Radial displacement vector

In the REC model, the ligand is modelled through an effective point charge situated in the axis described by the lanthanide-coordinated atom axis, at a distance R, which is smaller than the real metal-ligand distance (Figure 2.6). To account for the effect of covalent electron sharing, a radial displacement vector (Dr) is defined, in which the polar coordinate R is varied. At the same time, the charge value (q) is scanned in order to achieve the minimum deviation between calculated and experimental data, whereas 9 and cp remain constant. 

Fig. 78.—Radial distribution function W r) of the chain displacement vectors for the same polymer chains as in Fig. 77. W(r) is expressed in A"h...

This gives the relationship of the radial components of the displacement vector between the cases of the pure diffusion and the combination of the pure diffusion and shear relaxation. For pure diffusion, Eq. (3.5) becomes... 

Fig. 4.6 Surface velocities of the displacement vector indicating radial distribution and orientation in the x-direction (adapted from Martin and Hager, 1989a)...

With the help of models, the tangential and radial stresses can be calculated for the ideal case of curved surfaces with a constant radius of curvature. Introduction of the oxide displacement vector M is helpful here ... 

As shown in Figure 12.13, we have performed the pressure dependence study of the Ge-Ge modes of the NCs for one pressure cycle and there is no hysteresis observed. This confirms that the pressure-induced strain in the NCs is reversible. With this elastic behavior of our Ge/Si02 NCs-matrix system, we assume both the NCs and the matrix as isotropic elastic continua [46]. We modeled the NC as a sphere of radius ri in a spherical Si02 matrix of radius t2, where r2 2> ri. The system is subjected to a hydrostatic pressure P. Using spherical co-ordinates with the origin at the center of the NC sphere, our system has a spherical symmetry where the displacement vector u is everywhere radial and can be written d u = ar + b/r. The components of the strain tensor are Urr = a — and uee = =... 

The anticipated induced load created by the driven units determines orientation of the measurements. The primary (X-axis) radial measurement should be positioned in the same plane as the worst anticipated shaft displacement. The secondary (Y-axis) radial should be position at 90° in the direction of rotation to the primary point and oriented to permit vector analysis of actual shaft displacement. 

The reduced oxidation near sample corners is related to these stress effects, either by retarded diffusion or modified interfacial reactionsManning described these stresses in terms of the conformational strain and distinguished between anion and cation diffusion, and concave and convex surfaces. He defined a radial vector M, describing the direction and extent of displacement of the oxide layer in order to remain in contact with the retreating metal surface, where ... 

Two or more of these conditions can occur at the same time, resulting in asymmetric axial, radial and tangential velocity vectors. Some flowmeters are more sensitive than others to particular types of flow distortion, e.g. orifice meters are affected by pure swirl more than venturi meters are magnetic flowmeters are unaffected by changes in the radial velocity component whereas ultrasonic time-of-flight meters are highly susceptible thereto swirl and asymmetry have the least effect on positive displacement meters and the greatest effect on variable area meters. 

The strategy is as follows. We start by rewriting the equations in cylindrical coordinates (r, ,z). The variables we consider are the layer displacement u (now in the radial direction) from the cylindrical state, the director n, and the fluid velocity v. The central part of the cylinder, r < Ri, containing a line defect, is not included. It is not expected to be relevant for the shear-induced instability. We write down linearized equations for layer displacement, director, and velocity perturbations for a multilamellar (smectic) cylinder oriented in the flow direction (z axis). We are interested in perturbations with the wave vector in the z direction as this is the relevant direction for the hypothetical break-up of the cylinder into onions. The unperturbed configuration in the presence of shear flow (the ground state) depends on r and 0 and is determined numerically. The perturbations, of course, depend on all three coordinates. We take into account translational symmetry of the ground state in the z direction and use a plane wave ansatz in that direction. Thus, our ansatze for the perturbed variables are... 

Several structural properties are typically monitored to characterize water in these systems. Some of these are the density distribution of water molecules with respect to the surface, surface area per water molecule, the root-mean-square displacements from the optimal surface positions for corrugated surfaces, angular distributions of the dipolar and O—H bond vectors with respect to the surface normal, and moments of the angular distributions. Most of these characterize the water structure with reference to the metal surface. In contrast, the radial distribution function, number of nearest neighbors per molecule, and number of hydrogen bonds per molecule are used to characterize water-water interactions. 

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