System cartesian

Lagrangian-Eulerian (ALE) method. In the ALE technique the finite element mesh used in the simulation is moved, in each time step, according to a predetermined pattern. In this procedure the element and node numbers and nodal connectivity remain constant but the shape and/or position of the elements change from one time step to the next. Therefore the solution mesh appears to move with a velocity which is different from the flow velocity. Components of the mesh velocity are time derivatives of nodal coordinate displacements expressed in a two-dimensional Cartesian system as... 

In the absence of body force the equations of continuity and motion representing Stokes flow in a two-dimensional Cartesian system are written, on the basis of Equations (1.1) and (1.4), as... 

Coordinate Systems The commonly used coordinate systems are three in number. Others may be used in specific problems (see Ref. 212). The rectangular (cartesian) system (Fig. 3-25) consists of mutually orthogonal axes x, y, z. A triple of numbers x, y, z) is used to represent each point. The cylindricm coordinate system (/ 0, z Fig. 3-26) is frequently used to locate a point in space. These are essentially the polar coordinates (/ 0) coupled with the z coordinate. As... 

Microscopic Balance Equations Partial differential balance equations express the conservation principles at a point in space. Equations for mass, momentum, totaf energy, and mechanical energy may be found in Whitaker (ibid.). Bird, Stewart, and Lightfoot (Transport Phenomena, Wiley, New York, 1960), and Slattery (Momentum, Heat and Mass Transfer in Continua, 2d ed., Krieger, Huntington, N.Y., 1981), for example. These references also present the equations in other useful coordinate systems besides the cartesian system. The coordinate systems are fixed in inertial reference frames. The two most used equations, for mass and momentum, are presented here. 

Space. Some fixed reference system in which the position of a body can be uniquely defined. The concept of space is generally handled by imposition of a coordinate system, such as the Cartesian system, in which the position of a body can be stated mathematically. 

The calculation and combination of the components of particle motion requires imposition of a coordinate system. Perhaps the most commoi) is the Cartesian system illustrated in Figure 2-8. Defining unit vectors i, j, and k along the coordinate axes X, y, and z, the position of some point in space, P, can be defined by a position vector, r ... 

It is often convenient to use some other coordinate system besides the Cartesian system. In the normal/tangential system (Figure 2-10), the point of reference is not fixed in space but is located on the particle and moves as the particle moves. There is no position vector and the velocity and acceleration vectors are written in terms of... 

It is important to recognise the differences between scalar quantities which have a magnitude but no direction, and vector quantities which have both magnitude and direction. Most length terms are vectors in the Cartesian system and may have components in the X, Y and Z directions which may be expressed as Lx, Ly and Lz. There must be dimensional consistency in all equations and relationships between physical quantities, and there is therefore the possibility of using all three length dimensions as fundamentals in dimensional analysis. This means that the number of dimensionless groups which are formed will be less. 

To illustrate Equation (1.8), consider a solution of the forward and inverse problems in the simplest possible case, when the field is caused by an elementary mass. Suppose that a particle with mass m q) is situated at the origin of a Cartesian system of coordinates. Fig. 1.2a, and the field is observed on the plane z — h. Then, as follows from Equation (1.8), the components of the attraction field at the point p(x,y,h) are... 

Here d is the volume density at a point. For instance, at points where masses are absent div g = 0. Let us discuss the physical and mathematical content of these equations. The first one clearly shows that the attraction field does not have vortices and, correspondingly, the work done by this field is path independent. In other words, the circulation of the field is equal to zero. At the same time, the second equation demonstrates that the field g is caused by sources (masses) only. As illustration, consider the set of these equations in the Cartesian system of coordinates ... 

First, introduce a Cartesian system of coordinates with its origin at the middle of the layer and z-axis directed perpendicular to its surface. Let us note that the layer has infinite extension along the a and y axes, (Fig. 1.14a). At the beginning, suppose that the observation point is located outside the layer, that is, z >h/2. Then we mentally divide the layer into many thin layers which in turn are replaced by a system of plane surfaces with the density a — 5Ah, where Ah is the thickness of the elementary layer. Taking into account the infinite extension of the surfaces, the solid angle under which they are seen does not depend on the position of the observation point and equals either —2n or 2%. Correspondingly, each plane surface creates the same field ... 

Consider a rotation of the earth around the z-axis in which every particle, elementary volume, of the earth moves along the horizontal circle with the radius r. Our first goal is to find the distribution of forces inside the earth and with this purpose in mind we will derive an equation of motion for an elementary volume of the fluid. Let us introduce a Cartesian system of coordinates with its origin 0, located on the z-axis of rotation. Since this frame of reference is an inertial one, it does not move with the earth, we can write Newton s second law as... 

Assume that origins of two Cartesian systems of coordinates are located at the same point and the frame of reference P rotates about a point 0 of the frame P with constant angular velocity co. Let us imagine two planes, one above another, so that the upper plane P rotates and, correspondingly, unit vectors iiand ji change their direction, Fig. 2.2b. Consider an arbitrary point p, which has coordinates x, y on the plane P and xi, yi on P, and establish relationships between these pairs of coordinates. For the radius vector of the point p in both frames we have... 

