Coordinates, curvilinear

Vector equations can provide an elegant abstract formulation of physical laws, but, in order to solve problems, it is usually necessary to express these equations in a particular coordinate system Thus far, we have considered Cartesian coordinates almost exclusively. Another choice, for example, cylindrical or spherical coordinates, might prove more appropriate, reflecting the symmetry of the problem. 

We will consider the general class of orthogonal curvilinear coordinates, designated q, q2, and 73, whose coordinate surfaces always intersect at right angles. The Cartesian coordinates of a point in three-dimensional space can be expressed in terms of a set of curvilinear coordinates by relations of the form 

The components of the gradient vector represent directional derivatives of a function. For example, the change in the function p q, qi, qs) along the qi direction is given by the ratio of to the element of length Qi dqi. Thus, the gradient in curvilinear coordinates can he written as 

The divergence V A represents the limiting value of the net outward flux of the vector quantity A per unit volume. Referring to Fig. 11.19, the net flux of the component A in the 71 direction is given by the difference between the outward contributions Q2Q3A dq2dq2, on the two shaded faces. As the volume element approaches a point, this reduces to 

Adding the analogous contributions from the q2 and directions and dividing by the volume dr, we obtain the general result for the divergence in curvilinear coordinates  

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Let us analyse p near an interface. The Laplacian in tire curvilinear coordinates (ir )can be written such that (A3.3.71) becomes (near the interface)... 

Figure 5.21 Thin layer between curved surfaces and the general curvilinear coordinate system...

Let S o be a surface located at mid-channel between two smooth surfaces separated by a narrow gap. The curvilinear coordinate system, corresponding to this... 

For the example in Figure 2.14 it would be possible to perform the coordinate transformation analytically by introducing cylindrical coordinates. However, in general, geometries are too complex to be described by a simple analytical transformation. There are a variety of methods related to numerical curvilinear coordinate transformations relying on ideas of tensor calculus and differential geometry [94]. The fimdamental idea is to establish a numerical relationship between the physical space coordinates and the computational space curvilinear coordinates The local basis vectors of the curvilinear system are then given as... 

Computational space curvilinear coordinates Zeta potential... 

Section 5.3. There, the operator T> s d/dx was used in the solution of ordinary differential equations. In Chapter 5 the vector operator del , represented by the symbol V, was introduced. It was shown that its algebraic form is dependent on the choice of curvilinear coordinates. 

Although transformations to various curvilinear coordinates can be carried out relatively easily with the use of the vector relations introduced in Section 5.15, it is often of interest to make the substitutions directly. Furthermore, it is a very good exercise in the manipulation of partial derivatives. 

Local reptation regime For times t > xe we have to consider curvilinear Rouse motion along the spatially fixed tube. The segment displacement described by Eq. (18) (n = m) must now take the curvilinear coordinates s along the tube into consideration. We have to distinguish two different time regimes. For (t < xR), the second part of Eq. (19) dominates - when the Rouse modes... 

Here the subscript i denotes th set of the curvilinear coordinates. The local mean and Gaussian curvatures are determined by using nonlinear regression fitting after a number of sections at a given point has been made [this corresponds to different sets of the local coordinates (u, v)]. 

To derive an expression for divA in curvilinear coordinates the net outward flow through the surface bounding the volume dv is defined as divAdv. 

The Laplacian in curvilinear coordinates follows from the definition div gradF = V2V... 

Another curvilinear coordinate system of importance in two-centre problems, such as the diatomic molecule, derives from the more general system of confo-cal elliptical coordinates. The general discussion as represented, for instance by Margenau and Murphy [5], will not be repeated here. Of special interest is the case of prolate spheroidal coordinates. In this system each point lies at the intersection of an ellipsoid, a hyperboloid and and a cylinder, such that... 

Applications to Schroedinger s Equation, C. Cerjan, Ed., Kluwer, Dordrecht, The Netherlands,1993, pp. 1-23. Fast Pseudospectral Algorithm Curvilinear Coordinates. 

We first consider derivatives with respect to curvilinear coordinates of the determinant g of the metric tensor in the unconstrained space. Using Eq. (A.14) and definition (2.16) for yields... 

Fig. 3. Curvilinear coordinate system introduced by Marcus (19) and Light (20) showing the hatched region in which the coordinates x and s are not uniquely defined.

To obtain the anharmonic terms in the potential, on the other hand, the choice of coordinates is important 130,131). The reason is that the anharmonic terms can only be obtained from a perturbation expansion on the harmonic results, and the convergence of this expansion differs considerably from one set of coordinates to another. In addition it is usually necessary to assume that some of the anharmonic interaction terms are zero and this is true only for certain classes of internal coordinates. For example, one can define an angle bend in HjO either by a rectilinear displacement of the hydrogen atoms or by a curvilinear displacement. At the harmonic level there is no difference between the two, but one can see that a rectilinear displacement introduces some stretching of the OH bonds whereas the curvilinear displacement does not. The curvilinear coordinate follows more closely the bottom of the potential well (Fig. 12) than the linear displacement and this manifests itself in rather small cubic stretch-bend interaction constants whereas these constants are larger for rectilinear coordinates. A final and important point about the choice of curvilinear coordinates is that they are geometrically defined (i.e. independent of nuclear masses) so that the resulting force constants do not depend on isotopic species. At the anharmonic level this is not true for rectilinear coordinates as it has been shown that the imposition of the Eckart conditions, that the internal coordinates shall introduce no overall translation or rotation of the body, forces them to have a small isotopic dependence 132). 

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. 

Derive the transfomation metrics hr, he, and ha that are needed to represent the governing equations in the proposed curvilinear coordinates. 

The components of the velocity vector (v, v2, V3) align with the curvilinear-coordinate directions ( 1, 2, 3). 

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