Orthogonal-curvilinear coordinate

It is not unusual to encounter a problem that is not conveniently posed in one of the common coordinate systems (i.e., cartesian, cylindrical, or spherical). As an illustration consider the flow behavior for the system shown in Fig. 5.20. The analysis seeks to understand the details of the flow field and pressure drop in the narrow conical gap between the movable flow obstruction and the conical tube wall. Intuitively one can anticipate that the flow may have a relatively simple behavior, with the flow parallel to the gap. However, such simplicity can only be realized when the flow is described in a coordinate system that aligns with the gap. An orthogonal curvilinear coordinate system can be developed to model this problem. 

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. 

The differential forms of the conservation equations derived in the appendixes for reacting mixtures of ideal gases are summarized in Section 1.1. From the macroscopic viewpoint (Appendix C), the governing equations (excluding the equation of state and the caloric equation of state) are not restricted to ideal gases. Most of the topics considered in this book involve the solutions of these equations for special flows. The forms that the equations assume for (steady-state and unsteady) one-dimensional flows in orthogonal, curvilinear coordinate systems are derived in Section 1.2, where specializations accurate for a number of combustion problems are developed. Simplified forms of the conservation equations applicable to steady-state problems in three dimensions are discussed in Section 1.3. The specialized equations given in this chapter describe the flow for most of the combustion processes that have been analyzed satisfactorily. 

The conservation equations in orthogonal, curvilinear coordinate-systems may easily be derived from equations (l)-(4). Since the complete form is complicated and will not be required in subsequent problems, the... 

By utilizing the expression for the divergence of a diagonal tensor in orthogonal, curvilinear coordinate systems, one can show that equation (2) reduces to... 

Figure 18. A curvilinear reaction coordinate on a potential surface. Chemical reaction may be viewed as a steady flow that would exhange energy with the local coordinates orthogonal to the reaction coordinate.

The transport equations can be written in many different forms, depending on the coordinate system used. Generally, we may select the orthogonal curvilinear Cartesian-, cylindrical-, and spherical coordinate systems, or the non-orthogonal curvilinear coordinate systems, which may be fixed or moving. In reactor engineering we frequently apply the simple curvilinear... 

Considering a generalized orthogonal coordinate system, the orthogonal curvilinear coordinates are defined as qa- In this O-system the base vectors Gq, are defined as unit vectors along the coordinates. The position of the point P is given by the coordinates, or by the position vector r = r qa,t). 

C.2.4 Orthogonal Curvilinear Coordinate Systems and Differential Operators... 

In this section the relevant differential operators are defined for generalized orthogonal curvilinear coordinate systems. 

The resulting expression for the nabla operator (C.86) are then employed to deduce the transformation formulas for the gradient, divergence, and curl operators in any orthogonal curvilinear coordinate system [11] ... 

The metrics (or scale factors) for a large number of orthogonal curvilinear coordinate systems can be found in the appendix by Happel and Brenner (1973) J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics (Noordhoff International, Leyden, The Netherlands, 1973). 

An analytic solution for such a problem can thus be sought as a superposition of separable solutions of this equation in any orthogonal curvilinear coordinate system. The most convenient coordinate system for a particular problem is dictated by the geometry of the boundaries. As a general rule, at least one of the flow boundaries should coincide with a coordinate surface. Thus, if we consider an axisymmetric coordinate system (f, rj, ), then either f = const or r] = const should correspond to one of the boundaries of the flow domain. 

We use a local orthogonal curvilinear system of dimensionless coordinates f, ), ip, where 77 varies along and is normal to the surface of the particle. In the axisymmetric case, the azimuth coordinate tp varies from 0 to 27r in the plane case, it is supposed that 0 <

constant value = s. The dimensionless fluid velocity components can be expressed via the dimensionless stream function rp as follows ... 

In what follows, we present the basic differential operators in the orthogonal curvilinear coordinates x1, x2, x3. The corresponding unit vectors are denoted by ii, i2, and i3. The gradient of a scalar P is... 

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