Spherical coordinate

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 ... 

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... 

In the remainder of this section, we focus on the two lowest doublet states of Li3. Figures 3 and 4 show relaxed triangular plots [68] of the lower and upper sheets of the 03 DMBE III [69,70] potential energy surface using hyper-spherical coordinates. Each plot corresponds to a stereographic projection of the... 

In performing integration over all space, it is necessary to convert the multiple integral from cartesian to spherical coordinates ... 

In many applications, derivative operators need to be expressed in spherical coordinates. In converting from cartesian to spherical coordinate derivatives, the chain rule is employed as follows ... 

In addition, the volume element of interest is not the box dx dy dz shown in Fig. 1.6a but, rather, a spherical shell of radius r and thickness dr as shown in Fig. 1.6b. The result of expressing the volume element in spherical coordinates and integrating over all angles is the replacement... 

Sorption Rates in Batch Systems. Direct measurement of the uptake rate by gravimetric, volumetric, or pie2ometric methods is widely used as a means of measuring intraparticle diffusivities. Diffusive transport within a particle may be represented by the Fickian diffusion equation, which, in spherical coordinates, takes the form... 

V Radius cylindrical and spherical coordinate distance from midplane to a point in a body i i for inner wall of annulus Vo for outer wall of annulus for inside radius of tube for distance from midplane or center of a body to the exterior surface of the body m ft... 

One-Dimensional Conduction Many heat-condrrction problems may be formrrlated into a one-dimensional or pserrdo-one-dimensional form in which only one space variable is involved. Forms of the condrrction eqrration for rectangrrlar, cylindrical, and spherical coordinates are, respectively,... 

Table 5-12 provides material balances for Cartesian, cylindrical, and spherical coordinates. The generic form applies over a unit cross-sectional area and constant volume ... 

Point Sink A point sink is defined as a pcjint in space at which the fluid is continuously and uniformly drawn off. The radial velocity into the sink at a distance r from the sink is, in spherical coordinates,... 

Flow Past a Point Sink A simple potential flow model for an unflanged or flanged exhaust hood in a uniform airflow can be obtained by combining the velocity fields of a point sink with a uniform flow. The resulting flow is an axially symmetric flow, where the resulting velocity components are obtained by adding the velocities of a point sink and a uniform flow. The stream function for this axisymmetric flow is, in spherical coordinates. 

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.

Yet, Eq. (14) does not describe the real situation. It must also be taken into account that gas concentration differs in the solution and inside the bubble and that, consequently, bubble growth is affected by the diffusion flow that changes the quantity of gas in the bubble. The value of a in Eq. (14) is not a constant, but a complex function of time, pressure and bubble surface area. To account for diffusion, it is necessary to translate Fick s diffusion law into spherical coordinates, assign, in an analytical way, the type of function — gradient of gas concentration near the bubble surface, and solve these equations together with Eq. (14). 

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 ... 

For convenience, let us define a new velocity variable, %, and represent this variable in spherical coordinates ... 

The estimation of the diffusional flux to a clean surface of a single spherical bubble moving with a constant velocity relative to a liquid medium requires the solution of the equation for convective diffusion for the component that dissolves in the continuous phase. For steady-state incompressible axisym-metric flow, the equation for convective diffusion in spherical coordinates is approximated by... 

For swarms of spherical bubbles, the field may be expected to be approximately spherically symmetric when the origin of coordinates is fixed on the center of mass of a typical particle. Therefore, by using spherical coordinates and the initial condition ... 

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

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 ... 

Inasmuch as this system is not used so often as Cartesian, cylindrical, or spherical coordinates, let us describe it in some detail. First of all, we find a condition when a family of non-intersecting surfaces can be a family of equipotential surfaces. Suppose that the equation of the surfaces is... 

Earlier we solved the boundary value problem for the spheroid of rotation and found the potential of the gravitational field outside the masses provided that the outer surface is an equipotential surface. Bearing in mind that, we study the distribution of the normal part of the field on the earth s surface, where the position of points is often characterized by spherical coordinates, it is natural also to represent the potential of this field in terms of Legendre s functions. This task can be accomplished in two ways. The first one is based on a solution of the boundary value problem and its expansion into a series of Legendre s functions. We will use the second approach and proceed from the known formula, (Chapter 1) which in fact originated from Legendre s functions... 

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. 

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