Coordinate systems spherical

Use now this equation to describe liquid film flow in conical capillary. Let us pass to spherical coordinate system with the origin coinciding with conical channel s top (fig. 3). It means that instead of longitudinal coordinate z we shall use radial one r. Using (6) we can derive the total flow rate Q, multiplying specific flow rate by the length of cross section ... 

Therefore, a stream function T may be introduced in the meridian plane of the cyclone, i.e., the r, 9) plane in the spherical coordinate system ... 

Figure 2-12. Equations of motion in a spherical coordinate system.

The most important parameters of the ionic atmosphere are the charge density Qv r) and the electrostatic potential /(r) at the various points. Each of these parameters is understood as the time-average value. These values depend only on distance r from the central ion, not on a direction in space. For such a system it is convenient to use a polar (spherical) coordinate system having its origin at the point where the central ion is located then each point can be described by a single and unique coordinate, r. 

Consider the specific example of a spherical electrode having the radius a. We shall assume that diffusion to the spherical surface occurs uniformly from all sides (spherical symmetry). Under these conditions it will be convenient to use a spherical coordinate system having its origin in the center of the sphere. Because of this synunetry, then, aU parameters have distributions that are independent of the angle in space and can be described in terms of the single coordinate r (i.e., the distance from the center of the sphere). In this coordinate system. Pick s second diffusion law becomes... 

Although the foregoing example in Sec. 4.2.1 is based on a linear coordinate system, the methods apply equally to other systems, represented by cylindrical and spherical coordinates. An example of diffusion in a spherical coordinate system is provided by simulation example BEAD. Here the only additional complication in the basic modelling approach is the need to describe the geometry of the system, in terms of the changing area for diffusional flow through the bead. 

Diffusion in a sphere may be more common than that in a cylinder in the pharmaceutical sciences. The example we may think of is the dissolution of a spherical particle. Since convection is normally involved in solute particle dissolution in reality, the dissolution rate estimated by considering only diffusion often underestimates experimental values. Nevertheless, we use it as an example to illustrate the solution of the differential equations describing diffusion in the spherical coordinate system [1],... 

A sphere is assumed to be a poorly soluble solute particle and therefore to have a constant radius rQ. However, the solid solute quickly dissolves, so the concentration on the surface of the sphere is equal to its solubility. Also, we assume we have a large volume of dissolution medium so that the bulk concentration is very low compared to the solubility (sink condition). The diffusion equation for a constant diffusion coefficient in a spherical coordinate system is... 

The spherical coordinate system is useful for many of the moisture uptake problems encountered in this chapter. The application of the spherical coordinate system can be illustrated by the following example (see Fig. 5). 

The spherical coordinate system is convenient if there is a point of symmetry in the system. This point is taken as the origin and the coordinates (p, 0, 9) illustrated in Fig. 3-27. The relations are x =... 

Some Basics. The field theory of electrostatics expresses experimentally observable action-at-a-distance phenomena between electrical charges in terms of the vector electric field E (r, t), which is a function of position r and time t. Accordingly, the electric field is often interpreted as force per unit charge. Thus, the force exerted on a test charge q, by this electric field is qtE. The electric field due to a point charge q in a dielectric medium placed at the origin r = 0 of a spherical coordinate system is... 

As seen from Chapter 2, adsorbed molecules often form monolayers with chain orientational structures in which the chains with identically oriented molecules alternate (Fig. 2.4). Consider the Davydov splitting of vibrational spectral lines in such systems. Let molecular orientations be specified by the angles 6> and [Pg.67]

Let M V X tij/) and N = (V x M)/< with p given by Eq. (53), where r is the radius vector of the spherical coordinate system. Then both M and N satisfy the vector wave equation and both have zero divergence. In addition, M and N satisfy... 

Figure 1.15 Is, 2p, and 3d degenerate atomic orbitals (ADs) and corresponding wave functions (cf. eq. 1.125). Upper left spherical coordinate system. Adapted from Harvey and Porter (1976).

