Parabolic point

At the same time Chien, Rising, and Ottino (17) studied chaotic flow in two-dimensional cavity flows with a periodic moving wall, which is relevant to mixing of viscous polymeric melts. All two-dimensional flows, as pointed out by Ottino (18), consist of the same building blocks hyperbolic points and elliptic points. A fluid particle moves toward a hyperbolic point in one direction and away from it in another direction, whereas the fluid circulates around parabolic points, as shown in Fig. 7.12. 

In the vicinity of elliptic points, the surface can be fitted to an ellipsoid, whose radii of curvature are equal to those at that point (Fig. 1.5a). The surface lies entirely to one side of its tangent plane, it is "synclastic" and both curvatures have the same sign. About a parabolic point, the surface resembles a cylinder, of radius equal to the inverse of the single nonzero principal curvature (Fig. 1.5b). H3rperbolic ("anticlastic") points can be fitted to a saddle, whic is concave in some directions, flat in others, and convex in others (Fig. 1.5c). At hyperbolic points, the surface lies both above and below its tangent plane. 

In Fig. 2, a numerical example of the angular distribution of emission power radiated from the point source is presented for the case of a two-dimensional square lattice of air holes in a polymer. The energy flux is strongly anisotropic, showing a relatively small intensity in all directions except along directions associated with parabolic points of iso-frequency surface, where the intensity tends to infinity. Predictions of asymptotic analysis on far-field radiation pattern (2) are substantiated with FTDT calculations, revealing a reasonable agreement (Fig. 2, left). 

The second order tensor in 2D images was also used by Noble [31], who pointed out that the local image surface can be classified according to the Hessian matrix determinant as a planar point (zero determinant), parabolic point (zero determinant), hyperbolic point (negative determinant) and elliptic point (positive determinant). Points of interest are those which contain strong intensity variation such as the hyperbolic and elliptic points. 

If the small temis in p- and higher are ignored, equation (A2.5.4) is the Taw of the rectilinear diameter as evidenced by the straight line that extends to the critical point in figure A2.5.10 this prediction is in good qualitative agreement with most experiments. However, equation (A2.5.5). which predicts a parabolic shape for the top of the coexistence curve, is unsatisfactory as we shall see in subsequent sections. 

The integral under the heat capacity curve is an energy (or enthalpy as the case may be) and is more or less independent of the details of the model. The quasi-chemical treatment improved the heat capacity curve, making it sharper and narrower than the mean-field result, but it still remained finite at the critical point. Further improvements were made by Bethe with a second approximation, and by Kirkwood (1938). Figure A2.5.21 compares the various theoretical calculations [6]. These modifications lead to somewhat lower values of the critical temperature, which could be related to a flattening of the coexistence curve. Moreover, and perhaps more important, they show that a short-range order persists to higher temperatures, as it must because of the preference for unlike pairs the excess heat capacity shows a discontinuity, but it does not drop to zero as mean-field theories predict. Unfortunately these improvements are still analytic and in the vicinity of the critical point still yield a parabolic coexistence curve and a finite heat capacity just as the mean-field treatments do. 

By following Section II.B, we shall be more specific about what is meant by strong and weak interactions. It turns out that such a criterion can be assumed, based on whether two consecutive states do, or do not, form a conical intersection or a parabolical intersection (it is important to mention that only consecutive states can form these intersections). The two types of intersections are characterized by the fact that the nonadiabatic coupling terms, at the points of the intersection, become infinite (these points can be considered as the black holes in molecular systems and it is mainly through these black holes that electronic states interact with each other.). Based on what was said so far we suggest breaking up complete Hilbert space of size A into L sub-Hilbert spaces of varying sizes Np,P = 1,..., L where... 

All Np states belonging to the Pth sub-space interact strongly with each other in the sense that each pair of consecutive states have at least one point where they form a Landau-Zener type interaction. In other words, all j = I,... At/> — I form at least at one point in configuration space, a conical (parabolical) intersection. 

Colburn [Tran.s. Am. In.st. Chem. Eng., 35, 211 (1939)] has shown that when the equihbrium line is straight near the origin but curved slightly at its upper end, Nqc . u be computed approximately by assuming that the equihbrium curve is a parabolic arc of slope mo near the origin and passing through the point X, K X at the upper end. The Colburn equation for this case is... 

Proportional limit the point on the stress-strain curve at which will commence the deviation in the stress-strain relationship from a straight line to a parabolic curve (Figure 30.1). 

Of special interest is the case of parabolic barrier (1.5) for which the cross-over between the classical and quantum regimes can be studied in detail. Note that the above derivation does not hold in this case because the integrand in (2.1) has no stationary points. Using the exact formula for the parabolic barrier transparency [Landau and Lifshitz 1981],... 

At high temperatures (/S -r 0) the centroid (3.53) collapses to a point so that the centroid partition function (3.52) becomes a classical one (3.49b), and the velocity (3.63) should approach the classical value Uci- In particular, it can be directly shown [Voth et al. 1989b] that the centroid approximation provides the correct Wigner formula (2.11) for a parabolic barrier at T > T, if one uses the classical velocity factor u i. A. direct calculation of Ax for a parabolic barrier at T > Tc gives... 

Taking the point seriously that neighboring parabolas with tip position at Ztip > 0 cannot intersect further down the cold side than at z = 0, we can use the parabolic relation z p = A /82 for the intersection of two parabolas of radius 2 at z = 0, being a distance A apart from each other. The result for the tail instability to occur is... 

The aforementioned inconsistencies between the paralinear model and actual observations point to the possibility that there is a different mechanism altogether. The common feature of these metals, and their distinction from cerium, is their facility for dissolving oxygen. The relationship between this process and an oxidation rate which changes from parabolic to a linear value was first established by Wallwork and Jenkins from work on the oxidation of titanium. These authors were able to determine the oxygen distribution in the metal phase by microhardness traverses across metallographic sections comparison of the results with the oxidation kinetics showed that the rate became linear when the metal surface reached oxygen... 

Salt solutions When a zinc sheet is immersed in a solution of a salt, such as potassium chloride or potassium sulphate, corrosion usually starts at a number of points on the surface of the metal, probably where there are defects or impurities present. From these it spreads downwards in streams, if the plate is vertical. Corrosion will start at a scratch or abrasion made on the surface but it is observed that it does not necessarily occur at all such places. In the case of potassium chloride (or sodium chloride) the corrosion spreads downwards and outwards to cover a parabolic area. Evans explains this in terms of the dissolution of the protective layer of zinc oxide by zinc chloride to form a basic zinc chloride which remains in solution. 

For a parabolic mirror one conjugate is real and located in front of the mirror, the other is located at infinity. As shown in Fig. 2, light emitted from the one conjugate of the parabolic mirror will focus onto the other conjugate without any additional system aberration. For example, light emitted from an infinitely distant point will come to perfect focus at the finite conjugate. 

Suppose you are marching down the infamous tube and at step j have determined the temperature and composition at each radial point. A correlation is available to calculate viscosity, and it gives the results tabulated below. Assume constant density and Re = 0.1. Determine the axial velocity profile. Plot your results and compare them with the parabolic distribution. 

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