Now we demonstrate the system of coordinates, where the ellipsoids of rotation and hyperboloids of one sheet form two mutually orthogonal coordinate families of surfaces. First, we introduce the Cartesian system at the center of the mass and suppose that semi-axes of the ellipsoid of rotation obey the condition brelation between coordinates of the Cartesian and cylindrical... 

In order to find the vertical component of the field we can apply the same approach as before, namely, the integration over the volume of the spheroid, only in this case the polar axis of the spherical system should be directed along the z-axis. However, we solve this problem differently and will proceed from the second equation of the gravitational field. In Cartesian system of coordinates we have... 

It is simple to visualize this motion. Suppose that the initial position of mass is characterized by the angle 6q. At the instant t = 0 it begins to move. Since the force is directed along the motion, the velocity increases and at the lowest point, (x = 0 and z = /), it reaches the maximum value. As soon as a mass passes this point, the velocity begins to decrease because the force component and velocity have opposite directions. Finally, the particle stops and then the motion begins again but in opposite direction. Assuming that friction is absent, we may expect a periodic movement of the mass around the middle point. In fact, equations of a motion of this particle in the Cartesian system of coordinates are... 

First, we introduce a Cartesian system of coordinates with an origin 0 at the center of mass, Fig. 3.11b. The x and y axes are directed to north and east, respectively, and the z-axis is along the vertical. By definition, the moment of the force F(x, y, z) with components F, Fy, and F with respect to point 0 is... 

Suppose that a body is strongly elongated in some direction and with sufficient accuracy it can be treated as the two-dimensional. In other words, an increase of the dimension of the body in this direction does not practically change the field at the observation points. We will consider a two-dimensional body with an arbitrary cross section and introduce a Cartesian system of coordinates x, y, and z, as is shown in Fig. 4.5a, so that the body is elongated along the y-axis. It is clear that if at any plane y — constant the behavior of the field is the same. To carry out calculations we will preliminarily perform two procedures, namely,... 

It should be noted that in the above presentation of the combination of vectors by addition or subtraction, no reference has been made to their components, although this concept was introduced in the beginning of this chapter. It is, however, particularly useful in the definition of the product of vectors and can be further developed with the use of unit vectors. In the Cartesian system employed in Fig. 1 the unit vectors can be defined as shown in Fig. 4. 

Triple products involving vectors arise often in physical problems. One such product is (A x B) x C, which is clearly represented by a vector. It is therefore called the vector triple product, whose development can be made as follows. If, in a Cartesian system, the vector A is chosen to be coMnear with the x direction, A = Axi. The vector B can, without loss of generality, be placed in the x,y plane. It is then given by B = Bxi + Byj. The vector C is then in a general direction, as given by C = Cxi + Cyj + Czk, as shown in Fig. 6. Then, the cross products can be easily developed in the form A x B — AxByk and... 

In previous sections of this chapter, vectors have been described by their components in a Cartesian system. However, for most physical problems it is not the most convenient one. It is generally important to choose a system of coordinates that is compatible with the natural symmetry of the problem at hand. This natural symmetry is determined by the boundary conditions imposed on the solutions. 

Evidently, we would arrive at the same result if the problem were considered in the Cartesian system of coordinates, with accordingly different values of anharmonic coupling coefficients ... 

An affine manifold is said to be flat or Euclidean at a point p, if a coordinate system in which the functions Tl-k all vanish, can be found around p. For a cartesian system the geodesics become... 

There are many forms of statistical graphics (a partial list, classified by function, is presented in Table 22.7), and a number of these (such as scatter plots and histograms) can be used for each of a number of possible functions. Most of these plots are based on a Cartesian system (that is, they use a set of rectangular coordinates), and our review of construction and use will focus on these forms of graphs. 

In the Cartesian system used for an unwrapped screw, the rate of work can be represented in terms of the viscosity, the local velocities, and the screw geometry. 

This vector is sometimes represented as a one-row matrix or a column vector. Usually, because of context, there in no confusion that stems from these alternative representations. More discussion on this point can be found in Appendix A.) As long as the dimensions are sufficiently small, the orthogonal (z, r, 9) coordinate system becomes sufficiently close to a cartesian system. In fact the arguments that follow are identical to those made in a cartesian setting. The planes that are formed by the intersection of A with the coordinate axes have areas Az = nzA, Ar = nrA, and Aq = ngA. These four planes form a tetrahedron. The discussion that follows considers the limit of vanishingly small dimensions, that is, shrinking the tetrahedron to a point. 

In noncartesian coordinates the divergence of a second-order tensor cannot be evaluated simply as a row-by-row operation as it can in a cartesian system. Hence some extra, perhaps unexpected, terms (e.g., rrg/r) appear in the direction-resolved force equations. General expressions for V-T in different coordinate systems are found in Section A.ll. 

In these equations, the vector operators are understood to be nondimensional. For example in a one-dimensional cartesian system... 

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