Equations in Spherical Coordinates

Since the potential depends only upon the scalar r, this equation, in spherical coordinates, can be separated into two equations, one depending only on r and one depending on 9 and ( ). The wave equation for the r-dependent part of the solution, R(r), is... 

Difference schemes for an equation in spherical coordinates. If a solution to the 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 ... 

Thus, M and N have all of the properties required of the electromagnetic field. Furthermore, ij/ satisfies the scalar wave equation in spherical coordinates. 

The H2O diffusion equation in spherical coordinates is as follows (Equation 4-92) ... 

The radial part of the Schrodinger equation in spherical coordinates is... 

For the steady flow of an incompressible fluid, state the appropriate mass-continuity equation in spherical coordinates. What can be inferred from the reduced continuity equation about the functional form of of the circumferential velocity v 2... 

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... 

Many problems in nuclear physics and chemistry involve potentials, such as the Coulomb potential, that are spherically symmetric. In these cases, it is advantageous to express the time-independent Schrodinger equation in spherical coordinates (Fig. E.6). The familiar transformations from a Cartesian coordinate system (x, y, z) to spherical coordinates (r, 0, tp) are (Fig. E.6)... 

Munera and Guzman [56] obtained new explicit noncyclic solutions for the three-dimensional time-dependent wave equation in spherical coordinates. Their solutions constitute a new solution for the classical Maxwell equations. It is shown that the class of Lorenz-invariant inductive phenomena may have longitudinal fields as solution. But here, these solutions correspond to massless particles. Hence, in this framework a photon with zero rest mass may be compatible with a longitudinal field in contrast to that Lehnert, Evans, and Roscoe frameworks. But the extra degrees of freedom associated with this kind of longitudinal solution without nonzero photon mass is not clear, at least at the present state of development of the theory. More efforts are needed to clarify this situation. 

Dateo, C.E., Engel, V., Almeida, R., and Metiu, H. (1991). Numerical solutions of the time-dependent Schrodinger equation in spherical coordinates by Fourier transform methods, Computer Physics Communications 63, 435-445. 

The limiting step in the kinetics of ion exchange in the zeolite is the interdiffusion of the electrolyte ions A zi and ions of the species B [24], In the case where the solid ion-exchanger particle is spherical (see Figure 7.9) and the particle diffusion control is the rate-determining process, then Fick s second law equation in spherical coordinates is [47]... 

The radial diffusion equation in spherical coordinates may be written for constant diffusion coefficient as ... 

Consider a spherical elemental control volume of dimensions Ar, rsincontinuity equation in spherical coordinates. 

M.R. Hermann, J.A. Heck, Split-operator spectral method for solving the time-dependent Schrodinger-equation in spherical coordinates, Phys. Rev. A 38 (12) (1988) 6000-6012. 

The general heat conduction equations in spherical coordinates can be obtained from an energy balance on a volume element in spherical coordinates, shown in Fig, 2-24, by following the steps outlined above. It can al.so be obtained directly from Eq. 2-38 by coordinate transformation using the following relations between the coordinates of a point in rectangular and spherical coordinate systems ... 

Problem 1.2 (Worked Example) Compute the shear-rate profile in a cone-and-plate geometry. For a cone angle a of 0.1 radians, what percentage increase occurs in shear rate as one migrates from the plate to the cone Hint Look at the component of the momentum balance equation in spherical coordinates.)... 

We work in spherical coordinates and define the polar angle as 0 = nil — ot (see Fig. Al-2). Now we go to the momentum-balance equations in spherical coordinates in the Appendix. By symmetry, derivatives with respect to r and 0 are zero and Eq. (A-9) reduces to... 

In summary then, the leading-order problem is just the translation of a spherical drop through a quiescent fluid. The solution of this problem is straightforward and can again be approached by means of the eigenfunction expansion for the Stokes equations in spherical coordinates that was used in section F to solve Stokes problem. Because the flow both inside and outside the drop will be axisymmetric, we can employ the equations of motion and continuity, (7-198) and (7-199), in terms of the streamfunctions f<(>> and < l)), that is,... 

A simple example for which problems (9-16) and (9 17) can be solved easily is the case of a heated sphere. In this case, we may choose the sphere radius as the characteristic length scale for nondimensionalization, and the problem is to solve (9 16) subject to the boundary condition 0 = 1 at r = 1. This can be done easily. We may first note that a general solution of Laplace s equation in spherical coordinates is... 

The convective diffusion equation in spherical coordinates has the form... 

Substituting the production rate into Equation 7.14, expressing the equation in spherical coordinates, and assuming pellet symmetry in 9 and coordinates gives... 

Obvionsly, these requirements are satisfied by the mass transfer equation in spherical coordinates, given by (17-36). The parameter e governs whether the solution is given by ... 

The set of equations defining the PGM and MGM are the classic diffusion-reaction equations in spherical coordinates, with and without the pseudohomogenous approximation, respectively, but they must be solved under moving boundary conditions since the particle expands during polymerization these model equations have been explained in detail in a review dedicated to single-particle models [114]. 

---