Two-dimensional Cartesian coordinate system

Mathematical derivations presented in the following sections are, occasionally, given in the context of one- or two-dimensional Cartesian coordinate systems. These derivations can, however, be readily generalized and the adopted style is to make the explanations as simple as possible. 

In a fixed two-dimensional Cartesian coordinate system, the continuity equation for a free boundary is expressed as... 

In this section the governing Stokes flow equations in Cartesian, polar and axisymmetric coordinate systems are presented. The equations given in two-dimensional Cartesian coordinate systems are used to outline the derivation of the elemental stiffness equations (i.e. the working equations) of various finite element schemes. 

Governing equations in two-dimensional Cartesian coordinate systems... 

Teaching Thermodynamic Relations Using a Story and Two-Dimensional Cartesian Coordinate System... 

Abstract Thermodynamic energy functions are related to six variables such as volume, pressure, temperature, entropy, chemical potential and amount of substance. They are rather cumbersome and perplexing to undergraduates who start to leam their relations. With a story and the two-dimensional Cartesian coordinate system, most of thermodynamic relations could be obtained in addition to the Maxwell relations for a reversible change in a closed system only in the presence of pressure-volume work and heat. 

Various mnemonics have been reported to help students to be familiar with thermodynamic relations [2-5]. Most of them are rather direct notation and demand their memorization. Teaching the pertinent thermodynamic relations to them could be consummated with a simple story displayed in the two-dimensional Cartesian coordinate system for a reversible change in a closed system without composition change in the absence of any other work except pressure-volume work. 

Fig. 1 Thermodynamic energy functions with the four variables shown in the two-dimensional Cartesian coordinate system...

If we are plotting a pair of data, we can use a two-dimensional cartesian coordinate system. This is the most common application of visualization of data. 

In two-dimensional Cartesian coordinate system x,y) we consider magneto-convection, steady, laminar, electrically conduction, boundary layer flow of a viscoelastic fluid caused by a stretching surface in the presence of a uniform transverse magnetic field and a heat source. The x -axis is taken in the direction of the main flow along the plate and the y -axis is normal to the plate with velocity components u,v in these directions. 

Figure 10.1 illustrates a typical solidified control volume in two-dimensional Cartesian coordinate system. To integrate the conservation equations, a staggered grid is employed in which the velocity component (C/e, V, and Eg) are defined... 

In the mid-seventeenth century the French mathematician Rene Descartes proposed a simple method of relating pairs of numbers as points on a rectangular plane surface today called a rectangular Cartesian coordinate system. A typical two-dimensional Cartesian coordinate system consists of two perpendicular axes, called the coordinate axes. The vertical or y-axis is called the ordinate, while the horizontal or jc-axis is called the abscissa. The point of intersection between the two axes is called the origin. In designating a point on this coordinate system, the abscissa of the point always is given first. Thus, the notation (4,5) refers to the point whose abscissa is 4 and whose ordinate is 5, as shown in Fig. 1-1. 

In the above example, V, was considered to be a function of only a single variable t Such an equation, V, =/(f), can be represented by a series of points on a two-dimensional Cartesian coordinate system. Physicochemical systems, however, usually depend on more than one variable. Thus, it is necessary to extend the definition of function given above to include functions of more than one variable. For example, we find experimentally that the volume of a gas will vary with temperature according to Equation (2-2) only if the pressure of the gas is held constant. Thus, the volume of a gas is not only a function of temperature, but also is a function of pressure. Careful measurements in the laboratory will show that for most gases at or around room temperature and one atmosphere pressure the law relating the volume of a gas simultaneously to the temperature and the pressure of the gas is the well-known ideal gas law... 

The Laplace equation is first written in terms of a finite-difference approximation. For simplicity, a two-dimensional Cartesian coordinate system is assumed ... 

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