Fig. 3.1. Derivation of the tunneling matrix elements. A separation surface is placed between the tip and the sample. The exact position and the shape of the separation surface is not important. The coordinates for the Cartesian coordinate system and spherical coordinate system are shown, except y and 4>. (Reproduced from Chen, 1990a, with permission.)...

The angular integration in this final three dimensional integral is easily done if a spherical coordinate system is introduced with the z axis chosen along Rt ... 

In practice, the transformation of any operator to irreducible form means in atomic spectroscopy that we employ the spherical coordinate system (Fig. 5.1), present all quantities in the form of tensors of corresponding ranks (scalar is a zero rank tensor, vector is a tensor of the first rank, etc.) and further on express them, depending on the particular form of the operator, in terms of various functions of radial variable, the angular momentum operator L(1), spherical functions (2.13), as well as the Clebsch-Gordan and 3n -coefficients. Below we shall illustrate this procedure by the examples of operators (1.16) and (2.1). Formulas (1.15), (1.18)—(1.22) present concrete expressions for each term of Eq. (1.16). It is convenient to divide all operators (1.15), (1.18)—(1.22) into two groups. The first group is composed of one-electron operators (1.18), the first two... 

Since we will, see further below, introduce the Mellin transformation to analyze scale invariance in connection with the micro-macro problems, it will be natural to rewrite some of our formulas in a spherical coordinate system, i.e., in terms of (r,, ) = (r, 2) with (u ru)... 

Table 5.1 presents the continuity equation in the Cartesian, cylindrical and spherical coordinate systems. 

Table 5.5 presents the complete energy equation in the Cartesian, cylindrical and spherical coordinate systems. Table 5.6 defines the viscous dissipation terms for an incompressible Newtonian fluid. 

Equations 2.8-2 and 2.8-3 are coordinate-independent compact tensorial forms of the Newtonian constitutive equation. In any particular coordinate system these equations break up into nine (six independent) scalar equations. Table 2.3 lists these equations in rectangular, cylindrical and spherical coordinate systems. 

In this section we only need to work in the framework of a geometrical theory of space-time, in which the trajectories of light rays are assumed to be the null geodesics. Let s have a comoving spherical coordinates system (r, 9, ... 

To make the shield tensor DSij(d) explicit, we introduce spherical coordinate systems (r, 6, tp) with the polar axis in the x, direction, i = 1,2, 3 (Fig. 1). It then follows from Eq. (23) that DSij d) — Sy(d) — <)ijSrr has the values... 

Using the spherical coordinate system with the center of the sphere as the origin, the electrical potential outside the ion cloud is governed by Laplace s equation,... 

Fig. 3-11 Volume and resistance elements (a) cartesian, (b) cylindrical, and (c) spherical coordinate systems.

Fig. 8-10 Spherical coordinate system used in derivation of radiation shape factor.

The mass conservation equation in rectangular coordinates x, y, and z (Equation 3.31) is suitable for catalyst pellets with a parallelepiped shape, in particular when they can be considered as a plate. For particles with cylindrical surfaces and for spheres, the cylindrical and spherical coordinate systems are more natural. For ax-isymmetrical concentration distributions in cylindrical coordinates Equation 3.30 becomes... 

Similar equations apply to cylindrical and spherical coordinate systems. Finite difference, finite volume, or finite element methods are generally necessary to solve (5-15). Useful introductions to these numerical techniques are given in the General References and Sec. 3. Simple forms of (5-15) (constant k, uniform S) can be solved analytically. See Arpaci, Conduction Heat Transfer, Addison-Wesley, 1966, p. 180, and Carslaw and Jaeger, Conduction of Heat in Solids, Oxford University Press, 1959. For problems involving heat flow between two surfaces, each isothermal, with all other surfaces being adiabatic, the shape factor approach is useful (Mills, Heat Transfer, 2d ed., Prentice-Hall, 1999, p. 164). 

Trivial integration over spherical and prolate spherical coordinate systems. 

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