Ellipsoidal coordinate system

Problem 9-17. Heat Transfer From an Ellipsoid of Revolution at Pe S> 1. In a classic paper, Payne and Pell. J. Fluid Meek 7, 529(1960)] presented a general solution scheme for axisymmetric creeping-flow problems. Among the specific examples that they considered was the uniform, axisymmetric flow past prolate and oblate ellipsoids of revolution (spheroids). This solution was obtained with prolate and oblate ellipsoidal coordinate systems, respectively. 

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. 

With the new coordinate system only the three diagonal components axx, ayy, and olzz referred to as principal values of a are nonzero. The halfaxes of the ellipsoid are a 2, ay]/2, and aj1/2. If the polarizability ellipsoid... 

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... 

While Onsager s formula has been widely used, there have also been numerous efforts to improve and generalize it. An obvious matter for concern is the cavity. The results are very sensitive to its size, since Eqs. (33) and (35) contain the radius raised to the third power. Within the spherical approximation, the radius can be obtained from the molar volume, as determined by some empirical means, for example from the density, the molar refraction, polarizability, gas viscosity, etc.90 However the volumes obtained by such methods can differ considerably. The shape of the cavity is also an important issue. Ideally, it should be that of the molecule, and the latter should completely fill the cavity. Even if the second condition is not satisfied, as by a point dipole, at least the shape of the cavity should be more realistic most molecules are not well represented by spheres. There was accordingly, already some time ago, considerable interest in progressing to more suitable cavities, such as spheroids91 92 and ellipsoids,93 using appropriate coordinate systems. Such shapes... 

The other four integrals are also equal to one another, and this is a function of the distance, R, between the two atoms called the overlap integral, S R). The overlap integral is an elementary integral in the appropriate coordinate system, confocal ellipsoidal-hyperboloidal coordinates [27]. In terms of the function of Eq. (2.12) it has the form... 

We noted in the preceding section that the polarizability of an ellipsoid is anisotropic the dipole moment induced by an applied uniform field is not, in general, parallel to that field. This anisotropy originates in the shape anisotropy of the ellipsoid. However, ellipsoids are not the only particles with an anisotropic polarizability in fact, all the expressions above for cross sections are valid regardless of the origin of the anisotropy provided that there exists a coordinate system in which the polarizability tensor is diagonal. 

The state of stress in a flowing liquid is assumed to be describable in the same way as in a solid, viz. by means of a stress-ellipsoid. As is well-known, the axes of this ellipsoid coincide with directions perpendicular to special material planes on which no shear stresses act. From this characterization it follows that e.g. the direction perpendicular to the shearing planes cannot coincide with one of the axes of the stress-ellipsoid. A laboratory coordinate system is chosen, as shown in Fig. 1.1. The x- (or 1-) direction is chosen parallel with the stream lines, the y- (or 2-) direction perpendicular to the shearing planes. The third direction (z- or 3-direction) completes a right-handed Cartesian coordinate system. Only this third (or neutral) direction coincides with one of the principal axes of stress, as in a plane perpendicular to this axis no shear stress is applied. Although the other two principal axes do not coincide with the x- and y-directions, they must lie in the same plane which is sometimes called the plane of flow, or the 1—2 plane. As a consequence, the transformation of tensor components from the principal axes to the axes of the laboratory system becomes a simple two-dimensional one. When the first principal axis is... 

Fig. 1.1. Laboratory coordinate system x direction of flow (also 1-direction), y direction of velocity gradient (also 2-direction), I, II principal directions of stress, y orientation angle if stress-ellipsoid, vx velocity (in -direction), q velocity gradient...

In Fig. 7 we depict the mean square displacement of the director as a function of time for the prolate ellipsoids. At this state point it is very low. After ten time units, the square root of the MSD is only 4°. It is important to keep this figure in mind. This low reorientation rate means that a director-based coordinate system is an inertial frame to a very good approximation even if one does not apply the constraint equation, Eq. (2.31). This MSD was obtained from a simulation of 256 molecules. If the system size increases the MSD will be even smaller. 

In the coordinate system fixed to a molecule in the origin the pattern of fluid flow looks like Fig. 2. Since the force on the molecule is proportional to the relative fluid velocity one can expect that the chain shape will be ellipsoidal in the coordinate system. 

The discussion will be restricted to molecules which can be described as ellipsoids of revolution. We shall denote the semi-axis of revolution by a, the equatorial semi-axis by b. The orientation of a molecule may thus be completely described by the orientation of the a semiaxis the frame of reference is shown in Fig. 4. The center of the coordinate system is located in the center of gravity of the ellipsoid. The velocity gradient in the liquid, indicated in Fig. 4, tends to rotate the molecule clockwise. The orientation of the a semi-axis of the molecule is specified by the angles and 0, as defined in the figure and the accompanying legend. 

As discussed in Sect. 2.1.2, the index of an anisotropic medium is described by the index ellipsoid (Eq. 22). If the coordinate system is chosen such that the axes do not match with the principal symmetry axes of the crystal, the index ellipsoid is described by the more general expression [11]... 

