Spherical coordinates determination

Since the vector g is represented above in terms of the g-coordinate system (i i is) having — g as the i3 axis, it is necessary to determine the transformation to the (iI,iJ/,i2) coordinate system in which the particle velocities are written, in order to evaluate certain integrals. If we let be the spherical coordinate angles of the vector v2 — vlt in the v-coordinate system, then ... 

The relative velocity between the molecules not only determines whether A and B collide, but also if the kinetic energy involved in the collision is sufficient to surmount the reaction barrier. Velocities in a mixture of particles in equilibrium are distributed according to the Maxwell-Boltzmann distribution in spherical coordinates ... 

To calculate d4/d , we need to evaluate partial derivatives, such as U->4/7) , which measures the rate of change in energy with the order parameter. To do so we need to define generalized coordinates of the form ( , qi, , qN-1). Classical examples are spherical coordinates (r, 6, o), cylindrical coordinates (r, 0, z) or polar coordinates in 2D. Those coordinates are necessary to form a full set that determines... 

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 ... 

The wave function, V /(r), is a function of the vector position variable r. To determine it at every point in space it is convenient to take advantage of the fact that the potential V(r) depends only on the scalar interatomic distance r. In spherical coordinates (Figure 1.2), the Laplacian operator V2 has the form... 

As is the case with all differential equations, the boundary conditions of the problem are an important consideration since they determine the fit of the solution. Many problems are set up to have a high level of symmetry and thereby simplify their boundary descriptions. This is the situation in the viscometers that we discussed above and that could be described by cylindrical symmetry. Note that the cone-and-plate viscometer —in which the angle from the axis of rotation had to be considered —is a case for which we skipped the analysis and went straight for the final result, a complicated result at that. Because it is often solved for problems with symmetrical geometry, the equation of motion is frequently encountered in cylindrical and spherical coordinates, which complicates its appearance but simplifies its solution. We base the following discussion on rectangular coordinates, which may not be particularly convenient for problems of interest but are easily visualized. 

When is expressed in spherical coordinates as t i(r, 0, cp), then reflection through the origin is accomplished by replacing 0 and cp by (it — 0) and (it + cp), respectively. (r cannot change sign as it is just a distance.) In other words, the parity of the wave function is determined only by its angular part. For spherically symmetric potentials, the value of l uniquely determines the parity as... 

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]... 

A system of N spherical particles in an electrolyte solution with permittivity em is considered. Particle radii are denoted as ak, and their permittivities are denoted as ek (k = 1, 2,TV). We link the local polar spherical coordinates (rk, 0k, (pk) with the particle centers (rk is a polar radius, 6k is an azimuth angle, q>k is a polar angle). The arrangement of two arbitrarily chosen particles from the ensemble is shown in Figure 1 with corresponding coordinates indicated. Global coordinates (x,y,z) of an observation point P(x,y,z) are determined by vectors rk, r. in the local coordinates, and a distance between centers of the spheres is Rkj (Figure 1). 

To start with, let us determine the stress and the deformation of a hollow sphere (outer radius J 2, inner radius R ) under a sudden increase in internal pressure if the material is elastic in compression but a standard solid (spring in series with a Kelvin-Voigt element) in shear (Fig. 16.1). As a consequence of the radial symmetry of the problem, spherical coordinates with the origin in the center of the sphere will be used. The displacement, obviously radial, is a function of r alone as a consequence of the fact that the components of the strain and stress tensors are also dependent only on r. As a consequence, the Navier equations, Eq. (4.108), predict that rot u = 0. Hence, grad div u = 0. This implies that... 

The radial and transverse stresses can be determined from the stress-strain relationships. Owing to the orthogonality of the spherical coordinates, the formal structure of the generalized Hooke s law, given by Eq. (P4.11), is preserved, so that the nonzero components of the stress tensor are expressed in terms of the strain tensors as... 

The expression (10) is not suitable to determine collisional wavcfunctions because their asymptotic part is determined in Jacobi space-fixed coordinates specific for each arrangement A. The transformation from body-frame hypcrsphcrical democratic coordinates (p,6,(j),S) to space-fixed (Rx,rx) Jacobi coordinates is performed in several steps. It necessitates the introduction of Fock internal hvjier-spherical coordinates p,oJx., ii ) to analyze the rovibrational character of the surface functions in the fragmentation region at huge p. 

Dell Era. A., and Pasquali, M. 2009. Comparison between different ways to determine diffusion coefficient by solving Fick s equation for spherical coordinates. Journal of Solid State Electrochemistry 13, 849-859. 

Again, it is emphasized that though the form of this solution does depend on the fact that we have used eigenfunctions for a spherical coordinate system, it is strictly independent of the body geometry this will come into play only when we try to apply boundary conditions at the body surface to determine C and D . For now we simply leave these constants unspecified. 

To do this, we must first determine the form of streamfunction at large distances from the sphere when the velocity field has the asymptotic form (7 176). Specifically, if we transform (7—176) to spherical coordinates by using the general relationships... 

Determine the velocity and pressure fields in the liquid as well as the velocity of the bubble by means of a full eigenfunction expansion for spherical coordinates. Does this solution satisfy the normal-stress condition on the bubble surface ... 

Problem 7-20. Sphere in a Parabolic Flow. Use the general eigenfunction expansion for axisymmetric creeping-flows, in spherical coordinates, to determine the velocity and pressure fields for a sohd sphere of radius a that is held fixed at the central axis of symmetry of an unbounded parabolic velocity field,... 

This type of confinement is alternative to that by a circular cone in [22] and Section 4.2, using spheroconal coordinates instead of spherical coordinates. The radial coordinate is common to both sets, but the angular coordinates are different. Here we first identify the set of cones by fixed values of the coordinate xi with their common vertex as the origin, their common axis as the z-axis, and elliptical cross sections as determined next. In fact, by using the relationships... 

Although the secular determinant associated with the c/ coeffici-ents is sufficient for calculating the separation constants, it would be even more useful to have a set of coefficients that relate the eigenstates in spheroidal coordinates, designated as nam >, and the customary states in spherical coordinates, designated by n/m >, in the following simple manner ... 

---