Coordinate system three-dimensional, equations

In an axisymmetric flow regime all of the field variables remain constant in the circumferential direction around an axis of symmetry. Therefore the governing flow equations in axisymmetric systems can be analytically integrated with respect to this direction to reduce the model to a two-dimensional form. In order to illustrate this procedure we consider the three-dimensional continuity equation for an incompressible fluid written in a cylindrical (r, 9, 2) coordinate system as... 

Symmetry 50. Intercepts 50. Asymptotes 50. Equations of Slope 51. Tangents 51. Equations of a Straight Line 52. Equations of a Circle 53. Equations of a Parabola 53. Equations of an Ellipse of Eccentricity e 54. Equations of a Hyperbola 55. Equations of Three-Dimensional Coordinate Systems 56. Equations of a Plane 56. Equations of a Line 57. Equations of Angles 57. Equation of a Sphere 57. Equation of an Ellipsoid 57. Equations of Hyperboloids and Paraboloids 58. Equation of an Elliptic Cone 59. Equation of an Elliptic Cylinder 59. 

Examples. A Brownian particle, together with its surrounding fluid, is a closed isolated system. The variables xt are its three coordinates and Q is its mass. Pe x) is a constant. Equation (4.1) for this case is the three-dimensional analog of (VIII.3.1) = 0 and is constant. 

The concentration profile may be independent of time and vary in more than one dimension thus a two-dimensional or three-dimensional spatial problem results. Occasionally a system is encountered where rapid convection occurs perpendicular to the electrode surface so that diffusion is negligible in that coordinate. By rearranging equations (76) and (77) and normalizing the concentration, the mass transport to a ChE at steady state is given by (110). 

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. 

In Chapter I we found that curvilinear coordinates, such as spherical polar coordinates, are more suitable than Cartesian coordinates for the solution of many problems of classical mechanics. In the applications of wave mechanics, also, it is very frequently necessary to use different kinds of coordinates. In Sections 13 and 15 we have discussed two different systems, the free particle and the three-dimensional harmonic oscillator, whose wave equations are separable in Cartesian coordinates. Most problems cannot be treated in this manner, however, since it is usually found to be impossible to separate the equation into three parts, each of which is a function of one Cartesian coordinate only. In such cases there may exist other coordinate systems in terms of which the wave equation is separable, so that by first transforming the differential equation into the proper... 

Mathematicians1 have studied the conditions under which the wave equation is separable, obtaining the result that the three-dimensional wave equation can be separated only in a limited number of coordinate systems (listed in Appendix IV) and then only if the potential energy is of the form... 

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