To actually use these results, it is of course necessary to actually calculate the components of the resistance tensors. We have seen that it is necessary to solve only three problems for translation and three problems for rotation in the coordinate directions to specify all of the components of A, B, C, and D. It probably does not need to be said that the orientation of the coordinate axes should be chosen to take advantage of any geometric symmetries that can simplify the fluid mechanics problems that must be solved. For example, if we wish to determine the force and/or torque on an ellipsoid for motions of arbitrary magnitude and direction (with respect to the body geometry), we should specify the components of the resistance tensors with respect to axes that are coincident with the principal axes of the ellipsoid, as this choice will simplify the fluid mechanics problems that are necessary to determine these components. If arbitrary velocities U and ft are then specified with respect to these same coordinate axes, the Eqs. (7-22) will yield force and torque components in this coordinate system. 

On the other hand, for an ellipsoid of revolution with the origin of the coordinate system at the geometric center and coordinate axes parallel and perpendicular to the principal axes of the ellipse, it can be shown that... 

The thermal boundary-layer equation, (9-257), also apphes for axisymmetric bodies. One example that we have already considered is a sphere. However, we can consider the thermal boundary layer on any body of revolution. A number of orthogonal coordinate systems have been developed that have the surface of a body of revolution as a coordinate surface. Among these are prolate spheroidal (for a prolate ellipsoid of revolution), oblate spheroidal (for an oblate ellipsoid of revolution), bipolar, toroidal, paraboloidal, and others.22 These are all characterized by having h2 = h2(qx, q2), and either h2/hx = 1 or h2/hx = 1 + 0(Pe 1/3) (assuming that the surface of the body corresponds to q2 = 1). Hence the thermal boundary-layer equation takes the form... 

The study of the electronic structure of diatomic species, which can nowadays be done most accurately with two-dimensional numerical finite difference techniques, both in the non-relativistic [90,91] and the relativistic framework [92-94], is still almost completely restricted to point-like representations of the atomic nuclei. An extension to allow the use of finite nucleus models (Gauss-type and Fermi-type model) in Hartree-Fock calculations has been made only very recently [95]. This extension faces the problem that different coordinate systems must be combined, the spherical one used to describe the charge density distribution p r) and the electrostatic potential V(r) of each of the two nuclei, and the prolate ellipsoidal one used to solve the electronic structure problem. 

The polarizability tensor can be visualized easily if we draw a polarizability ellipsoid by plotting l/Va in any direction from the origin. This gives a three-dimensional surface such as is shown in Fig. 1-8. If we orient this ellipsoid with its principal axes along the X, V, Z axes of the coordinate system, Eq. 5.8 is simplified to... 

The surfaces of constant are prolate ellipsoids, those of constant r are hyperbolic sheets. The system is illustrated in Fig. 7.1. If we now take a particular pair of the STO functions comprising the minimal basis for H2O, say Isb and one of the 2po functions we can transform them to the new coordinate system and see what is involved in the integral calculation. 

Besides the well-known cartesian system there are a lot of other systems in use in science. These include polar cylindrical coordinates, spherical polar coordinates, ellipsoidal coordinates, parabolic cylindrical coordinates, and cylindrical bipolar coordinates. 

Here, the focal length d defines a family of coordinate systems that vary from spherical polar when d = 0 to cylindrical polar in the limit when d oo. A surface of constant transmural coordinate A (Figure 54.1) is an ellipse of revolution with major radius a = d cosh A and minor radius b = d sinh A. In an ellipsoidal model with a truncation factor of 0.5, the longitudinal coordinate fi varies from zero at the apex to 120° at the base. Integrating the Jacobian in prolate spheroidal coordinates gives the volume of the wall or cavity ... 

First, elastic properties of each impregnated yam segment are calculated using homogenisation formulae (e.g. the aforementioned Chamis formulae) for the unidirectional array of fibres, using local fibre volume fraction at the segment, properties of the fibres and elastic properties of the matrix. The result is the stiffuess matrix C], expressed in the local 123 coordinate system. Then each yarn segment is represented by an ellipsoidal inclusion with axis... 

The eigenvalues were obtained by diagonalization of the Hessian. Sueh diagonalization corresponds to a rotation of the local coordinate system (cf., p. 359). Imagine a two-dimensional surface that at the minimum could be locally approximated by an ellipsoidal valley. The diagonalization means such a rotation of the coordinate system x, y that both axes of the ellipse coincide with the new axes x, y (as discussed in Chapter 7). On the other hand, if our surface locally resembled a cavalry saddle, diagonalization would lead to such a rotation of the coordinate system that one axis would be directed along the horse, and the other across. ... 

We will start with the general index ellipsoid for a crystal in its simplest form. This will allow us to determine the direction of polarization as well as the corresponding refractive indices of the crystal. The index ellipsoid is given in the principal coordinate system as [27,29,74]... 

---