Rectangular coordinate system

In the relative velocity system of Fig. 1-4, a rectangular coordinate system is set up as follows the i3 axis is the line antiparallel to g, through the origin the ix axis is the line in the reference plane from which e is measured the i2 axis is in the reference plane, perpendicular... 

So far in this chapter, consideration has been given to transfer taking place in a single direction of a rectangular coordinate system. In many applications of mass transfer, one of the fluids is injected as approximately spherical droplets into a second immiscible fluid, and transfer of the solute occurs as the droplet passes through the continuous medium. 

To use Fourier s law of heat conduction, a thermal balance must first be constructed. The energy balance is performed over a thin element of the material, x to x + Ax in a rectangular coordinate system. The energy balance is shown in equation 13 ... 

There is an important relationship between vectors and skew-symmetric tensors. Suppose A and B are two vectors in a three-dimensional rectangular coordinate system whose components are connected by... 

In a rectangular coordinate system, the rotation matrix n given by... 

Once a radial coordinate has been invariantly calibrated, it is a matter of routine to define a rectangular coordinate system based on this radial calibration this is taken as done for the remainder of this chapter. 

We note that the coordinate X directly reflects the distance between atoms one and two, whereas the coordinate X2 reflects a combination of both distances. Therefore, a knowledge of the two coordinates does not directly tell us what the distances are between the involved atoms. Also, for the potential energy function in a collinear collision, the natural variables will be the distances between atoms A and B and atoms B and C. These variables appear as the components along a new set of coordinate axes, if instead of a rectangular coordinate system we use a mass-weighted skewed angle coordinate system. 

Let Xi and Ab be the coordinates in a rectangular Cartesian coordinate system with X1 along the ordinate axis and X2 along the abscissa axis. One of the axes in the new coordinate system is now chosen to be collinear with the abscissa axis in the rectangular coordinate system, and we let the coordinate along this axis be the first term in the expression for X2 in Eq. (D.22), namely 2(ay — ay). The situation is sketched in Fig. D.1.1. The other axis with a coordinate proportional to the other distance ay — a 1 forms an angle 4> with the first. This angle is determined from the requirements that the projections of this coordinate on the X coordinate axis is a (a 2 — ay) and on the X2 coordinate axis is aym ay — ay )/.sy. If we let the proportionality constant of X2 — ay be / , then we have... 

Now Eq. 2.5-2 is a vectorial equation that has three components, reflecting the fact that linear momentum is independently conserved in the three spatial directions. For a rectangular coordinate system, Eq. 2.5-2 becomes ... 

In this example, we consider the viscous, isothermal, incompressible flow of a Newtonian fluid between two infinite parallel plates in relative motion, as shown in Fig. E2.5a. As is evident from the figure, we have already chosen the most appropriate coordinate system for the problem at hand, namely, the rectangular coordinate system with spatial variables x, y, z. 

The concentric gap is c = r2 — r, and clearly a < c. The gap is very small, and locally we can assume flow between parallel plates. Thus we define a rectangular coordinate system X, YfZ located on the surface of the journal such that X is tangential to the journal, as indicated in Fig. E2.9a. The gap between the journal and bearing is denoted as B ) and is well approximated as a function of angle 6 by the following expression ... 

The flow is viscometric because there is only one velocity component, vz(y), which is changing only in one spatial direction, y. Adopting a rectangular coordinate system, we find in analogy to Example 3.3 that vy — vx — 0, and therefore the equation of motion reduces to... 

Fig. 6.22 The nip region of the two-roll geometry, with radii R. A rectangular coordinate system is placed at the midplane in the gap between the rolls connecting the two roll centers.

The velocity distribution for a fully developed, isothermal drag flow between parallel plates separated by a distance of H and with the upper plate moving at constant velocity, Vq, in a rectangular coordinate system located at the stationary plate, is vx = yVo/H, and the flow rate is q = VqH/2. The fraction of exiting flow rate between... 

For one-dimensional dispersion in soils, lire describing equation for a conservative species and/or pollutant, c, is a cartesian (rectangular) coordinate system moving with velocity Vx is... 

The graph of a linear equation, in a rectangular coordinate system, is a straight line, hence the term linear. The graph of simultaneous linear equations is a set of lines, one corresponding to each equation. The solution to a simultaneous system of equations, if it exists, is the set of numbers that correspond to the location in space where all the lines intersect in a single point. 

A molecule is the smallest fundamental group of atoms of a chemical compoimd that can take part in a chemiccd reaction. The atoms of the molecule are organized in a 3D structure the molecular matrix M is a rectangular matrix Ax3 whose rows represent the molecule atoms and the columns the atom Cartesian coordinates x, y, z) with respect to any rectangular coordinate system with axes X, Y, Z. The cartesian coordinates of a molecule usually correspond to some optimized molecular geometry obtained by the methods of -> computational chemistry. The molecular geometry can also be obtained from crystallographic coordinates or from 2D-3D automatic converters. 

Figure A2.1 (a) Schematic representation of a coiled polymer molecule showing the end-to-end distance, (b) Diagram showing a coiled polymer molecule of end-to-end distance r in a rectangular coordinate system with one chain end fixed at the origin.

To calculate the tangential stress pT = p, it is convenient to define a rectangular coordinate system (x, y, z) with the (xt y) plane parallel to the transition layer and the 2-axis normal to the surface and directed from liquid to vapor. The tangential stress is the sum of the momentum transport and the force transmitted across a strip of unit width, perpendicular to the transition layers and extending from 2 — — 1/2 in the interior of the liquid to 2 = 1/2 in the gas phase. Kirkwood and Buff expressed this stress in terms of the intermolecular force and the pair distribution function. Thus they obtained... 

To describe the two-dimensional problem, we use the rectangular coordinate system X, Y, where the X-axis is directed oppositely to the temperature gradient on the lower surface and the Y-axis is directed vertically upward. The origin is chosen to be in the middle of the layer therefore, -htemperature fields are described by the equations [142, 143]... 

The analysis of this model is similar to that of the well-known random-walk model, which was first developed to describe the random movement of molecules in an ideal gas. The only difference now is that for the freely jointed chain, each step is of equal length 1. To analyze the model one end of the chain may be fixed at the origin O of a three-dimensional rectangular coordinate system, as shown in Fig. A2.1(b), and the probability, P(x,y,z), of finding the other end within a small volume element dx.dy.dz at a particular point with coordinates x,y,z) may be calculated. Such calculation leads to an equation of the form (Young and Lovell, 1990) ... 

Let the displacement of each nucleus be expressed in terms of rectangular coordinate systems with the origin of each system at the equilibrium position of each nucleus. Then the kinetic energy of an A-atom molecule would be expressed... 

The control volume. When deriving the conservation equations it is necessary to select a control volume. The derivation can be performed for a volume element of any shape in a given coordinate system, although the most convenient shape is usually assumed for simplicity (e.g., a rectangular shape in a rectangular coordinate system). For illustration purposes, different coordinate systems are shown in Fig. 1.11. In selecting a control volume we... 

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