Physical-space diffusion

As will be shown for the CD model, early mixing models used stochastic jump processes to describe turbulent scalar mixing. However, since the mixing model is supposed to mimic molecular diffusion, which is continuous in space and time, jumping in composition space is inherently unphysical. The flame-sheet example (Norris and Pope 1991 Norris and Pope 1995) provides the best illustration of what can go wrong with non-local mixing models. For this example, a one-step reaction is described in terms of a reaction-progress variable Y and the mixture fraction p, and the reaction rate is localized near the stoichiometric point. In Fig. 6.3, the reaction zone is the box below the flame-sheet lines in the upper left-hand corner. In physical space, the points with p = 0 are initially assumed to be separated from the points with p = 1 by a thin flame sheet centered at... 

When using moment methods for inhomogeneous systems, the moment set is transported in physical space due to advection, diffusion, and free transport. Since the moment-transport equations are derived from a transport equation for the NDE, the problem of moment transport is closely related to the problem of transporting the NDF. Denoting the NDE by n(t, X, ), the process of spatial transport involves changes in n(t, x, ) for fixed values... 

For r > 0, a conservative scheme (for ttansport in physical space) can be developed by correctly treating the terms due to advection and diffusion in physical space. On rewriting the PBE in Eq. (D.l) as... 

The most convenient means of solving Eq. (4.6) is via the Lagrangian Monte Carlo procedure [4]. With the Lagrangian procedure, the FMDF is represented by an ensemble of computational stochastic elements (or particles ) which are transported in the physical space by the combined actions of large-scale convection and diffusion (molecular and subgrid). In addition, transport in the composition space occurs due to chemical reaction and SGS mixing. All of... 

Synchronization or entrainment is a key concept to the understanding of selforganization phenomena occurring in the fields of coupled oscillators of the dissipative type. We may even say that Part II is devoted to the consideration of this single mode of motion in various physical situations. Specifically, Chap. 6 is concerned with wave phenomena and pattern formation, which may be viewed as typical synchronization phenomena in distributed systems. In contrast, we shall study in Chap. 7 turbulence in reaction-diffusion systems, which is caused by desynchronization among local oscillators. Chapter 5 deals with self-synchronization phenomena in the discrete populations of oscillators where the way they are distributed in physical space is not important (for reasons stated later). We shall introduce some kind of randonmess by assuming that the oscillators are either different in nature from each other or at best statistically identical. One may then expect phase-transition-like phenomena, characterized by the appearance or disappearance of collective oscillations in the oscillator communities. In describing such a new class of phase transitions. Method I turns out to be very useful. 

The system [2] is nothing more than a local balance of matter, written with the hypothesis that Fick s second law appropriately takes diffusion transport into account. As distinct from [1], we have here a system of partial differential equations, whose solution poses a far more substantial mathematical problem (43). The dimension of physical space considered is of prime importance, as much from a theoretical standpoint as in experiment, even when we make the hypothesis, as we have here, that convection does not intervene (see photos (c) and (d)). 

The resulting electric field will not be completely equipotential along the planes parallel to the membrane, but will be slowly delocalized by the diffusion of protons in the bulk phases and compensated by ion redistribution. The resulting voltage profile will be a function of the physical spacing between the coupling units, the electrical conductivity and the exact location of the proton diffusion barriers,and the conductivity of the bulk-to-bulk leaks (cf. Zimanyi,Garab 1982). 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Electronic-Grade MMCs. Metal-matrix composites can be tailored to have optimal thermal and physical properties to meet requirements of electronic packaging systems, eg, cotes, substrates, carriers, and housings. A controUed thermal expansion space tmss, ie, one having a high precision dimensional tolerance in space environment, was developed from a carbon fiber (pitch-based)/Al composite. Continuous boron fiber-reinforced aluminum composites made by diffusion bonding have been used as heat sinks in chip carrier multilayer boards. 

Identification of the pollutant source and installation of the local exhaust is critically important. For example, an improperly designed local exhaust can draw other contaminants through the occupied space and make the problem worse. The physical layout of grilles and diffusers relative to room occupants and pollutant sources can be important. If supply diffusers are all at one end of a room and returns are all at the other end, the people located near the supplies may be provided with relatively clean air while those located near the returns breathe air that has already picked up contaminants from all the sources in the room that are not served by local exhaust. 

A system is the region in space that is the subject of the thermodynamic study. It can be as large or small, or as simple or complex, as we want it to be, but it must be carefully and consistently defined. Sometimes the system has definite and precise physical boundaries, such as a gas enclosed in a cylinder so that it can be compressed or expanded by a piston. However, it may be also something as diffuse as the gaseous atmosphere surrounding the earth. 

The physical meaning of the g (ion) potential depends on the accepted model of an ionic double layer. The proposed models correspond to the Gouy-Chapman diffuse layer, with or without allowance for the Stem modification and/or the penetration of small counter-ions above the plane of the ionic heads of the adsorbed large ions. " The experimental data obtained for the adsorption of dodecyl trimethylammonium bromide and sodium dodecyl sulfate strongly support the Haydon and Taylor mode According to this model, there is a considerable space between the ionic heads and the surface boundary between, for instance, water and heptane. The presence in this space of small inorganic ions forms an additional diffuse layer that partly compensates for the diffuse layer potential between the ionic heads and the bulk solution. Thus, the Eq. (31) may be considered as a linear combination of two linear functions, one of which [A% - g (dip)] crosses the zero point of the coordinates (A% and 1/A are equal to zero), and the other has an intercept on the potential axis. This, of course, implies that the orientation of the apparent dipole moments of the long-chain ions is independent of A. 

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