Azimuthal coordinate

It is convenient to employ two sets of coordinate systems. Spherical polar coordinates r, Q, A) are defined with the origin at the vertex of the cone the axis is 0=0, the surface of the conical portion of the cyclone is the cone 0 = 0% and the azimuthal coordinate is A. Using the same origin, cylindrical polar coordinates (R, A, Z) are defined, where R = r sin 0 and the Z-axis coincides with the axis 0=0. 

Then, taking into account that the distribution of masses inside the spheroid is independent of the azimuth coordinate cp, we have for Laplace s equation... 

With a view to any simulation, a few important items have to be addressed. First of all, it has to be decided whether the flow to be simulated is 2-D, 21/2-D, or 3-D. When the flow is, e.g., axis-symmetrical and steady, a 2-D simulation may suffice. For a flow field in which all variables, including the azimuthal velocity component, may not depend on the azimuthal coordinate, a 21/2-D simulation may be most appropriate. Most other cases may require a full 3-D simulation. It is tempting to reduce the computational job by casting the 3-D flow field into a 2-D mode. The experience, however, is that in 2-D simulations the turbulent viscosity tends to be overestimated in this way, the flow... 

Fig. 2.20 Illustration of how the cylindrical-coordinate unit vectors er and eg depend on the azimuthal coordinate 0.

Figure 4. Example of an azimuthal scan. The microdensitometer readings Ds are plotted against an azimuthal coordinate (0-360°). Two symmetry-related equatorial...

We represent the phase-volume element dV in the form dT = dhdl dtp0, where h and canonically conjugated variables, (p0 is an initial instant that enters in addition to time (p in the law of motion of a dipole. Another pair of canonical variables is /, 0 we omit differential df>() in dT, since the variables we use do not depend on the azimuthal coordinate < )0. 

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]

Here the upper index of the C coefficients corresponds to the azimuthal number m of the spherical harmonic c""°. Operator L now includes the azimuthal coordinate and takes the form... 

Fig. 6. Binding energies of PF in turnstile rotation model situations, from CNDO/2 calculations. Z-axis as TR-axis reaction coordinates relative change, A, in the azimuth coordinates of the pair and trio ligands.

We now discuss the analysis of the x-ray intensities. The atoms of the C6o molecule are placed at the vertices of a truncated icosahedron. - The x-ray structure factor is given by the Fourier transform of the electronic charge density this can be factored into an atomic carbon form factor times the Fourier transform of a thin shell of radius R modulated by the angular distribution of the atoms. For a molecule with icosahedral symmetry, the leading terms in a spherical-harmonic expansion of the charge density are Koo(fl) (the spherically symmetric contribution) and KfimCn), where ft denotes polar and azimuthal coordinates. The corresponding terms in the molecular form factor are proportional to SS ° (q)ac jo(qR)ss n(qR)/qR and... 

We note that any pure state in is coherent. The interpretation of the parameter a is very simple its module is proportional to the polar coordinate, while its argument cp is the azimuthal coordinate of the representative Poincare sphere point. 

Statement of the problem. Exact solution. In infinite space, we consider the flow caused by a liquid jet discharging from a thin tube. We treat the jet source as a point source, since the size and shape of the nozzle section are unessential remote from the source. The jet is axisymmetric about the flow direction. If there is no rotation of the fluid, then the motion considered in the spherical coordinates (R, 6, (p) is independent of the azimuth coordinate ip. and moreover, the condition V, = 0 must be satisfied. 

We use a local orthogonal curvilinear system of dimensionless coordinates f, ), ip, where 77 varies along and is normal to the surface of the particle. In the axisymmetric case, the azimuth coordinate tp varies from 0 to 27r in the plane case, it is supposed that 0 <

constant value = s. The dimensionless fluid velocity components can be expressed via the dimensionless stream function rp as follows ... 

The authors considered analytical and numerical solutions for the differential equations by using the three main overpotentials charge transfer, mass transport, and ohmic drop. We discuss only the case of a two-dimensional system, where the current distribution varies with the radial and the azimuthal coordinates. 

Cylindrical coordinates are a generalization of polar coordinates to three dimensions, obtained by augmenting r and 9 with the Cartesian z coordinate. (Alternative notations you might encounter are r or p for the radial coordinate and 9 or 4> for the azimuthal coordinate.) The 3x3 Jacobian determinant is... 

Assuming that all the variables in the physical problem are independent of the azimuthal coordinate (axisymmetry), the projections of Table 11.3 model equations along the three coordinate axes are given in Table 11.4 together with the boundary conditions chosen at the four peripheral boundaries of the porous medium z = 0, z = L, r=0,r= R. We will assume without proof, uniqueness of solution for the system of equations describing this ferrohydrodynamic model. 

However, a solution can be found in spherical polar coordinates. This solution can be represented as a product of functions each dependent on one coordinate. These are the coordinates r,6,(p, where r is the length of the segment connecting the electron with the nucleus (the origin), 0 and (p are the polar and azimuthal coordinates, respectively. Because of this simplification of the potential, it is possible to carry out the separation of variables in the time-independent Schrodinger equation. This now can be written instead of (3.2) as... 

A circular waveguide with inner radius a is shown in Fig. 4.14. Here the axis of the waveguide is aligned with the z axis of a circular-cylindrical coordinate system, where p and are the radial and azimuthal coordinates, respectively. If the walls are perfectly conducting and the dielectric material is lossless, the equations for the TE , modes are... 

To calculate the flow rate Q for the elliptic channel, we need to evaluate a 2D integral in an elliptically shaped integration region. This is accomplished by coordinate transformation. Let p, 4>) be the polar coordinates of the unit disk, that is, the radial and azimuthal coordinates, which obey 0 < p < 1 and 0 < (p < 2n, respectively. The physical coordinates (y, z) and the velocity field u can then be expressed as functions of p, [Pg.33]

Investigated problems dare in axis5mimetric statement (dependence on azimuthal coordinate tp is not considered), and the liquid current is supposed to be turbulent and is presented by the system of the operating equations in the dimensional formulation. 

The electrostatic problem is again to solve the Laplace equation but now we must consider the spheroidal geometry. If we let the particle be a prolate spheroid, it is convenient to solve the Laplace equation in prolate spheroidal coordinates 17 and (p. This is an orthogonal coordinate system where and 7 are orthogonal spheroidal coordinates and (p is an azimuthal coordinate. At the surface of the spheroidal particle, = fo = = 1/[1 ( / ) ] and... 

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