Spatially nonuniform systems

Calculating the Thermodynamic Forces in Spatially Nonuniform Systems... 

To calculate the thermodynamic forces in spatially nonuniform systems, mathematical notion of divergence is essential. The divergence of a vecto rial field a(M) in point (x,y,z) is determined as a scalar quantity... 

This observation constitutes the basic idea of the local equilibrium model of Prigogine, Nicolis, and Misguich (hereafter referred to as PNM). One considers the case of a spatially nonuniform system and deduces from (3) an integral equation for the pair correlation function that is linear in the gradients. This equation is then approximated in a simple way that enables one to derive explicit expressions for all thermal transport coefficients (viscosities, thermal conductivity), both in simple liquids and in binary mixtures, excluding of course the diffusion coefficient. The latter is a purely kinetic quantity, which cannot be obtained from a local equilibrium hypothesis. 

A gas is not in equilibrium when its distribution function differs from the Maxwell-Boltzman distribution. On the other hand, it can also be shown that if a system possesses a slight spatial nonuniformity and is not in equilibrium, then the distribution function will monotonically relax in velocity space to a local Maxwell-Boltzman distribution, or to a distribution where p = N/V, v and temperature T all show a spatial dependence [bal75]. 

Real systems are typically spatially nonuniform even if they are inhomoge neous. Such systems are obviously nonequibbrium, since the nonuniform in respect to temperature, pressure, concentrations, and so on generates the matter and/or heat fluxes. This section considers some mathematical tools for calculating thermodynamic forces and fluxes in such systems, as well as the relation of these quantities with conventional thermo dynamic parameters. 

In real systems, especially in heterogeneous catalytic and biological sys terns, the reactants are often arranged irregularly in space. Therefore, an arising instability may cause simultaneous diffusion of substances from one point to another inside the system to make the reactant concentration oscillations arranged in a certain manner in space during the occurrence of nonlinear chemical transformations. As a result, a new dissipative structure arises with a spatially nonuniform distribution of certain reac tants. This is a consequence of the interaction between the process of diffusion, which tends to create uniformity of the system composition, and local processes of the concentration variations in the course of nonlinear... 

ON THE THEORY OF THE ORIGIN OF SPATIALLY NONUNIFORM STATIONARY STATES (DISSIPATIVE STRUCTURES) IN HETEROGENEOUS CATALYTIC SYSTEMS... 

In this work, the numerical experiment was carried out for y = 1, i.e., in the region where uniform oscillations in system (3,4) are impossible. Can the distributedness of the system act, under such conditions, as a factor capable of inducing a spatially nonuniform self-oscillatory process There is no strict criterion at the present time for the existence of this new type of self-oscillatory instability. Numerical experiments, however, provide a positive answer to this question. Figure 36 shows a time display of a self-oscillatory process discovered in the course of DS transformation as the catalytic element length / was varied. Other parameters were ... 

On The Theory Of The Origin Of Spatially Nonuniform Stationary States (Dissipative Structures) In Heterogeneous Catalytic Systems 551... 

In the sections that follow, illustrations of homogeneous and nonho-mogeneous dislocation nucleation are presented. The former case implies dislocation formation in a material system that is otherwise spatially uniform nucleation is equally likely at all locations. Nonhomogeneous nucleation, on the other hand, implies that spatial nonuniformity arising through configuration, material structure, or material defects renders certain sites in the structure far more susceptible to dislocation nucleation than other sites. 

The conclusion follows directly from superposition of linear elastic fields, without the need for detailed calculation. The system under discussion is depicted in part (a) of Figure 8.34. The figure shows a relatively thick elastic substrate with a film of some uniform thickness bonded to its surface. The film supports a spatially uniform mismatch strain, presumably due to the constraint of epitaxy. The elastic properties of the film material may be different from those of the substrate material. Some of the film material has gathered into an isolated epitaxial island on the surface of the strained layer. The island material also supports the same mismatch strain. The lateral faces of the island are free of applied traction. As a result, the elastic strain field is spatially nonuniform and the elastic strain energy is... 

To answer this question, consider the change in system free energy corresponding to the spatially nonuniform perturbed composition... 

In uniform systems spiral waves usually appear as a pair when a break of a wave front creates two open ends of the front [2, 3] in nonuniform systems single spirals emerge frequently [44]. Only one open end of the wave front can form a spiral under some asymmetric spatial distributions of parameters (Figure 3). Single spirals can emerge at nonuniformities situated near a boundary of the system. In other cases, one spiral from a pair can be brought by gradient-induced drift to the medium boundary where it perishes. 

Epstein has also reviewed the design of pH-regulated oscillators and described a general model for such systems. Noyes has also reviewed the current state of chemical oscillators. Oscillation and spatial nonuniformities in membranes have been reviewed. The proceedings of the 1989 Conference on Exotic Phenomena held at Hajduszoboszlo, Hungary, have been published and Gray and Scott have edited an issue of Philosophical Transactions devoted to this topic. A number of theoretical papers have appeared. 

The thermal parameters for comfort should be relatively uniform both spatially and temporally. Variations in heat flow from the body make the physiological temperature regulation more difficult. Nonuniform thermal conditions can lead to nonuniform skin temperatures. The active elements of the regulatory system may need to make more adjustments and work harder in order to keep thermal skin and body temperatures stable. To avoid discomfort from environmental nonuniformities, the temperature difference between feet and head should be less than about 3 °C (Fig. 5.9) and the mean surface temperature or radiant difference from one side of the body to the other should not he greater then about 10 °C. 

Such expressions can be extended to permit the evaluation of the distribution of concentration throughout laminar flows. Variations in concentration at constant temperature often result in significant variation in viscosity as a function of position in the stream. Thus it is necessary to solve the basic expressions for viscous flow (LI) and to determine the velocity as a function of the spatial coordinates of the system. In the case of small variation in concentration throughout the system it is often convenient and satisfactory to neglect the effect of material transport upon the molecular properties of the phase. Under these circumstances the analysis of boundary layer as reviewed by Schlichting (S4) can be used to evaluate the velocity as a function of position in nonuniform boundary flows. Such analyses permit the determination of material transport from spheres, cylinders, and other objects where the local flow is nonuniform. In such situations it is not practical at the present state of knowledge to take into account the influence of variation in the level of turbulence in the main stream. 

The transient gas-particle dynamics of the earlier prototypes were found to deliver microparticles with a range of velocities and a nonuniform spatial distribution. For targeted delivery, however, especially in the area of gene and peptide delivery, the system should deliver particles with a narrow and controllable velocity range and a uniform spatial distribution. This was achieved with a certain embodiment called the contoured shock tube configured to achieve uniform particle impact conditions by entraining particles within a quasi-steady gas flow (Kendall et al. 2002). 

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