Cartesian coordinates Cross-section

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 ... 

Suppose that a body is strongly elongated in some direction and with sufficient accuracy it can be treated as the two-dimensional. In other words, an increase of the dimension of the body in this direction does not practically change the field at the observation points. We will consider a two-dimensional body with an arbitrary cross section and introduce a Cartesian system of coordinates x, y, and z, as is shown in Fig. 4.5a, so that the body is elongated along the y-axis. It is clear that if at any plane y — constant the behavior of the field is the same. To carry out calculations we will preliminarily perform two procedures, namely,... 

Since the density is invariant to the velocity coordinates (Cartesian or polar) used, one has, expressing the density in terms of differential cross-section,... 

To sum up, the basic idea of the Doppler-selected TOF technique is to cast the differential cross-section S ajdv3 in a Cartesian coordinate, and to combine three dispersion techniques with each independently applied along one of the three Cartesian axes. As both the Doppler-shift (vz) and ion velocity (vy) measurements are essentially in the center-of-mass frame, and the (i j-componcnl, associated with the center-of-mass velocity vector can be made small and be largely compensated for by a slight shift in the location of the slit, the measured quantity in the Doppler-selected TOF approach represents directly the center-of-mass differential cross-section in terms of per velocity volume element in a Cartesian coordinate, d3a/dvxdvydvz. As such, the transformation of the raw data to the desired doubly differential cross-section becomes exceedingly simple and direct, Eq. (11). 

The theory of a straight and uniform MNF with a circular cross-section of radius a and a step-index profile is well developed59-61. A section of an MNF is shown in Fig. 13.3 in Cartesian coordinates (pc,y,z). The distribution of refractive index along the radius r = Jx2 + y2 of such MNF is... 

Figure 7.2 Transforming the Cartesian reference frame a) cylindrical cross section of the screw and barrel with flow out of the surface of the page, and b) the unwound rectangular channel with a stationary barrel and the Cartesian coordinate frame positioned on the screw, is the velocity of the screw core in the z direction and it is negative...

FIGURE 9.2 Sketch of far-field detection of CARS waves generated in the focal volume. The figure shows the yz cross-section of the excitation volume. O is the origin of the coordinate system and also the center of the focal volume and O is a far-field point on the optical axis. The coordinates of the points in the near-field are represented by (x,y,z) and those of the points in the far-field by (X,Y,Z) in cartesian coordinate system. The outgoing arrows around the near-field points represent the CARS waves generated in the focal volume. The incoming arrows at the far-field points represent the contributions from the individual points in the focal volume. 

Note that the dimensionality of the flow also depends on the choice of coordinate system and its orientation. The pipe flow discussed, for example, is one-dimensional in cylindrical coordinates, but two-dimensional in Cartesian coordinates illustrating the importance of choosing the most appropriate coordinate system. Also note that even in this simple flow, the velocity cannot be uniform across the cross section of the pipe because of the no-slip condition. However, at a well-rounded entrance to the pipe, the velocity profile may be approximated as being nearly uniform across the pipe, since the velocity is nearly constant at all radii except very close to the pipe wall. 

Semiclassical perturbation theory [56, 59] is applicable to describe TPA as a second-order phenomenon. Light can be seen as an electromagnetic wave perturbing the stationary wavefunctions of a molecule. Quantities related to these wavefunctions are the absorption cross sections rxgi and <7if (Fig. 3.1). Furthermore, high excitation power/photon density requires additional higher order terms according to perturbation theory. The spatial (Cartesian coordinates, r) and time (f) dependent wavefunction i//g2) (r, t) is shown for second-order perturbation in Eq. (2) [23] ... 

Here the space variables r and s in the Cartesian coordinate frame of the projection have been replaced by the Cartesian coordinates x and y in the laboratory frame (cf. Fig. 5.4.1), and

rotation angle between both frames. This equation is another formulation of the projection cross-section theorem (cf. eqn (5.4.12)), which states that the Fourier transform p(k, cp) of a projection P(r, (p) is defined on a line p kcos(p, k sin p) at an angle

[Pg.203]

Closely related to the pressure-driven unidirectional flow between two parallel plane surfaces is the pressure-driven motion in a straight tube of circular cross section. This is the famous problem studied experimentally as a model for blood flow in the arteries by Poiseuille in 1840.6 Although this problem could be solved by use of Cartesian coordinates, withz being the axial direction, it is always much simpler to use a coordinate system in which the boundaries of the flow domain are coincident with a line or surface of the coordinate... 

