Discontinuous surfaces

Early theoretical treatments of bluff-body stabilized flame spreading have been based, in general, on the assumption that the flame is a discontinuous surface separating gas streams of different densities and temperatures [1, 15-17]. These theories neglect the finite thickness of turbulent flame zone and predict the increase of the spreading rate both with the density ratio across the flame, and with the increase in the laminar flame velocity of fuel-air mixture. This does not correspond to experimental observations (e.g., [8, 10]). 

It is obvious that the entropy change will be positive in the region Mi > 1 and negative in the region Mi < 1 for gases with 1 < y < 1-67. Thus, Eq. (1.46) is valid only when Ml is greater than unity. In other words, a discontinuous flow is formed only when Ml > 1. This discontinuous surface perpendicular to the flow direction is the normal shock wave. The downstream Mach number, Mj, is always < 1, i. e. subsonic flow, and the stagnation pressure ratio is obtained as a function of Mi by Eqs. (1.37) and (1.41). The ratios of temperature, pressure, and density across the shock wave are obtained as a function of Mi by the use of Eqs. (1.38)-(1.40) and Eqs. (1.25)-(1.27). The characteristics of a normal shock wave are summarized as follows ... 

There are two types of discontinuity surfaces contact surfaces and shock fronts. There is no flow between regions separated by a contact surface, while shock fronts are crossed by the flow. A contact surface moves with the fluid and separates two zones of different density and temperature, but the same pressure. The normal component of the flow velocity is the same on both sides of a contact discontinuity... 

Addition of oxidizing agent to soy flour resulted in a less-puffed or undertexturized extrudates. The extrudate showed rough, discontinuous surface and small dieimeter. This result further proves that the change of disulfide linkages is an essential reaction for texture formation. 

The flow of combustion products behind the flame front has a non-zero vorticity. When the gas crosses the flame front, which represents the gas-dynamic discontinuity surface where the velocity, pressure, density and gas temperature are step-like changing, the vorticity of combustion products is generated (Zeldovich, 1944, 1966, 1979, 1980 Borisov, 1978 Zeldovich et al. 1979 Tsien, 1951 and Chernyi, 1954). 

Starting from the flame front the intensity of the vortices remains constant along each streamline, so that the region filled by combustion products is a rotational one. In some of the previous works mentioned, however, the existence of the stagnation zone behind the flame front has not been accounted for, so that the quantitative conclusions diflier essentially from those of the hydrodynamic model presently under consideration. It should be noted that the boundary streamline of the stagnation zone is a tangential velocity component discontinuity surface or a vortex sheet. As a consequence of the... 

The developments of impedance spectroscopy described in this paper highlight that precise investigations of polarization effects under conditions of monotonously distributed potentials are possible. Further development of experimental and theoretical bases of the method will allow direct studies of discontinuity surface states that are of great fundamental and applied interest. 

Steady-State Permeation Through Discontinuous Surface Coatings... 

Figure 6. A, one symmetry unit (% lattice repeat) in a lattice consisting of a membrane with a discontinuous surface coating. B, the same unit with appropriate boundary conditions. (Reproduced with permission from Ref. 5. Copyright 1986 Elsevier.)...

Before we discuss the measurement of particle surface, it must be pointed out that this process is variously defined. We may consider a surface in a physical sense as a continuous array of perfectly smooth, infinitesimal surfaces without discontinuities, or as a chemical surface composed of an infinite number of discontinuous surfaces which are molecules of its constituent material. The surface measured in the first instance will always be less than in the second. The methods of surface measurement described in the present chapter include both types of surface measurement. Fortunately, the differences between the methods presented are not great, but it must be kept in mind that surface as ordinarily understood applies only to indirect statistical measures, as outlined in Chapter 3. In experimental methods of surface measurement, the kind of surface determined must be understood distinctly. 

In order to establish the depositional history of the speleothems, outcrop-scale features such as discontinuities were mapped. Four discontinuity surfaces were found within the NG flowstone (Figure 4). The discontinuities are laterally continuous over the entire outcrop of the specimen sampled. Debris such as broken soda straws and stalactites lie horizontally on these surfaces (Figure 4). Although columnar calcite crystals with rhombohedral terminations protrude from these sub-horizontal surfaces, they are... 

The Late Miocene reflector represents a regional discontinuity surface produced by a relative sea level drop, and corresponds to the boundary... 

