Ellipsoid of revolution oblate

Figure 9.3 Intrinsic viscosity according to the Simha theory in terms of the axial ratio for prolate and oblate ellipsoids of revolution.

At 37°C the viscosity of water is about 0.69 X 10"3 kg m" sec" the difference between this figure and the viscosity of blood is due to the dissolved solutes in the serum and the suspended cells in the blood. The latter are roughly oblate ellipsoids of revolution in shape. 

The quantity riV/RT is equal to six times the rotational period. The rotational relaxation time, p, should he shorter than the fluorescence lifetime, t, for these equations to apply. It is possible to perform calculations for nonspherical molecules such as prolate and oblate ellipsoids of revolution, but in such cases, there are different rotational rates about the different principal axes. 

Figure 4.17 (a) Prolate and (b) oblate ellipsoids of revolution, showing the relationship... 

Figure 4.12 Three-dimensional appearance of prolate and oblate ellipsoids of revolution representing three-dimensional protein structures. (Reproduced with permission from Bezkorovainy A. Basic Protein Chemistry. Springfield, IL Thomas, p. 83, 1970.)...

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. 

Closer examination of (10-291) shows that the bubble is deformed into an oblate ellipsoid of revolution. A sketch of the bubble shape for several small values of We is given in Figure 10-12. 

Oblate ellipsoid of revolution. Let us consider an oblate ellipsoid of revolution (on the left in Figure 2.5) with semiaxes a and b a>b) in a translational Stokes flow with velocity U[. We assume that the fluid viscosity is equal to p. We pass from the Cartesian coordinates X, Y, Z to the reference frame... 

The dependence of drop deformation on the Weber number and the vorticity inside the drop was studied in [336]. It was shown that the drop is close in shape to an oblate ellipsoid of revolution with semiaxis ratio > 1 If there is no vortex inside the drop, then this dependence complies with the function We(x) given in (2.8.3). The ratio x decreases as the intensity of the internal vortex increases. Therefore, the deformation of drops moving in gas is significantly smaller than that of bubbles at the same Weber number We. The vorticity inside an ellipsoidal drop, just as that of the Hill vortex, is proportional to the distance TZ from the symmetry axis,... 

Figure 4.1. Shape factor ratio against perimeter-equivalent factor for particles of various shape in a stagnant medium 1, circular cylinder 2, oblate ellipsoid of revolution 3, prolate ellipsoid of revolution 4, cube...

In the quest for a universal feature in the short-to-intermediate time orientational dynamics of thermotropic liquid crystals across the I-N transition, Chakrabarti et al. [115] investigated a model discotic system as well as a lattice system. As a representative discotic system, a system of oblate ellipsoids of revolution was chosen. These ellipsoids interact with each other via a modified form of the GB pair potential, GBDII, which was suggested for disc-like molecules by Bates and Luckhurst [116]. The parameterization, which was employed for the model discotic system, was k = 0.345, Kf = 0.2, /jl= 1, and v = 2. For the lattice system, the well-known Lebwohl-Lasher (LL) model was chosen [117]. In this model, the particles are assumed to have uniaxial symmetry and represented by three-dimensional spins, located at the sites of a simple cubic lattice, interacting through a pair potential of the form... 

Most experiments cannot reveal the shape of a general ellipsoid, so that most data are interpreted in terms of the ellipsoids of revolution. Two cases are possible, the prolate and oblate ellipsoids of revolution. These shapes are offen referred to as prolate or oblate ellipsoids. In a prolate ellipsoid, the unique axis is longer than the other two equal axes a>b c). Prolate ellipsoids are elongated along the symmetry axis. A typical prolate ellipsoid is DPH (Figure 12.2). For an oblate ellipsoid, the unique axis is shorter than the other two eqmval t axes (a < b = c). (%late ellipsoids are shaped like flattened spheres. Perylene is an oblate ellipsoid. 

The disk can be regarded as a flattened sphere oblate ellipsoid of revolution) and the needle as a stretched sphere (prolate ellipsoid of revolution). A spheroid is the same as an ellipsoid of revolution. 

Computations were also performed for simple cubic arrays of oblate ellipsoids of revolution with aspect ratio 5/1 and with solid concentrations varying from 0.02 up to 0.524. The double layer thickness was kept equal to kR = 2.81. The geometry is... 

This technique has been applied to estimate stability-of-solution of the inverse problem in the case of oblate ellipsoids of revolution (Khlebtsov et al., 1978ab), to the influence of particle and medium substance dispersion (Ramazanov and Shchyogolev, 1979), to the experimental error of t (Khlebtsov et al., 1978b). 

None of the prolate dipolar systems mentioned above exhibit (global) ferroelectric ordering. A stable ferroelectric phase has, however, been shown to exist in fluids of oblate ellipsoids of revolution, with a central dipole moment along the axis of revolution, in the range of breadth-to-height ratio... 

Simha provided an equation for the viscosities of ellipsoids of revolution. The prolate ellipsoids of revolution are cigar-shaped while the oblate ellipsoids of revolution are disc-shaped (see Figure 5.3). According to derivations by Einstein and later by R. Simha,... 

Figure 16.32. Variation of the average shape parameter of prolate and oblate ellipsoids (of revolution) as a function of the axial ratio...

The Triton X-100 micelle is considered by many workers to be spherical [70,71]. From geometrical considerations, Robson and Dennis [72] have shown, however, that a spherical micelle would be possible only if several oxyethylene chains were embedded in the hydrophobic core (Fig. 3.3a). These authors consider that an oblate (Fig. 3.3b) rather than prolate (Fig. 3.3c) micelle would be most consistent with intrinsic viscosity measurements and volume calculations. Small-angle X-ray scattering measurements [73], conductivity and viscosity measurements [74] were also more consistent with an oblate ellipsoid of revolution rather than a prolate equivalent. 

From these results it is clear that the picture of the hemoglobin molecule as a prolate ellipsoid is to be discarded. On the other hand the conclusion that it is a right-circular cylinder 34 A. high and 57 A. in diameter with slightly convex ends accords reasonably well with other data. Such a cylinder may be approximated by an oblate ellipsoid of revolution having an axial ratio of about 2.0. This, assuming ilf(l -t- h) = 97,000, should have two relaxation times / (for a moment parallel to the axis of revolution) = 10 sec., and t (for a moment parallel to the transverse axis) = 1.1 X 10 sec. Either component of the moment... 

Figs. 2 and 3, Viscosity factor for prolate and oblate ellipsoids of revolution with axial ratio p. is the volume fraction of ellipsoids, % the rate of shear (see eq. 2, p, 343). Drot, the constant of rotadonal diffussion of the ellipsoids, is for very long ellipsoids app-... 

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