Azimuthal radial

In classical geosteering the sensors for inclination, azimuth, drilling parameters, and logging are located above the mud motor and the distances may be in the order of those shown in Figure 4-296 that is 30 ft or more above the drill bit. Although radial measurements can be performed to verify that the borehole is being drilled in the pay zone, it is often too late to make a correction and the borehole leaves the pay zone. 

The framework we adopted for measuring the scaling behavior from AFM images is the following. The 2-D power spectral density (PSD) of the Fast Fourier Transform of the topography h(x, y) is estimated [541, then averaged over the azimuthal angle

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From equations (8), (9) and (10) it is evident that the path of the electron in the ith region is a segment of the KepleT ellipse defined by the segmentary radial and the azimuthal quantum numbers n and k, so that it can be described by the known equations... 

The shell theory has had great success in accounting for many nuclear properties (3). The principal quantum number n for nucleons is usually taken to be n, + 1, where nr, the radial quantum number, is the number of nodes in the radial wave function. (For electrons n is taken to be nr + / +1 / is the azimuthal quantum number.) Strong spin-orbit coupling is assumed,... 

The index n is the radial degree and the index m is the azimuthal frequency. The function R r) is a radial function given by... 

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 ... 

Motivation and Principle. Broadened reflections are characteristic for soft matter. The reason for such broadening is predominantly both the short range of order among the particles in the structural entities, and imperfect orientation of the entities themselves. A powerful method for the separation of these two contributions is Ruland s streak method [30-34], Short range of order makes that the reflection is considerably extended in the radial direction of reciprocal space - often it develops the shape of a streak. This makes it practically possible to measure reflection breadths separately on several11 nested shells in reciprocal space. As a function of shell diameter one of the contributions is constant, whereas the other is changing12. If the measurement is performed on spheres (azimuthal), the orientation component is constant. 

The three quantum numbers may be said to control the size (n), shape (/), and orientation (m) of the orbital tfw Most important for orbital visualization are the angular shapes labeled by the azimuthal quantum number / s-type (spherical, / = 0), p-type ( dumbbell, / = 1), d-type ( cloverleaf, / = 2), and so forth. The shapes and orientations of basic s-type, p-type, and d-type hydrogenic orbitals are conventionally visualized as shown in Figs. 1.1 and 1.2. Figure 1.1 depicts a surface of each orbital, corresponding to a chosen electron density near the outer fringes of the orbital. However, a wave-like object intrinsically lacks any definite boundary, and surface plots obviously cannot depict the interesting variations of orbital amplitude under the surface. Such variations are better represented by radial or contour... 

In order to understand the physics behind the observed super-high sensitivity, we investigated the optical field distribution in the microtube using cylindrical coordinates z, r, and time-independent field distribution for the resonant mode can be separated into a radial-dependent mode component and an azimuthal-dependent phase term, T lYjexpfiw J, where / is the amplitude of the axial magnetic (TE) or electric (TM) modal field and m is the azimuthal quantization... 

Fig. 8.33 Calculated sensitivity in bulk index sensing for different radial mode numbers with the same azimuthal number m 700...

Fig. 8.34 Surface sensing sensitivity of different radial order modes are simulated by using the perturbation method with the same azimuthal number m 700.The adsorbed polymer layer is assumed to have a refractive index of 1.46...

Fig. 17.1 Illustrations of whispering gallery modes (WGM) in a spherical optical resonator. The WGM modes are classified in terms of their radial quantum number p as well as by their angular momentum quantum number / and the azimuthal quantum number m that can have (21+ 1) values, meaning that the resonance frequency ( ,/ has a (2/ + 1) degeneracy...

The projection of T,p on each of the radial unit vectors can be evaluated in terms of the basic angular functions which make up the vector spherical harmonics.(27) Although these functions are associated Legendre polynomials for an arbitrarily oriented donor dipole, for the case of full azimuthal symmetry shown in Figure 8.19 the angular functions are ordinary Legendre functions, P (i.e., w = 0). Under these circumstances,... 

Fig. 24. Contour plot of the structure factor (the kinematic LEED intensity) of a x y/i monolayer in a triangular lattice gas with nearest-neighbor repulsion, at a temperature k TIi = 0.355 (about 5% above T ) and a chemical potential // = 1.5 (0c = 0.336 at the transition temperature.) Contour increments are in a (common) logarithmic scale separated by 0.1, starting with 3.2 at the outermost contour. Center of the surface Brillouin zon is to the left k, and k the radial and azimuthal components of kH, are in units of nlXla, a being the lattice spacing. Data are based on averages over 2x10 Monte Carlo steps per site. (From Bartelt et...

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 effect of heating on the azimuthal and streamwise components of the vorticity field is shown in Fig. 11.1 the effect on the radial component is comparable to that on the azimuthal component, and is therefore not separately shown. The vorticity distributions at the same nondimensional time t = 35 are plotted side by side for the unheated and heated case for each component. The positive and negative values of vorticity are shown by solid and dotted lines, respectively. 

To quantify the vorticity increases due to heating, the total enstrophy and also the enstrophies corresponding to the azimuthal, streamwise, and radial components of vorticity are examined. Computed values are shown using a linear-log scale in Fig. 11.2. In the absence of heating, the total as well as the component enstrophies all fall beyond time t = 25, as would be expected in a fully developed turbulent jet. When heat is applied, there is a virtually exponential rise of the enstrophies after some time. At t = 35, the enstrophies are one order of magnitude higher with heating than without. 

Figure 11.2 Comparison of evolution of enstrophy in the heated (thick lines) and unheated (thin lines) jet. The enstrophy components are total (solid line), azimuthal (dashed), radial (dotted), and streamwise (long-dashed). Note logarithmic scale on j/-axis... 

Along a radial line at z = 0.017586207 m, graph the azimuthal vorticity. Qualitatively explain the observed behavior in terms of the flow field and the signs of the vorticity. Where is the flow rotating clockwise and counterclockwise, and why ... 

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