The fluid pressure P R=a at the bubble boundary can be determined from the condition on the jump of normal stresses on the discontinuity surface, that is, the bubble boundary [24, 430]. Under the homobaric assumptions, the gas in the bubble does not move, which implies that... 

Introduction to the Thermodynamics of the Discontinuity Surface in a Single Component System... 

The existence of the free surface energy can be explained by the presence of unsaturated bonds between the molecules at the interface. The formation of a new interface requires work to be performed, in order to bring molecules to the interface from the bulk. The intermolecular interactions at the interface and in the bulk of a phase are substantially different. In the vicinity of an interface, and at distances comparable with molecular dimensions, the composition and properties of individual phases are no longer continuous. This means that a nonuniform layer exists between the phases, within which a transition from properties characteristic of one phase to those characteristic of another occurs. Such a nonuniform transition layer is referred to as the physical interface of discontinuity, or simply the discontinuity surface, according to G ibbs [ 1 -3 ]. 

The thermodynamics of the discontinuity surface can be examined by analyzing how the density of free energy f changes upon transition from one phase to another. From thermodynamics one can establish the relationship between the free energy, F, the isobaric-isothermal potential, f/, and the chemical potential, p, for a single component system ... 

Fig. 1-3. Changes in the free energy density within the discontinuity surface...

The integration limits in eq. (1.2) can be set as -8 and +8", respectively, since free energy densities are identical to their bulk values outside of the discontinuity surface. Equation (1.2) can thus be written as... 

The definition of a invariant with respect to positioning of the dividing surface can be worked out, if one analyzes trends in the/z)-pc(z) function within the discontinuity surface. The specified quantity has the same value in the bulk of both phases, equal to the negative external pressure (Fig. 1-4). Within the discontinuity surface, pressure p has a tensor nature, making Pascal s law invalid. Meanwhile, the concentration and pressure dependence of the surface energy density,/ given by eq. (1.1), is valid only in the regions where Pascal s law holds, i.e., where pressure is a scalar quantity (direct summation of a scalar and a tensor within the same equation is not permitted). 

The quantity pT can be regarded as the tangential pressure , acting in a plane parallel to the interface and tending to decrease an interfacial area. Taking into account that the difference between pT and p is significant only within the discontinuity surface, the Bakker equation can be written as... 

Following the method established originally by Rehbinder, let us relate the surface energy to the internal pressure. The latter is the other quantity used to characterize the intermolecular interactions. To make things simpler, let us assume that the liquid phase is non-volatile (/ " / ), and that the free energy density,/ changes linearly from the bulk value f to some value/, within the entire discontinuity surface of thickness 8 = 8 (see Fig. 1-7). Let us also treat the surface tension a as the work, required to bring molecules contained in the volume of 1 m2 x 8 m = 8 m3 (i.e. 8 n molecules, provided that n is a number of molecules per 1 m3) from the bulk to the surface. Such treatment allows us to write... 

In a two-phase system consisting of two or more components the composition of the discontinuity surface (see Chapter I) may significantly differ from that of a bulk of both phases in contact. Primarily the components that lower the system s free surface energy are expected to accumulate within the discontinuity surface this spontaneous concentration of substances is referred to as adsorption. The quantitative measure of the adsorption of the /-th component, T was introduced by Gibbs, and is also referred to as the adsorption, or the surface excess ofthe amount of substance. This measure has a meaning of the molar excess of a particular component per unit interfacial area ... 

Let us consider a model two-phase two-component system consisting of a solution of hexyl alcohol (component 2) in water (component 1) at equilibrium with their own vapors. A schematic change in the concentration of water c,(z) and that of hexanol c2(z) across the discontinuity surface is shown in Fig. II-l. In the regions below and above the discontinuity surface the concentrations of both components are constant and equal c, and c2 in the liquid phase, and c" and c2 in the vapor phase, respectively. Furthermore, due to low vapor densities c, c" and c2 c". 

Within the discontinuity surface the concentration of water monotonously decreases from c/ to c", which are the concentrations in the liquid and gas phases respectively, while the hexyl alcohol behaves differently its concentration increases and substantially exceeds both c2 and... 

To examine the relationship between the adsorption of a second component and the distribution of the latter within the discontinuity surface, let us draw a prism of cross-sectional area S in the direction perpendicular to the discontinuity surface (Fig. II-1). Let us then compare the amount of substance accumulated within such a prism in the real system and in an idealized one, for which in the z = 0 plane the concentration increase from c2 to c2" has the form of a step function. 

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