Oblate spheroid coordinates

The governing equation is therefore identical with that for the irrotational flow of an ideal fluid through a circular aperture in a plane wall. The stream lines and equipotential surfaces in this rotationally symmetric flow turn out to be given by oblate spheroidal coordinates. Since, from Eq. (157), the rate of deposition of filter cake depends upon the pressure gradient at the surface, the governing equation and boundary conditions are of precisely the same form as in the quasi-steady-state approximation... 

The first one, on the binding of an electron by a confined polar atom, is a challenging alternative to [4,5]. The second one is an alternative in the choice of boundary to [43] for the molecular hydrogen close to a confining plane. Section 5.4 emphasizes the importance of using the complete harmonic expansions, outside and inside the sources, of the electrostatic potential of atoms and molecules. In particular, the dipole fields in spherical, prolate and oblate spheroidal coordinates define other alternatives to [4,5]. 

As a specific experience, we can point out that our works on the helium atom confined by paraboloids [46,47] were preceded by variational calculations for the hydrogen atom in the respective confinement situations [49] and by the construction of the paraboloidal harmonic expansion of the Coulomb potential [50]. The last reference also includes the corresponding expansions in prolate and oblate spheroidal coordinates. 

A. Strong, G. Schneider, and M. M. Yovanovich, Thermal Constriction Resistance of a Disc with Arbitrary Heat Flux—Finite Difference Solution in Oblate Spheroidal Coordinates, AIAA-74-690, AIAA/ASME1974 Thermophysics and Heat Transfer Conference, Boston, MA, July 15-17,1974. 

Spheroids are of special interest, since they represent the shape of such naturally occurring particles as large hailstones (C2, L2, R4) and water-worn gravel or pebbles. The shape is also described in a relatively simple coordinate system. A number of workers have therefore examined rigid spheroids. Disks are obtained in the limit for oblate spheroids as E 0. The sphere is a special case where E = I. Throughout the following discussion. Re is based on the equatorial diameter d = 2a (Fig. 4.2). 

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... 

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. 

V elliptic cylinder, bicylinder, oblate and prolate spheroidal coordinate... 

V r, V, z r, V- 6 Tl. V, z Tl, V, z T1,0, V 0 0 0,1 0, oo OQ 1,0 1, OO 1,2 1,2,3 12 ID constant volume condition cylindrical coordinates spherical coordinates elliptical cylinder coordinates bicylinder coordinates oblate and spheroidal coordinates zero thickness limit based on centroid temperature zeroeth order, first order value on the surface and at infinity infinite thickness limit first eigenvalue value at zero Biot number limit first eigenvalue value at infinite Biot number limit solids 1 and 2 surfaces 1 and 2 cuboid side dimensions net radiative transfer one-dimensional conduction... 

Lavagnini 1, Pastore P, Magno F, Amatore CA (1991) Performance of a numerictil method based on the hopscotch algorithm and on an oblate spheroidal space coordinate- expanding time grid for simulation of voltammetric curves at an inlaid disk microelectrode. J Electroaneil Chem 316 37-47... 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

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