Curvature radius calculations

In this section we present the system of quasi-one-dimensional equations, describing the unsteady flow in the heated capillary tube. They are valid for flows with weakly curved meniscus when the ratio of its depth to curvature radius is sufficiently small. The detailed description of a quasi-one-dimensional model of capillary flow with distinct meniscus, as well as the estimation conditions of its application for calculation of thermohydrodynamic characteristics of two-phase flow in a heated capillary are presented in the works by Peles et al. (2000,2001) and Yarin et al. (2002). In this model the set of equations including the mass, momentum and energy balances is ... 

The shape of the metal specimen considered is obviously related to the kind of system to be modelled. For SERS and the others SE phenomena, the presence of curved surfaces, with a curvature radius on the nanometric scale, is fundamental for the enhancement. Thus, spheres, ellipsoids, ensembles of spheres, spheres close to planar metal surfaces and planar metal surfaces with random roughness have been considered. We refer to the review by Metiu [57], which describes most of these analytically solvable models. More recently, the modelling of the electric field acting on the point molecule has moved to more realistic shapes (including fractal metal specimen) [59] which require numerical methods to be tackled. The aim of these approaches is usually to calculate the total electric field around the metal particle, and the molecule does not even appear explicitly in the calculations. Interested readers are referred to some recent reviews on the subject [60] (see also Chapters 2 and 5 of ref. [56]). 

The advantage of DI for analyzing not only qualitatively, but also quantitatively the droplet geometry at the air/water surface is that the height of the hat at various local positions may be determined and used to calculate the curvature radius at these local positions. Agreement, within 10% (+ 2ym) of the curvature radii then indicates spherical shape. The curvature radius may then be used with the radius of the contact to determine the angle a (Figure 7). 

Low-temperature nitrogen adsorption. By taking capillary condensation as the reason for the hysteresis loops of the sorption isotherms (Fig. 7.26), it is possible to calculate the pore size distribution in accordance with the well-known Kelvin equation. This equation relates the relative pressure of vapor above a concave meniscus ofliquid in a capilLary to the curvature radius... 

Calculated results of the curvature radius with the parameters obtained from the microscopic images for satin 6 and satin 8/satin 4 are shown in Table 8.1. It has... 

Figure 8.9. Comparison between capillary energy lyt/r, r the average radius of grain boundary curvature) and calculated coherency strain energies stored in a (Bao.8Pho,2)Ti03 layer on BaTiOs. Specific grain boundary energy is assumed to be 0.5...

Humphrey, J. A. C., Numerical Calculation of Developing Laminar Flow in Pipes of Arbitrary Curvature Radius, Can J. Chem. Engg., 56, 151-164 (1978). 

FIGURE 2.22 Relation between pnll-off force and curvature radius of asperity peak measured on silicon asperity arrays of various groove depths. The pull-off forces were derived from fig. 2.10. The curvature radii were calculated from the AFM data by using a hemisphere approximation program. 

As a simplification, spatial curvature of spiral jet path is modeled with two radii of curvature. We calculate the radius of curvature in the rotational plane and the radius of curvature in the plane in which the jet descents Cbq. Cbq describes the curvature in the b-a-plane. bq indicates that a movement in negative q-direction results in a movement in positive a-direction of same amount. 

For a specific drilled length, the radius of curvature is calculated using the following formula ... 

Sliding experiments were carried out at room temperature in a UHV condition of less than 1x10 Pa throughout the experiment A normal load of 250 mN was applied on the Si wafer by a spherical diamond slider (3 mm curvature radius), and the sliding speed was 0.1 mm/sec controlled by a pulse-drive motor. The sUder was driven at 100 pulses per second. In this contact condition, approximately 350 MPa of Hertzian pressure was applied at maximum on a circular contact area 32 pm in diameter, as determined by our calculations. In the observed tracks, the width was less than 40 pm, which was close to the calculated width after 1000 cycles of reciprocal sliding, so that the lateral fluctuation of the contact point was stable enough to assume that the iterated sliding was in contact approximately on the same track. 

At the middle of the capillary where the eflect of the walls on chemical potential is negligible, the radius of curvature will be equal to r as calculated by the Kelvin equation (3.20) but it will become progressively larger as the wall is approached. 

Fig. 3.21 The effect of meniscus curvature on surface tension. Plot of /) against r . y is the surface tension of the meniscus having the mean radius of curvature and y that of a plane surface of liquid, according to Melrose. The value of y/y was calculated by the equation V = /x(l - with a . = 3 a.

The force p needed to compress a single asperity and the displacement 8 of its tip relative to the undeformed region of the substrate was calculated using JKR theory and determined to be related to the radius of curvature of the asperity and the contact radius a by... 

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