Azimuthal approximation

On the basis of tbe above analyses, it follows that them is no need to compute multidimensional potential surfaces if one wishes to handle the R-T effect in the framework of the model proposed. In spite of that, such computations were carried out in [152] in order to demonstrate the reliability of the model for handling the R-T effect and to estimate the range in which it can safely be applied in its lowest order (quadratic) approximation. The 3D potential surfaces involving the variation of the bending coordinates pi, p2 and the relative azimuth angle 7 = 4 2 1 were computed for both component of the state. 

The infrared ellipsometer is a combination of a Fourier-transform spectrometer (FTS) with a photometric ellipsometer. One of the two polarizers (the analyzer) is moved step by step in four or more azimuths, because the spectrum must be constant during the scan of the FTS. From these spectra, the tanf and cosd spectra are calculated. In this instance only A is determined in the range 0-180°, with severely reduced accuracy in the neighborhood of 0° and 180°. This problem can be overcome by using a retarder (compensator) with a phase shift of approximately 90° for a second measurement -cosd and sind are thereby measured independently with the full A information [4.315]. 

It has been found possible to evaluate s0 theoretically by means of the following treatment (1) Each electron shell within the atom is idealised as a uniform surface charge of electricity of amount — zte on a sphere whose radius is equal to the average value of the electron-nucleus distance of the electrons in the shell. (2) The motion of the electron under consideration is then determined by the use of the old quantum theory, the azimuthal quantum number being chosen so as to produce the closest approximation to the quantum... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

The transition from laminar to turbulent flow on a rotating sphere occurs approximately at Re = 1.5 4.0 x 104. Experimental work by Kohama and Kobayashi [39] revealed that at a suitable rotational speed, the laminar, transitional, and turbulent flow conditions can simultaneously exist on the spherical surface. The regime near the pole of rotation is laminar whereas that near the equator is turbulent. Between the laminar and turbulent flow regimes is a transition regime, where spiral vortices stationary relative to the surface have been observed. The direction of these spiral vortices is about 4 14° from the negative direction of the azimuthal angle,. The phenomenon is similar to the flow transition on a rotating disk [19]. 

For turbulent flow on a rotating sphere or hemisphere, Sawatzki [53] and Chin [22] have analyzed the governing equations using the Karman-Pohlhausen momentum integral method. The turbulent boundary layer was assumed to originate at the pole of rotation, and the meridional and azimuthal velocity profiles were approximated with the one-seventh power law. Their results can be summarized by the... 

Figure 9.7. Separation of misorientation (Bg) and extension of the structural entities (1/ (L)) for known breadth of the primary beam (Bp) according to Ruland s streak method. The perfect linearization of the observed azimuthal integral breadth measured as a function of arc radius, s, shows that the orientation distribution is approximated by a Lorentzian with an azimuthal breadth Bs... 

Whittaker [3] has given an expression for the azimuthal angle of the spherical pendulum in terms of elliptic functions of the third kind, so that, not surprisingly, there has been very little numerical discussion of its motion. Instead, we see if our approximate theory can be used to obtain a simpler picture of the motion. 

In the texture pattern described in this chapter, the background was approximated by a Gaussian curve from three different radial profiles, since there can be some deviations from the average radial profile in different directions which can lead to over- or underestimations of the background. Each profile was averaged over 3° in the azimuthal direction the directions... 

The position and boundary of each spot are determined interactively from contour maps, using contour heights selected by taking radial and azimuthal scans through the approximate centre as described in section2. 

Here t is a part of the azimuthal coordinate cj), periodically changing in time and which on the conditions of the RP approximation vanishes, since in this approximation [Pg.189]

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

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