Oblate spheres

It is possible to wind integrally most of the bodies of revolution, such as spheres, oblate spheres, and torroids. Each application, however, requires a study to insure that the winding geometry satisfies the membrane forces induced by the configuration being wound. 

Detailed calculations on reactants visualized as oblate spheres have shown that Hel also depends on the shape and nodal character of overlapping orbitals [56]. Thus, Hel for planar n-conjugated systems is predicted to vary depending on the relative orientation of the molecules (Fig. 15 and 16). Orbital character has special importance in PET where overlap between orbitals of the quencher and the rapidly oscillating wavefunctions of the excited reactant determines the extent of electron coupling. 

Four volumetric defects are also included a spherical cavity, a sphere of a different material, a spheroidal cavity and a cylinderical cavity (a side-drilled hole). Except for the spheroid, the scattering problems are solved exactly by separation-of-variables. The spheroid (a cigar- or oblate-shaped defect) is solved by the null field approach and this limits the radio between the two axes to be smaller than five. 

Equation (3-33) shows how the inertia term changes the pressure distribution at the surface of a rigid particle. The same general conclusion applies for fluid spheres, so that the normal stress boundary condition, Eq. (3-6), is no longer satisfied. As a result, increasing Re causes a fluid particle to distort towards an oblate ellipsoidal shape (Tl). The onset of deformation of fluid particles is discussed in Chapter 7. 

FINITE CYLINDER C TOUCHING SPHERES O OBLATE SPHEROID- PROLATE SPHEROID THIN DISK CUBE... 

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). 

Fig. 6.6 Ratio of drag on oblate spheroid or disk to drag on sphere of same equatorial radius.

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

Since most irregular particles of practical concern tend to be oblate, lenticular, or rod-like with moderate aspect ratio, these comparisons generally support Heywood s approach. Combining this observation with the fact that the volumetric shape factor is more readily determined than sphericity, we conclude that Heywood s approach is preferred for the intermediate range. For convenience in estimating Uj, Table 6.4 gives correlations, fitted to Heywood s values, for 0.1 < k < 0.4 at specific values of Since is relatively insensitive to interpolation for at other values of is straightforward. In common with Heywood s tabulated values, the correlations in Table 6.4 do not extrapolate to = 1 for a sphere k = 0.524). 

Transfer from large bubbles and drops may be estimated by assuming that the front surface is a segment of a sphere with the surrounding fluid in potential flow. Although bubbles are oblate ellipsoidal for Re < 40, less error should result from assumption of a spherical shape than from the assumption of potential flow. 

The hydrodynamic shape factor and axial ratio are related (see Eigure 4.18), but are not generally used interchangeably in the literature. The axial ratio is used almost exclusively to characterize the shape of biological particles, so this is what we will utilize here. As the ellipsoidal particle becomes less and less spherical, the viscosity deviates further and further from the Einstein equation (see Eigure 4.19). Note that in the limit of a = b, both the prolate and oblate ellipsoid give an intrinsic viscosity of 2.5, as predicted for spheres by the Einstein equation. 

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). 

A glance at the curves in Fig. 11.15 reveals extinction characteristics similar to those for spheres at small size parameters there is a Rayleigh-like increase of Q a with x followed by an approximately linear region broad-scale interference structure is evident as is finer ripple structure, particularly in the curves for the oblate spheroids. The interference structure can be explained... 

Calculated and measured values of P = —Sn/Su, the degree of linear polarization, for several nonspherical particles are shown in Fig. 13.9. The prolate and oblate spheroids, cubes, and irregular quartz particles have made their appearance already (Fig. 13.8) a new addition is NaCl cubes. Also shown are calculations for equivalent spheres. 

Noctilucent cloud particles are now generally believed to be ice, although more by default—no serious competitor is still in the running—than because of direct evidence. The degree of linear polarization of visible light scattered by Rayleigh ellipsoids of ice is nearly independent of shape. This follows from (5.52) and (5.54) if the refractive index is 1.305, then P(90°) is 1.0 for spheres, 0.97 for prolate spheroids, and 0.94 for oblate spheroids. 

It is not surprising, however, that Mie theory is inadequate in this instance the indium particles are not spheres, they are more nearly oblate spheroids with (average) major and minor diameters of about 1390 and 368 A. 

The absorption spectrum of isolated indium spheres differs from that of closely packed oblate spheroids in that the peak shifts from 2230 A to 4100 A about half of this shift is attributable to particle shape and half to particle interaction. Indium particles on immunological slides are not identical, however, but are distributed in size and shape about some mean this tends to broaden the spectrum. 

Conformation of the Macromolecule. In solution, macromolecules can have a wide variety of shapes or conformations. The simplest is the solid sphere or Einstein sphere. It is a round ball, impermeable to solvent. The ball may be stretched into a prolate ellipsoid like a football or flattened into an oblate ellipsoid like a flying saucer. Many soluble proteins have conformations that approximate ellipsoids. If a prolate ellipsoid is stretched enough, it becomes a rod. Certain virus macromolecules are rodlike. 

Suppose that the particles in Figure 1.8 were actually oblate ellipsoids (all in their preferred orientation) rather than spheres. Would their volume be over- or underestimated if the particles were assumed to be spheres In terms of their axial ratio, calculate the factor by which the mass is under- or overestimated when the particles are assumed to be spheres. (Consult a handbook for the volume of an ellipsoid.)... 

At relatively low concentrations of surfactant, the micelles are essentially the spherical structures we discussed above in this chapter. As the amount of surfactant and the extent of solubilization increase, these spheres become distorted into prolate or oblate ellipsoids and, eventually, into cylindrical rods or lamellar disks. Figure 8.8 schematically shows (a) spherical, (b) cylindrical, and (c) lamellar micelle structures. The structures shown in the three parts of the figure are called (a) the viscous isotropic phase, (b) the middle phase, and (c) the neat phase. Again, we emphasize that the orientation of the amphipathic molecules in these structures depends on the nature of the continuous and the solubilized components. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

Figure 6.10 Schematic representations of the prolate and oblate deformations of a uniform sphere. A prolate deformation corresponds to the stretching of the distribution along only one axis while the distribution shrinks equally along the other two axes. An oblate deformation corresponds to the compression of the distribution along one axis with increases along the other two axes.

To the extent that the X-ray structure can be represented by an ellipsoid of revolution, it appears to be an oblate ellipsoid with an axial ratio no greater than 1 2. The approximate dimensions are 25 X 45 X 45 A. From molecular weight and free diffusion data the frictional ratio /// is 1.26. This corresponds to an unhydrated prolate ellipsoid with an axial ratio of 5.2 or an oblate ellipsoid with a ratio 0.18. If the hydration is assumed to be 0.34 g H20/g protein, the axial ratio values would be 3.0 and 0.33, respectively. The maximum hydration for a sphere would be 0.7 or 0.55 g/g for a prolate ellipsoid of axial ratio 2. 

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