction [34], Here we reproduce the construction for one spatial dimension. The three dimensional case can be treated in analogy to Ref. [35]. Given the densities o(x) and jo x) we define the following functions... 

For a solid-catalyzed reaction to take place, a reactant in the fluid phase must first diffuse through the stagnant boundary layer surrounding the catalyst particle. This mode of transport is described (in one spatial dimension) by the Stefan-Max well equations (see Appendix C for details) ... 

By transforming to the frequency domain (Stepisnik, 1993) it may be shown that the displacement in one spatial dimension is linked to spectrum of velocity correlation through... 

Two different sandstone samples are used to demonstrate the methodology developed in Sections 2.1-2.3 in one spatial dimension. The first sample is a Bentheimer sandstone sample we have labeled KBE, which is saturated with oil. The second sample is a Brown sandstone sample, labeled MCD, that is saturated with water. 

Exactly the same problem can be approached by using the equation for molecular diffusion in one spatial dimension, x. 

Chemical engineering processes involve the transport and transfer of momentum, energy, and mass. Momentum transfer is another word for fluid flow, and most chemical processes involve pumps and compressors, and perhaps centrifuges and cyclone separators. Energy transfer is used to heat reacting streams, cool products, and run distillation columns. Mass transfer involves the separation of a mixture of chemicals into separate streams, possibly nearly pure streams of one component. These subjects were unified in 1960 in the first edition of the classic book. Transport Phenomena (Bird et al., 2002). This chapter shows how to solve transport problems that are one-dimensional that is, the solution is a function of one spatial dimension. Chapters 10 and 11 treat two- and three-dimensional problems. The one-dimensional problems lead to differential equations, which are solved using the computer. 

Charge carriers in semiconductors can be confined in one spatial dimension (ID), two spatial dimensions (2D), or three spatial dimensions (3D). These regimes are termed quantum films, quantum wires, and quantum dots as illustrated in Fig. 9.1. Quantum films are commonly referred to as single quantum wells, multiple quantum wells or superlattices, depending on the specific number, thickness, and configuration of the thin films. These structures are produced by molecular beam epitaxy (MBE) and metalorganic chemical vapor deposition (MOCVD) [2j. The three-dimensional quantum dots are usually produced through the synthesis of small colloidal particles. 

The next slightly more complicated situation concerns a fluid confined to a nanoscopic slit-pore by structured rather than unstructured solid surfaces. For the time being, we shall restrict the discussion to cases in which the symmetry of the external field (represented by the substrates) i)reserves translational invarianee of fluid properties in one spatial dimension. An example of such a situation is depicted in Fig. 5.7 (see Section 5.4.1) showing substrates endowed with a chemical structm e that is periodic in one direction (x) but quasi-infinite (i.e., macroscopically large) in the other one (y). 

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