Average normal curvature

The parallel surface method (PSM) has been invented to measure the average interface curvature (and the Euler characteristic) from the 3D data images [222]. First, a parallel surface to the interface is formed by translating the original interface along its normal by an equal distance everywhere on the surface (see Fig. 33). The change of the surface area at the infinitely small parallel shift of the surface is... 

Ocular Response Analyzer. The ocular response analyzer applies force to the cornea in the form of a collimated air pulse, with an electrooptical system used to monitor changes in curvature during corneal deformation. The cornea moves inward with the air pulse and then returns to normal curvature. The curvature detection system records two pressure values at inward and outward applanation events. Corneal biomechanical properties create a damping effect that manifests as a difference between the two pressures. Averaging these two pressures provides a Goldmann-correlated lOE The difference between these two pressure values is referred to as corneal hysteresis. 

For a minimal surface, the average value of the Gaussian curvature, , is simply related to the normal curvature, Kn and the geodesic torsion, Xg (section 1.6) ... 

The normal curvature of curves on the surface linking adjacent T atoms is set by the Euclidean distances (1) between these atoms and the angles (A) subtended by the straight lines joining adjacent T atoms. The magnitude of this curvature is equal to the reciprocal of the radius of a circle that contains three vertices spanned by two adjacent bonds. The average value of the normal curvature of the surface over the region of surface sampled by the two bonds is equal to ... 

As the normalized mismatch strain Cm is increased above the value 0.3, the curvature distribution becomes increasingly nonuniform. The general trend is that the curvature assumes values substantially below the average curvature for portions of the substrate near its center, and it takes on values substantially above the average value near the periphery of the substrate. For example, for a normalized mismatch strain of Cm = 2, the normalized curvature varies from about 0.7 at the substrate center to a value of about... 

We can see that in addition to the average surface curvature, the expression of the 4-divergence of the normal vector 4N also contains a term representing interfacial acceleration. When V N c, i.e. in non-relativistic cases, we find ... 

Fig. 18.10 Average normal force Fn = Ft /m on a single particle as a function of the wire-C7Stal distance ho for different edge curvature radii of the indenting particles.

VOF or level-set models are used for stratified flows where the phases are separated and one objective is to calculate the location of the interface. In these models, the momentum equations are solved for the separated phases and only at the interface are additional models used. Additional variables, such as the volume fraction of each phase, are used to identify the phases. The simplest model uses a weight average of the viscosity and density in the computational cells that are shared between the phases. Very fine resolution is, however, required for systems when surface tension is important, since an accurate estimation of the curvature of the interface is required to calculate the normal force arising from the surface tension. Usually, VOF models simulate the surface position accurately, but the space resolution is not sufficient to simulate mass transfer in liquids. 

If, on Ae other hand, Ae double-Aamond symmetry were found in a normal cubic phase, i.e., with mean curvature on Ae average toward the hyAocarbon regions, Aen one would expect to find Aat Ae polar / apolar interfacial surface shown in Figure la would look instead like interconnected cylindrical rods, because Ae necks and bulges in Figure la would not correspond to water channels but raAer to channels occupi by surfactant tails wiA a preferred stretch Astance. Thus far, such a normal cubic phase has not been observed with this symmetry, but has with another symmetry Ascussed below ( 230), and Ae principles are exactly Ae same. 

These results provide additional confirmation for the mechanism of pyrolysis of simple polyolefins. The absence of monomer in the volatile products, the maxima in the rate curves, and the sharp decrease in the intrinsic viscosity for linear polymethylene (29) and polypropylene (2, 6, 13, 30) all point to an essentially random scission, due to pronounced intermolecular chain transfer, Equation 2. However, deviations appear when a, the fraction of bonds broken, or, what amounts to the same, the number average DP is examined as a function of time. For small a, the former relation should be one of simple proportionality and hnearity in 1/P. Instead, for both polypropylene (6) and polymethylene [see Figure 5, in (29)] curvature appears, indicating a reduction of the scission rate after an initial period of rapid degradation. For polypropylene this has been interpreted as a breaking of weak and normal bonds. Between 250° and 280° C., one weak link per 2.4 X 10 is found (6). At 295° C., the existence of more than two types of bonds would have to be postulated. 

The oxidation rates measured by the respirometer were used to calculate acceleration factors for the oxidative aging of the polyurethane foam. The oxidation rates at each temperature were averaged and used to calculate acceleration factors that were normalized to 1.0 at 95°C. The results versus inverse temperature are shown in Figure 7. The acceleration factor plot exhibits some curvature in the lower temperature region of the curve. The activation energy of the relatively linear region between 50 and 110°C is 92 kJ/mol. 

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