One-dimensional interfaces

Linear, one-dimensional interfaces. Perhaps the most interesting theory of promoter action is that the linear boundary on the surface between two surfaces of different composition is the seat of the catalysis, i.e. the active patch. There is abundant evidence that such linear interfaces often possess unusual reactive powers. They are undoubtedly... 

One result of a dissociation occurring most easily at the boundary between the two solid phases is that it is often difficult to start dissociation, or the reverse process of recombination, when the solid surface consists entirely of one or the other constituent. Faraday7 found that hydrated sodium sulphate and some other salts do not commence to effloresce until the surface is scratched, thus starting a one-dimensional interface on the surface by mechanical means, from which the loss of water could occur... 

The remarkable effect of a minute trace of sulphur on the surface of a silver bromide grain, referred to in 16, indicates that the instability of a one-dimensional interface may be fundamental to the speed of the modem photographic plate. 

This production of a peculiar degree of instability in the silver bromide, close to the atoms of sulphur in the sulphide, seems closely analogous to the other cases of promoter action, and the effects of a one-dimensional interface in a solid surface, which were considered in 3. It may be an effect of the same nature as the increased ease of decomposition of the calcium carbonate group, when this has calcium oxide groups adjacent to it. There is some evidence that the silver sulphide crystal lattice is rather more easily disorientated than the silver bromide, but since silver sulphide is, alone, not particularly sensitive to light, it seems certain that the sensitizing action of the sulphide speck must be due to a boundary action between the sulphide and the bromide.8... 

Radial) chord length distribution (CLD) (One-dimensional) interface distribution function (IDF) Image data format returned by image plate scanners... 

A basic concept is then the interfacial stiffness and the description in terms of the capillary wave Hamiltonian (Privman, 1992). To introduce these terms, we consider the one-dimensional interface z = h(x) of a two-dimensional system for simplicity. Noting that in lattice systems the interfacial energy jnt will depend on the angle 9 between the tangent to the interface and the x-axis, we write 9 = arctanfdft/dx)]... 

Show that the equilibrium roughness of a one-dimensional interface is qualitatively larger than that of a two-dimensional interface. Show that the one-dimensional interface is always rough i.e., the roughening transition temperature is zero). 

For simplicity, imagine a two-dimensional drop (with a one-dimensional interface between the fluid drop and the vapor). We assume that the substrate area is larger or equal to the size of the drop and that it therefore does not... 

Another function used to obtain structure information is the one-dimensional interface distribution function, g(x) (18,19). This is simply the second derivative of fee onedimensional correlation Action, or g(x) = y"(x). This function gives the probability that two interfaces will be separated by a distance, x. In an ideal two-phase system, the phases would have constant d and L throughout fee scattering volume. The interfece distribution function would be a series of delta functions. Real polymo systems have a spread cf values of d and L. This causes g(x) to be a smooth curve wife broad peaks located at d, L-d, L, L+d, etc. The peak locations and feeir breadths can be analyzed, and it has been shown(18) that g(x) provides a more reliable estimation cf d and L than Y(x), when the material contains broad distributions of thicknesses. 

The divergence of is due to the low-wave-number modes. In two dimensions (one-dimensional interface), at g = 0 and L = oc, the contribution to made by all capillary waves of wave number greater than any prescribed q is kThmq. In three dimensions, again at g = 0 and L = x, it... 

We shall see later that, both in theory and by experiment, the tension T of a three-phase line may be of either sign. It is to be distinguished from the physically quite different tension of a one-dimensional interface between two surface phases, which we mentioned in passing in 8.5, and which, like any two-phase boundary tension, is necessarily positive. The reason such a boundary tension must be positive at equilibrium is that if it were negative the interface between the phases (the two-dimensional interface in three dimensions or the one-dimensional interface in two... 

The gel-to-fluid chain-melting transition in pseudo-two-dimensional lipid bilayer membranes induces formation of lipid domains of gel-like lipids in the fluid phase and and fluid-like lipids in the gel phase. The average domain size and in particular the average length of the one-dimensional interfaces between lipid domains and bulk have a dramatic temperature dependence with anomalies at the transition temperature. These anomalies are related to similar anomalies in response functions. The interfacial area may be modulated by intrinsic impurities which are interfacially active molecules such as cholesterol [1,2]. The properties of the interfacial area provide a means for understanding aspects of the functioning of certain biological membrane processes like the passive permeability of small ions and the activity of some membrane enzymes. 

Figure Bl.9.12. The schematic diagram of the relationships between the one-dimensional electron density profile, p(r), correlation fiinction y (r) and interface distribution fiinction gj(r).

Test data are available for two experiments at different impact velocities in this configuration. In one of the tests the projectile impact velocity was 1.54 km/s, while in the second the impact velocity was 2.10 km/s. This test was simulated with the WONDY [60] one-dimensional Lagrangian wave code, and Fig. 9.21 compares calculated and measured particle velocity histories at the sample/window interface for the two tests [61]. Other test parameters are listed at the top of each plot in the figure. 

