Coordinate systems Eulerian

In this chapter the general equations of laminar, non-Newtonian, non-isothermal, incompressible flow, commonly used to model polymer processing operations, are presented. Throughout this chapter, for the simplicity of presentation, vector notations are used and all of the equations are given in a fixed (stationary or Eulerian) coordinate system. 

The starting point for obtaining quantitative descriptions of flow phenomena is Newton s second law, which states that the vector sum of forces acting on a body equals the rate of change of momentum of the body. This force balance can be made in many different ways. It may be appHed over a body of finite size or over each infinitesimal portion of the body. It may be utilized in a coordinate system moving with the body (the so-called Lagrangian viewpoint) or in a fixed coordinate system (the Eulerian viewpoint). Described herein is derivation of the equations of motion from the Eulerian viewpoint using the Cartesian coordinate system. The equations in other coordinate systems are described in standard references (1,2). 

While the Eulerian system has intuitive appeal, it is the Lagrangian coordinate system that is more convenient mathematically and in many practical applications. In this system, the coordinate is fixed to the material and moves with it. It is sometimes called the material coordinate system. In Fig. 2.2, the boxcars can be numbered, so the position of a car in this system never changes. By convention, the Lagrangian coordinate (h) is chosen so that it is equal to the Eulerian coordinate (x) at some time t = 0. Figure 2.10(b) illustrates a Lagrangian h-t diagram of the same system as shown in Fig. 2.10(a) with the Eulerian system. Because the flow is independent of the coordinate system chosen to describe it, both systems must lead to the same results. 

The form of the jump conditions also depends on the coordinate system. Substituting (2.40) into the general Eulerian form of the momentum jump condition (Table 2.1) yields the Lagrangian jump condition... 

The properties required of a material in order for it to support a stable shock wave were listed and discussed. Rarefaction, or release waves were defined and their behavior was described. The useful tool of plotting shocks, rarefactions, and boundaries in the time-distance plane (the x-t diagram) was introduced. The Lagrangian coordinate system was defined and contrasted to the more familiar Eulerian coordinate system. The Lagrangian system was then used to derive conservation equations for continuous flow in one dimension. 

Eulerian coordinates The coordinate system in which the spatial position (x) and time (t) are the independent variables. The dependent variables... 

Equation (12.43) is called an Eulerian approach because the behavior of the species is described relative to a fixed coordinate system. The equation can also be considered to be a transport equation for particles when they are... 

In Chapter 4 (Sections 4.7 and 4.8) several examples were presented to illustrate the effects of non-coincident g- and -matrices on the ESR of transition metal complexes. Analysis of such spectra requires the introduction of a set of Eulerian angles, a, jS, and y, relating the orientations of the two coordinate systems. Here is presented a detailed description of how the spin Hamiltonian is modified, to second-order in perturbation theory, to incorporate these new parameters in a systematic way. Most of the calculations in this chapter were first executed by Janice DeGray.1 Some of the details, in the notation used here, have also been published in ref. 8. 

In Section 5.1, we noted that to a good approximation the nuclear motion of a polyatomic molecule can be separated into translational, vibrational, and rotational motions. If the molecule has N nuclei, then the nuclear wave function is a function of 3/V coordinates. The translational wave function depends on the three coordinates of the molecular center of mass in a space-fixed coordinate system. For a nonlinear molecule, the rotational wave function depends on the three Eulerian angles 9, principal axes a, b, and c with respect to a nonrotating set of axes with origin at the center of mass. For a linear molecule, the rotational quantum number K must be zero, and the wave function (5.68) is a function of 6 and only only two angles are needed to specify the orientation of a linear molecule. Thus the vibrational wave function will depend on 3N — 5 or 3N — 6 coordinates, according to whether the molecule is linear or nonlinear we say there are 3N — 5 or 3N — 6 vibrational degrees of freedom. 

Sunderland and Grosh (Sll) use an explicit numerical scheme for solving the Landau problem. An Eulerian coordinate system is used with the origin at the melt interface. As noted by Landau, the numerical integration is simplified by appropriate choice of the ratio of the space and time intervals. Extension to time-dependent heat flux by either a numerical or a graphical technique is indicated. 

Let us consider the electron-vibrational matrix element. As is usually done, we consider two coordinate systems, the origins of which are located at the center of mass of the molecule. The first coordinate system is fixed in space, while the second system (the rotational one) is fixed to the molecule. For describing the orientation of the rotational system with respect to the fixed frame we use the Eulerian angles 6 = a, / , y. In the Born-Oppenheimer approximation, the motion of nuclei may be decomposed into the vibrations of the nuclei about their equilibrium position and the rotation of the molecule as a whole. Accordingly, the wave function of the nuclei X (R) is presented as a product of the vibrational wave function A V(Q) and the rotational wave function... 

Recent efforts to distinguish between the terms burning velocity and flame speed on the basis of Eulerian and Lagrangian coordinate systems appear to introduce confusion. Therefore, the terms are used interchangeably here, as synonyms for such terms as deflagration velocity, wave speed, and propagation velocity. They all refer to velocities measured with respect to the gas ahead of the wave. 

Ansumali et al.196 propose that the grand potential, written in the Eulerian coordinate system as... 

Other regional transport models, such as the Regional Lagrangian Model of Air Pollution (RELMAP Eder et al., 1986), use a different computational scheme than Eulerian models. In a Lagrangian model, the coordinate system moves with a parcel of air and mass balance of pollutant concentrations is computed on a parcel as it moves through space. 

Eulerian coordinate system A coordinate system that is fixed in space. 

We have shown that all the balance laws for Eulerian CVs can be cast in the same standard form, applicable in any coordinate system. The coordinate system is chosen to proceed with the solution for the problem in question in a convenient way. 

In the above, g are the Eulerian rotation factors necessary to rotate the local coordinate system used to evaluate the overlap back to the molecular coordinate system, and /are factors that Jaffe and Del Bene set to ... 

This Lagrangian should be thought of as dependent on 3Ne + 6 generalized coordinates, qt, and velocities, g, respectively. These are the 3Ng coordinates be, Ce which describe the relative positions of the Ne electrons with respect to the nuclear frame three coordinates Xo, Vo and Zo which describe the position of the molecular center of mass as referred to the laboratory coordinate system, and three Eulerian angles 6, and x which describe the instantaneous orientation of the molecular coordinate system with respect to the space fixed X-, Y- and Z-axes. There are numerous ways of specifying Eulerian angles. Because of later reference we will follow the choice used by Wilson et where and 6 are the ordinary polar coordinates of the molecular c-axis O d n 0 < < 2n) and x is the angle between the nodal line N and the positive b axis as is illustrated in Fig. IV.2. x is positive for clockwise rotation about the c axis. 

Fig. IV.2. Eulerian angles used to describe the instantaneous orientation of the nuclear frame with respect to the space fixed coordinate system...

Using this choice, the direction cosines between the basis vectors of the two coordinate systems and the instantaneous angular velocities about the molecular axes — Uo, oib, o)c — are related to the Eulerian angles and their time derivatives (f>, 6 and X through... 

Similarly the quadrupole interaction coordinate system can be related to the new one by an Eulerian angle transformation which will enable 1, 1, ly to be expressed in terms of S, Sy. In total, the angles d and Eulerian angles a, fi, and y are required to define the geometry uniquely. 

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