Channel coordinate systems

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 ... 

Let S o be a surface located at mid-channel between two smooth surfaces separated by a narrow gap. The curvilinear coordinate system, corresponding to this... 

As mentioned in the introduction to this chapter this is a necessary condition when approximating the cylindrical screw in the Cartesian coordinate system. The screw rotation theory, New Theory line, predicts that the rate should constantly increase as the channel gets deeper. When a fixed positive pressure occurs for the screw rotation model, the New Theory with Pressure line, the predictions fits the data very well for all H/Ws. Thus for modern screw designs with deeper channels, reduced energy dissipation, and lower discharge temperatures, the screw rotation model would be expected to provide a good first estimation of the performance of the extruder regardless of the channel depth for Newtonian polymers. 

A coordinate system that is natural for the conical channel can be established as illustrated in right-hand panel of Fig. 5.20. The origin of the new coordinate system begins on the tube wall at the entrance of the conical section. The x coordinate aligns with the surface of the tube wall and the y coordinate measures the distance across the channel and is normal to the tube wall. The 4> coordinate measures the circumferential angle around the conical... 

Figure 5.24 illustrates an elbow section in a cylindrical channel where the radius of curvature of the section R is comparable to the channel radius r,-. Analysis of the flow field in this section may be facilitated by the development of a specialized orthogonal curvilinear coordinate system, (r, 6, a). The unit vectors are illustrated in the figure. Referenced to the cartesian system, the angle 6 is measured from the x axis in the x-y plane. The angle a is measured from and is normal to the x-y plane. The distance r is measured radially outward from the center of the toroidal channel. 

Fig. 1. Coordinate systems for common electrode geometries, (a) Cylindrical symmetry (i) ring—disc electrodes, (ii) tubular electrodes (b) Cartesian symmetry channel electrodes (c) spherical symmetry dropping mercury electrode.

Fig. 5 a. The development of the concentration profile due to a plug of protein solution entering a buffer-primed, thin plate flow channel. Note that a bullet-shaped concentration profile develops with time (assuming no diffusion) b. the geometry and coordinate system used in the convection-diffusion treatment... 

Consider the flame in a plane channel, and choose the cartesian coordinate system (x,y) moving with flame at a constant velocity equal to... 

Fig. 13. Flow field before the flame front in the laboratory coordinate system. (Region 1—potential flow, region 2—vortex flow, region 3—stagnation zone, region 4—channel wall).

Suppose that we have two different illuminants. Each illuminant defines a local coordinate system inside the three- dimensional space of receptors as shown in Figure 3.23. A diagonal transform, i.e. a simple scaling of each color channel, is not sufficient to align the coordinate systems defined by the two illuminants. A simple scaling of the color channels can only be used if the response functions of the sensor are sufficiently narrow band, i.e. they can be approximated by a delta function. 

Figure 3.23 Two different illuminants define two coordinate systems within the space of receptors. A simple diagonal transform, i.e. scaling of the color channels, is not sufficient to align the two coordinate systems.

The output from the red-green and the blue-yellow color opponent cells defines a two-dimensional coordinate system. A cell that responds maximally to red light can be constructed using the outputs from the red-green as well as the blue-yellow channel. Let x be the output of the red-green channel and let y be the output of the blue-yellow channel. A cell, which computes its output z according to... 

This coupling potential is smooth everywhere, which allows numerical calculations with high precision. There is no nonadiabatic coupling since the basis functions [0< )( 2C) are independent of p in each sector. The solution I Wf/o, 2C) is connected smoothly, in principle, from sector to sector by a unitary frame transformation from the /th set of channels to the (/ + l)st set [97-99]. The coordinate system is transformed from the hyperspherical to the Jacobi coordinates at some large p, beyond which the conventional close-coupling equations are employed for determining the asymptotic form of the wavefunction appropriate for the scattering boundary condition [100]. 

The results of HSCC calculations have proved much more rapid convergence with the number of coupled channels than the conventional close-coupling equations in terms of the independent-particle coordinates or the Jacobi coordinates based on them. This is considered to be because of the particle-particle correlations considerably taken into account already in the choice of the hyperspherical coordinate system. The results suggest an approximate adiabaticity with respect to the hyperradius p, even when the mass ratios might appear to violate the conditions for the adiabaticity, for example, for Ps- with three equal masses. Then, it makes sense to study an adiabatic approximation with p adopted as the adiabatic parameter. 

If we consider the potential energy as a function of the Jacobi coordinates X and X2 and draw the energy contours in the X1-X2 plane, then the entrance and exit valleys will asymptotically be at an angle to one another and in the mass-weighted skewed angle coordinate system parallel to its axes. So the idea with this coordinate system is that it allows us to directly determine the atomic distances as they develop in time and that it shows us the asymptotic directions of the entrance and exit channels. 

In the present analysis it will be assumed that both walls of the channel are heated to the same uniform temperature and that the flow is therefore symmetrical about the channel center line. The coordinate system shown in Fig. 8.16 will therefore be used in the analysis and, because of the assumed symmetry, the solution will only be obtained for y values between 0 and W/2, IV being, as indicated in Fig. 8.16, the full width of the channel. 

Exploiting a four-dimensional rotation group analysis, the transformation between harmonic expansions in the two coordinates systems was given explicitly [32], as well as the most general representation in terms of Jacobi functions [2], In practice, however, the two representations are in one form or another those being used in all applications and specifically in recent treatments of the elementary chemical reactions as a three-body problem [11,33-36]. For example, Eqs. (29)-(31) and Eqs. (47)-(49) permitted to establish [37] the explicit connection between coordinates for entrance and exit channels to be used in sudden approximation treatments of chemical reactions [38],... 

Consider then an adiabatic well in the hyperspherical coordinate system. Classically, the motion of the periodic orbit at the well would be an oscillation from a point on the inner equipotential curve in the reactant channel to a point on the same equipotential curve in the product channel. This is qualitatively the motion of what are termed "resonant periodic orbits" (RPO s). For example the RPO s of the IHI system are given in Fig. 5. Thus, finding adiabatic wells in the radial coordinate system corresponds to finding RPO s and quantizing their action. Note that in Fig. 5 we have also plotted all the periodic orbit dividing surfaces (PODS) of the system, except for the symmetric stretch. By definition, a PODS is a periodic orbit that starts and ends on different equi-potentials. Thus the symmetric stretch PODS would be an adiabatic well for an adiabatic surface in reaction path coordinates. However, the PODS in the entrance and exit channels shown in Fig. 5 may be considered as adiabatic barrieres in either the radial or reaction path coordinate systems. Here, the barrier in radial coordinates, has quantally a tunneling path between the entrance and exit channels. 

Let us briefly discuss the idea of and the results obtained from coupled-channel calculations with momentum eigenfunctions as it has been employed recently (Tenzer et al. 2000b). The collision is considered in a coordinate system, where the scattering ions have equal but opposite velocities v = voez. With respect to this system the total time-dependent Hamiltonian H is decomposed into the unperturbed (free) Dirac Hamiltonian... 

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