First, however, it is important to recognize that the form of equations (10 28), (10-30), and (10 32) is independent of the geometry of the body (i.e., independent of the cross-sectional shape for any 2D body). Although we started our analysis with the specific problem of flow past a circular cylinder, and thus with the equations of motion in cylindrical coordinates, the equations for the leading-order approximation in the inner (boundary-layer) region reduce to a local, Cartesian form with Y being normal to the body surface and x... 

Let us consider a laminar steady-state fluid flow in a rectilinear tube of constant cross-section. The fluid streamlines in such systems are strictly parallel (we neglect the influence of the tube endpoints on the flow). We shall use the Cartesian coordinates X, Y, Z with Z-axis directed along the flow. Let us take into account the fact that the transverse velocity components of the fluid are zero and the longitudinal component depends only on the transverse coordinates. In this case, the continuity equation (1.1.1) and the first two Navier-Stokes equations in (1.1.2) are satisfied automatically, and it follows from the third equation in (1.1.2) that... 

The preceding presentation describes how the collision impact parameter and the relative translational energy are sampled to calculate reaction cross sections and rate constants. In the following, Monte Carlo sampling of the reactant s Cartesian coordinates and momenta is described for atom + diatom collisions and polyatomic + polyatomic collisions. Initial energies are chosen for the reactants, which corresponds to quantum mechanical vibrational-rotational energy levels. This is the quasi-classical model [2-4]. 

Choosing initial Cartesian coordinates for the polyatomic reactants follows the procedures outlined above for an atom + diatom collision and for normal-mode sampling. If the cross section is calculated as a function of the rotational quantum numbers J and K, the components of the angular momentum are found from... 

Consider a fluid material body enclosing an infinitesimal volume element dV at the point p, as shown in Fig. 5.1. Choose any plane cutting through the volume element and let the cross section be denoted by dS. The direction perpendicular to the plane is regarded as the direction of the plane, indicated by the unit vector n. If there are forces applied to the body, a surface force f will be exerted on the plane of the volume element at the point p. In general, the directions of f and n are different. By dividing f by dS, we obtain the stress t (= fn/dS) exerted on the plane at the point p. The stress can be separated into the component perpendicular to the plane (the normal stress) and those parallel to the plane (the shear stresses). At the point p, we choose a Cartesian coordinate system with n as one direction and m and 1 in the plane as the other two. Then t may be expressed as... 

We fix a local Cartesian coordinate system (e, e, e ) to every contour point of an elastic wire of finite thickness. In so doing, the C axis is taken in the direction of the unit tangent vector u of the chain centroid (or contour), and the and axes in the directions of the principal axes of inertia for the cross-section of the wire at the same point. When the wire is deformed (bent and twisted), the local coordinate system at contour point 5 4- is related to that at contour point s by an infinitesimal rotation dQ. The deformed state of the wire is then represented by a vector 0 (0, 0, 0 ) as a function of s, where 0 is defined by dil/ds. 

Cartesian coordinates. Furthermore, the following analyses are restricted to microchannels of a rectangular cross section, 2w x 2 h (width by height), which is close to the real shape of microchannels made using the microfabrication technologies. With these considerations, the model, based on the slip velocity approach, describing the velocity field of such flow can be further simplified to... 

Pressure-Driven Single-Phase Liquid Flows, Fig. 1 Schematic of the elliptical cross section and of the Cartesian coordinate system... 

For a rectangular channel with an aspect ratio P = hla (see Fig. 3a), for which a Cartesian system of coordinates x, y, z is assumed, with its origin in the left bottom comer of the inlet rectangular cross section (y along the side with length h and z perpendicular to the cross section), the analytical solution of Eqs. 11 and 12 gives the following velocity distribution ... 

Fig. 3 Schematic of the rectangular cross section with the Cartesian coordinate system used in (a) Eq. 25 and (b) Eq. 28...

Both the Doppler slice and the ion TOP measurement are essentially in the centre-of-mass system. Therefore the measurement directly maps out the desired 3D centre-of-mass distribution, i.e. d aj v dv dO = I 6,v) v in Cartesian velocity coordinates (d (r/diVx dvy dv ). Thus, the double differential cross-section I 6,v) is obtained by multiplying the measured density distribution in the centre-of-mass velocity space by and then transforming from the Cartesian to the polar coordinate system. This procedure has to be contrasted against the conventional neutral TOP technique (either in the universal machine or by the Rydberg-tagging method), for which the laboratory to centre-of-mass transformation must be performed, or against the 2D ionimaging technique, which involves 2D to 3D back transformation. 

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