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Background Principles
Published in Richard Leach, Stuart T. Smith, Basics of Precision Engineering, 2017
When there is a cylindrical or spherical symmetry present in geometry, it is often more convenient to work in cylindrical or spherical coordinate systems. Suppose that a point P in three-dimensional space is identified by the rectangular coordinates (x1,x2,x3). In a cylindrical coordinate system, the same point is identified by the coordinates (r,θ,z) (Figure 3.3a). The radial coordinate r(≥0) is the distance (from the origin O) of the projection of the point P onto the x1 – x2 coordinate. The coordinate z is the same as x3 and the azimuthal angle θ varies over a range of 2π, typically between 0 and 2π or from –π to π.
Orthogonal Coordinate Systems
Published in Sivaji Chakravorti, Electric Field Analysis, 2017
Spherical coordinates are particularly useful for analyzing fields having spherical symmetry. In the spherical coordinate system, the coordinates of any point in space are the ordered triplet (r,θ,ϕ), as shown in Figure 3.12. The coordinate r measures the radial distance from the origin to the point P. The coordinate θ is the angle that the r vector makes with the positive direction of the z-axis, whereas the coordinate ϕ is the azimuthal angle with respect to the positive direction of the x-axis spinning around the z-axis in counterclockwise sense. The angle ϕ is defined on the x–y plane. The ranges of the values of the three coordinates are 0 ≤ r < ∞, 0 ≤ θ ≤ π and 0 ≤ ϕ ≤ 2π.
Structural Analysis of Optics
Published in Anees Ahmad, Handbook of Optomechanical Engineering, 2017
Symmetry is defined as a balanced arrangement of structure about a point (spherical symmetry), a line (axisymmetry), or a plane (reflective symmetry). For a structure to be symmetric, both the structure and its boundary conditions must possess the same degree of symmetry. Applied loads may be nonsymmetric, although there are additional efficiencies when the load is also symmetric. A full discussion of using symmetry techniques in analysis is given by Doyle et al.1
Achieving geometrical enhancement of fields in chiral nanoplasmonics using fractional calculus
Published in Waves in Random and Complex Media, 2021
Tariq Mahmood, Qaisar Abbas Naqvi
As discussed in above section, the characteristic function is discontinuous in both media. The next step in our consideration is averaging Equation (5). The procedure leads to integration and evaluation of the integral with the polarization charge density term . For this purpose, a spherical volume of radius r is considered, such that , and . We suppose that a fractional mass within the considered spherical volume is , where D is the fractal dimension. Therefore, is the averaged density of the metallic phase. As random fractal behave as an isotropic scatterer i.e. . It is feasible to suppose that changes in the responding electric field due to the considered geometry are the same property. Any change in the potential in azimuth and inclination angle direction is neglected due to the spherical symmetry: . Only the radial component is considered i.e.
Mesh-Free Simulation of Spatially Inhomogeneous Aerosols in Arbitrary Geometries
Published in Nuclear Science and Engineering, 2019
Matthew Boraas, Sudarshan K. Loyalka
To further verify the DSMC results, they can also be compared to results generated by the FEM technique (see Fig. 4). The close correspondence between the two series of data points indicate that DSMC can be used as a close approximation of the FEM technique. We carried out the FEM calculations without using spherical symmetry, as we were interested in its usefulness for an arbitrary three-dimensional geometry. We found that the FEM approach is computationally very demanding both in terms of the data storage and the computational time as very fine meshes near the surfaces are needed and the usefulness of the approach (within the applicability of the Mathematica built-in FEM program that we used) was limited (it was also not applicable to nonlinear cases that would coagulate). Hence, we have used the FEM program for verification in a limited number of cases and for deposition only (see the Appendix).
On magnetostrophic dynamos in annular cores
Published in Geophysical & Astrophysical Fluid Dynamics, 2020
Paul H. Roberts, Cheng-Chin Wu
Basic to this paper is the magnetostrophic approximation in which the velocity, , in Earth's core is unaffected by viscous and inertial forces (in the reference frame rotating with the Earth). Dimensionless variables, defined in section 1, were used. In the fluid core, is created by the buoyancy force, , of an adverse temperature gradient produced by a difference, , in the temperature, , of the inner core and outer core surfaces; the angular velocity, ω, of the Earth's inner core, has been sought too. The motion is opposed by the Lorentz force of a magnetic field, , and its associated electric current density, . We have developed a procedure for finding , ω, and for a specified . We believe that this procedure is self-consistent, but the demonstration has inevitably been complicated for the reason given in section 1: the competition between dynamics dominated by Coriolis forces and geometry controlled by the near spherical symmetry of the Earth. The procedure for advancing the solution in time is summarised in table 1.