Cylindrical polar systems

This expression is very similar to (B 1.9.49). If the scattering system is anisotropic, equation (Bl.9.54) can then be expressed in cylindrical polar coordmates (see figure Bl.9.5 ... 

It is convenient to employ two sets of coordinate systems. Spherical polar coordinates r, Q, A) are defined with the origin at the vertex of the cone the axis is 0=0, the surface of the conical portion of the cyclone is the cone 0 = 0% and the azimuthal coordinate is A. Using the same origin, cylindrical polar coordinates (R, A, Z) are defined, where R = r sin 0 and the Z-axis coincides with the axis 0=0. 

Cartesian and cylindrical polar atomic coordinates of the structural repeating unit of 31 polysaccharide helices are provided in Tables A1 to A31. Errors, if any, in the original publications have been corrected. The coordinates of hydrogen atoms are given in a majority of structures. If missing, they are not available in the references cited in Table I. Each table caption contains the structure number and polymer name assigned in Table I. Refer to Table II for its chemical repeating unit. Cartesian (x, y, z) and cylindrical (r, , z) coordinates are related by x r cost ), y = r sin<(> and z is the same in both systems. 

Fig. 4. Schematic diagram of the layered model for a pore (47). The two nuclear spins diffuse in an infinite layer of finite thickness d between two flat surfaces. The M axes are fixed in the layer system. The L axes are fixed in the laboratory frame, with Bq oriented at the angle P from the normal axis n. The cylindrical polar relative coordinates p, (p, and z are based on the M axis. The smallest value of p corresponding to the distance of minimal approach between the two spin bearing molecules is 5.

Figure 8.2 Cylindrical polar coordinate system. The z axis lies along the axis of the infinite cylinder.

Equation (11) is written in the form of Newton s second law and states that the mass times acceleration of a fluid particle is equal to the sum of the forces causing that acceleration. In flow problems that are accelerationless (Dx/Dt = 0) it is sometimes possible to solve Eq. (11) for the stress distribution independently of any knowledge of the velocity field in the system. One special case where this useful feature of these equations occurs is the case of rectilinear pipe flow. In this special case the solution of complex fluid flow problems is greatly simplified because the stress distribution can be discovered before the constitutive relation must be introduced. This means that only a first-order differential equation must be solved rather than a second-order (and often nonlinear) one. The following are the components of Eq. (11) in rectangular Cartesian, cylindrical polar, and spherical polar coordinates ... 

Coordinate Systems The commonly used coordinate systems are three in number. Others may be used in specific problems (see Ref. 212). The rectangular (cartesian) system (Fig. 3-25) consists of mutually orthogonal axes x, y, z. A triple of numbers x, y, z) is used to represent each point. The cylindrical coordinate system (r, 0, z Fig. 3-26) is frequendy used to locate a point in space. These are essen-daUy the polar coordinates (r, 0) coupled with the z coordinate. As... 

This is a two-dimensional problem handled in cylindrical polar coordinates in which the longitudinal z axis is a symmetry axis for the system. The relevant coordinates are (r, 0) and the gradient is applied along the polar axis direction (i.e., across a diameter). The relaxing boundary is at a radial distance r = a from the cylinder center ... 

There are three possibilities corresponding to the dimension of the distribution. The first is a ID concentration distribution (d = 1), in which the diffusing species spreads evenly in the z directions from an initial line pulse at z = 0 on the xz plane. In this case, the variable r in (6-37) is the Cartesian variable z. The second case is a circularly symmetric distribution for c (d = 2), which evolves by diffusion on a plane from an initial compact planar pulse. In this case, r in (6 37) is the radial component of a polar (or cylindrical) coordinate system that lies in the diffusion plane. The third case is a spherically symmetric distribution corresponding to d = 3, which evolves at long times from a compact 3D pulse that diffuses outward into the frill 3D space. In this case, r is the radial variable of a spherical coordinate system. To obtain the long-time form of the distribution we must solve (6-37), but subject to the integral constraint that the total amount of the diffusing species is constant, independent of time ... 

Starting with (7-51) and (7-52), we can also obtain a general solution for xj/ in terms of a circular cylindrical coordinate system, and this solution is more immediately applicable to real problems. The governing equations in polar cylindrical coordinates are... 

Problem 17-1. The equation for the free particle is separable in many coordinate systems. Using cylindrical polar coordinates, set up and separate the wave equation, obtain the solutions in

recursion formula for the coefficients in the series solution of the p equation. Hint In applying the polynomial method, omit the step of finding the asymptotic solution. 

The cylindrical polar coordinate system is another three-dimensional coordinate system. It uses the variables p, 4>, and z, already defined and shown in Fig. 2.11. The equations needed to transform from Cartesian coordinates to cylindrical polar coordinates are Eqs. (2.39) and (2.40). The third coordinate, z, is the same in both Cartesian and cylindrical polar coordinates. Equations (2.37) and (2.38) are used for the reverse transformation. 

Coordinate systems such as spherical polar or cylindrical polar coordinates are called orthogonal coordinates, because an infinitesimal displacement produced by changing only one of the coordinates is perpendicular (orthogonal) to a displacement produced by an infinitesimal change in any one of the other coordinates. 

Figure B.4 Diagram (a) defining r, , and in the spherical polar coordinate system and (b) defining / , 0, and Z in the circular cylindrical coordinate system.