Fig. 6.7. The predicted, one-dimensional, mean-bulk temperatures versus location at various times are shown for a typical powder compact subjected to the same loading as in Fig. 6.5. It should be observed that the early, low pressure causes the largest increase in temperature due to the crush-up of the powder to densities approaching solid density. The "spike in the temperature shown on the profiles at the interfaces of the powder and copper is an artifact due to numerical instabilities (after Graham [87G03]).

Solving the one-dimensional Poisson equation with the charge density profile pc z), the electrostatic potential drop near the interface can be calculated according to... 

The extension of generic CA systems to two dimensions is significant for two reasons first, the extension brings with it the appearance of many new phenomena involving behaviors of the boundaries of, and interfaces between, two-dimensional patterns that have no simple analogs in one-dimensional systems. Secondly, two-dimensional dynamics make it an easy (sometimes trivial) task to compare the time behavior of such CA systems to that of real physical systems. Indeed, as we shall see in later sections, models for dendritic crystal growth, chemical reaction-diffusion systems and a direct simulation of turbulent fluid flow patterns are in fact specific instances of 2D CA rules and lattices. 

The quasi-one-dimensional model of two-phase flow in a heated capillary slot, driven by liquid vaporization from the interface, is described in Chap. 8. It takes... 

Below we consider a quasi-one-dimensional model of flow and heat transfer in a heated capillary, with hydrodynamic, thermal and capillarity effects. We estimate the influence of heat transfer on steady-state laminar flow in a heated capillary, on the shape of the interface surface and the velocity and temperature distribution along the capillary axis. 

The present model takes into account how capillary, friction and gravity forces affect the flow development. The parameters which influence the flow mechanism are evaluated. In the frame of the quasi-one-dimensional model the theoretical description of the phenomena is based on the assumption of uniform parameter distribution over the cross-section of the liquid and vapor flows. With this approximation, the mass, thermal and momentum equations for the average parameters are used. These equations allow one to determine the velocity, pressure and temperature distributions along the capillary axis, the shape of the interface surface for various geometrical and regime parameters, as well as the influence of physical properties of the liquid and vapor, micro-channel size, initial temperature of the cooling liquid, wall heat flux and gravity on the flow and heat transfer characteristics. 

Chapter 8 consists of the following in Sect. 8.2 the physical model of the process is described. The governing equations and conditions of the interface surface are considered in Sects. 8.3 and 8.4. In Sect. 8.5 we present the equations transformations. In Sect. 8.6 we display equations for the average parameters. The quasi-one-dimensional model is described in Sect. 8.7. Parameter distribution in characteristic zones of the heated capillary is considered in Sect. 8.8. The results of a parametrical study on flow in a heated capillary are presented in Sect. 8.9. 

The quasi-one-dimensional model of flow in a heated micro-channel makes it possible to describe the fundamental features of two-phase capillary flow due to the heating and evaporation of the liquid. The approach developed allows one to estimate the effects of capillary, inertia, frictional and gravity forces on the shape of the interface surface, as well as the on velocity and temperature distributions. The results of the numerical solution of the system of one-dimensional mass, momentum, and energy conservation equations, and a detailed analysis of the hydrodynamic and thermal characteristic of the flow in heated capillary with evaporative interface surface have been carried out. 

Two-phase flows in micro-channels with an evaporating meniscus, which separates the liquid and vapor regions, have been considered by Khrustalev and Faghri (1996) and Peles et al. (1998, 2000). In the latter a quasi-one-dimensional model was used to analyze the thermohydrodynamic characteristics of the flow in a heated capillary, with a distinct interface. This model takes into account the multi-stage character of the process, as well as the effect of capillary, friction and gravity forces on the flow development. The theoretical and experimental studies of the steady forced flow in a micro-channel with evaporating meniscus were carried out by Peles et al. (2001). These studies revealed the effect of a number of dimensionless parameters such as the Peclet and Jacob numbers, dimensionless heat transfer flux, etc., on the velocity, temperature and pressure distributions in the liquid and vapor regions. The structure of flow in heated micro-channels is determined by a number of factors the physical properties of fluid, its velocity, heat flux on... 

The developed theory of two-phase laminar flow with a distinct interface which is based on a one-dimensional approximation, takes into account the major features of the process the inertia, gravity, surface tension and friction forces and leads to the physically realistic pattern of a laminar flow in a heated micro-channel. This allows one to use the present theory to study the regimes of flow as well as optimizing a cooling system of electronic devices with high power densities. 

The quasi-one-dimensional model used in the previous sections for analysis of various characteristics of fiow in a heated capillary assumes a uniform distribution of the hydrodynamical and thermal parameters in the cross-section of micro-channel. In the frame of this model, the general characteristics of the fiow with a distinct interface, such as position of the meniscus, rate evaporation and mean velocities of the liquid and its vapor, etc., can be determined for given drag and intensity of heat transfer between working fluid and wall, as well as vapor and wall. In accordance with that, the governing system of equations has to include not only the mass, momentum and energy equations but also some additional correlations that determine... 

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