Spherical coordinates for

As a last example in this section, let us consider a sphere situated in a solution extending to infinity in all directions. If the concentration at the surface of the sphere is maintained constant (for example c — 0) while the initial concentration of the solution is different (for example c = c°), then this represents a model of spherical diffusion. It is preferable to express the Laplace operator in the diffusion equation (2.5.1) in spherical coordinates for the centro-symmetrical case.t The resulting partial differential equation... 

If the relative velocity is sufficiently low, the fluid streamlines can follow the contour of the body almost completely all the way around (this is called creeping flow). For this case, the microscopic momentum balance equations in spherical coordinates for the two-dimensional flow [vr(r, 0), v0(r, 0)] of a Newtonian fluid were solved by Stokes for the distribution of pressure and the local stress components. These equations can then be integrated over the surface of the sphere to determine the total drag acting on the sphere, two-thirds of which results from viscous drag and one-third from the non-uniform pressure distribution (refered to as form drag). The result can be expressed in dimensionless form as a theoretical expression for the drag coefficient ... 

In our discussion of spherical harmonics we will use an expression of the three-dimensional Laplacian in spherical coordinates. For this we need spherical coordinates not just on but on all of three-space. The third coordinate is r, the distance of a point from the origin. We have, for arbitrary (x, y, zY e... 

As in Fig. 11.13, the loop can be represented by an array of point sources each of length R0. Using again the spherical-sink approximation of Fig. 11.126 and recalling that d Rl Ro, the quasi-steady-state solution of the diffusion equation in spherical coordinates for a point source at the origin shows that the vacancy diffusion field around each point source must be of the form... 

In this study the numerical simulations were performed with a 3-D mechanistic global Cologne Model of the Middle Atmosphere (COMMA) based on the primitive equations expressed in spherical coordinates for the horizontal and log-pressure coordinates in the vertical direction. The model equations are solved on the basis of an explicit numerical scheme (leapfrog) with a fixed time step of 450 sec. To avoid separate evolution at even and odd time steps, a Robert time filter is used. 

The overall emission rate of photons with energy tku at time t reads (note the use of spherical coordinates for k and the abbreviation of the solid angle integration by f do)... 

To write down Eq. (4.53) as well as (4.51), we used, after Ref. 54, the distribution function moments presented as Cartesian tensors. However, when solving the orientational problem, it is more natural to use the set of spherical functions. Choosing spherical coordinates for the unit vectors e, it, and h as (Q, cp), (0,0), (v(/,0), respectively, that is, taking as the polar axis of the framework, one gets... 

These boundary conditions are in spherical coordinate for the spherical particles. The first condition corresponds to symmetry at the center of die pore. The second condition corresponds to the mass transfer at the interface between mobile and stationary phase. 

The ZA(f) are chosen so that they tend to 1 in the vicinity of atom A but drop to zero in the direction of all other nuclei. Thus, even for integrals involving atomic basis functions of two different atoms the integrand of each contribution la has no more than one singular point. The integration can be further simplified by suitable transformations to intrinsic coordinates, e.g. elliptic-hyperbolic coordinates for diatomic molecules or spherical coordinates for polyatomic systems. 

In general, for 2D flows, 3 can be identified with a Cartesian variable z, orthogonal to the plane of motion, and /13 = 1. However, for axisymmetric flows, c/3 represents the azimuthal angle

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Figure 7.9 Isothermal effectiveness (spherical coordinates) for the CO oxidation reaction. [After T.G. Smith, J. Zahradnik and J.J. Carberry, Chem. Eng. Sci., 30, 763, with permission of Pergaman Press, Ltd., London, England, (1975).]...

This section refers to methods based on wavefunctions of the types illustrated in equation (7) or (8). Although some work has been carried out on nonseparable fimctions of the r,y [59,60], we focus here on the simpler basis functions that correspond to the majority of the recent work. We describe the wavefunctions now under discussion as orbitals centered on particle N, multiplied by polynomial correlation factors involving the, where i and j are both less than N. It has been usual to use spherical coordinates for the orbitals (in our present notation with coordinates r jv), and to expand the correlation factors in terms of the orbital coordinates. This approach makes it unnecessary to analyze the kinetic energy as was done earlier in this work, as each term in the expansion of the r,-, is just a conventional orbital product. 

The orientation of a single fiber can be represented by two angular values (0 < 0 < r and 0 < (jxlTt, see Fig. 3.32) in spherical coordinates. For SMC compression molding processes, the usual part thickness is much smaller than the fiber length. Hence, the orientation state in compression molding processes can be regarded as two-dimensional or planar (i.e., 0=0). 

The integral over all orientations of one segment can be evaluated. Adopting spherical coordinates for a ( a = as,i ,( ), with the axis (i = 0) oriented parallel to g, we have... 

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