We shall next consider the rigorous theory of Poiseuille flow, i.e., the steady laminar flow, caused by a pressure gradient, of an incompressible fluid through a tube of circular cross-section.< > We shall suppose that the tube is of infinite length so that end effects can be ignored. Let us choose a cylindrical polar coordinate system r(pz with z along the axis of the tube. In the steady state the only component of the velocity gradient is = dv/dr. It is natural to expect that the director is everywhere in the rz plane... 

Here, the focal length d defines a family of coordinate systems that vary from spherical polar when d = 0 to cylindrical polar in the limit when d oo. A surface of constant transmural coordinate A (Figure 54.1) is an ellipse of revolution with major radius a = d cosh A and minor radius b = d sinh A. In an ellipsoidal model with a truncation factor of 0.5, the longitudinal coordinate fi varies from zero at the apex to 120° at the base. Integrating the Jacobian in prolate spheroidal coordinates gives the volume of the wall or cavity ... 

Fig. 9.1. The (r, z, ) cylindrical polar coordinate system used to model a microdisc electrode. The disc radius is re.

The unit cell and coordinates are illustrated in Figure 10.14(b). As with the array of microdiscs model, the unit cell is cylindrically symmetrical about an axis that passes through the centre of the pore, perpendicular to the electrode surface. The problem may thus be reduced from a three-dimensional one to a two-dimensional one. As with the microdisc electrode, this is a two-dimensional cylindrical polar coordinate system, and Pick s second law in this space is given by Eq. (9.6). The simulation space for the unit cell with its attendant boundary conditions is shown in Figure 10.15. 

In the following chapters we use a very familiar form of representation of stresses by a double-subscript notation in which the first subscript represents the direction of the normal of the area across which the stress is acting and the second is the direction of action of the stress. In the Cartesian rectangular system of coordinates, the subscripts i and j stand for x, y, z or xi,X2,X. In the cylindrical, polar coordinate system, the subscripts stand for r, 6, z. 

The transport equations describing the instantaneous behavior of turbulent liquid flow are three Navier-Stokes equations (transport of momentum corresponding to the three spatial coordinates r, z, in a cylindrical polar coordinate system) and a continuity equation. The instantaneous velocity components and the pressure can be replaced by the sum of a time-averaged mean component and a root-mean-square fluctuation component according to Reynolds. The resulting Reynolds equations and the continuity equation are summarized below ... 

A variety of approximations provides insight into the qualitative and quantitative behavior. The Derjaguin approximation, for example, is applicable for separations small compared with the radius of the spheres (Derjaguin, 1934). Under such conditions, elements on each sphere interact as parallel plane elements at the same separation the total interaction is a sum over the infinitesimal elements. To proceed formally, we adopt a polar cylindrical coordinate system with its axis joining the centers of the spheres of radius a, separated by the distance h and centered at midpoint. r is the interparticular distance on the midplane and z the distance on the center to center axis. A sphere surface is defined by ... 

The discussion here follows directly from that in Section 2.2. A film of thickness h is bonded to a substrate of thickness hs, with no restrictions on the thickness ratio. The stress and deformation fields are referred to a cylindrical coordinate system with polar coordinates in the plane of the system and with the z—direction normal to the interface the origin of coordinates lies in the substrate midplane. The equi-biaxial stress components are referred to polar coordinates, but they could equally well be expressed in rectangular coordinates. As long as the response is in the range of geometrically linear behavior, the common shape of the film and substrate in plan view is immaterial. As in Section 2.2, the biaxial elastic moduli of the film and substrate are Mf and Mg, respectively, and the corresponding coefficients of linear thermal expansion are af and dg, respectively. 

Cylindrically symmetric shells are considered which are of the form r=r (z) in a system of cylindrical polar coordinates r, 6, z). Inextensible fibers are wound on and bonded to this shell in such a way that at any point on it equal numbers of fibers are inclined at angles a and n-a. to the line of latitude (z=constant) passing through that point. At a point (r, d, z) of the shell there are n, n 2. .. Up fibers, per unit length measured perpendicular to the length of the fibers, with positive inclinations and an equal number of fibers with... 

A 10.5 Choice of Coordinate System for Hj Cylindrical Polar Coordinates... 

Figure A10.4 (a) The coordinate system and definition of an infinitesimal volume c/r for cylindrical polar coordinates, the factor u enters dx because of the definition of an arc length using angles measured in radians, (b) A slice through the H—H bond showing the charge density on a plane perpendicular to the H2 Cco axis, (c) The geometry used to obtain the distances r and /-j from each nuclear centre these are the varieJales used in the s-orbital basis functions, since they define the nuclear-electron separations.

Alternative coordinate systems. We have seen how coordinate systems can be cleverly exploited to advantage. For example, consider the elementary log r and 0 solutions for point sources and vortexes obtained in cylindrical polar coordinates. In Chapters 2 and 3, they were rewritten in (x,y) coordinates in order to develop solutions for line fractures and shales. Or consider the conformal mappings introduced in Chapter 5 there, the simple solutions in Chapters 2 and 3 were extended to flows in complicated geometries. A newer, more powerful approach involves the use of boundary-conforming grid sytems that wrap around wells and fractures in the nearfield and at the same time conform to the external boundaries of the farfield. The simplest example is provided by cylindrical coordinates, used to model circular wells concentrically located in circular reservoirs. Another is furnished by elliptical coordinates, used to model flows into straight, finite-length fractures in infinite systems. 

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