Spheroid eccentricity

Relationship between density, angular velocity, and spheroid eccentricity... 

Now we focus our attention on the conditions of equilibrium for a fluid spheroid rotating about a constant axis. In this case the mutual position of fluid particles does not change and all of them move with the same angular velocity, a>. As is well known, there is a certain relationship between the density, angular velocity, and eccentricity of an oblate spheroid in equilibrium. In studying this question we will proceed from the equation of equilibrium of a fluid, described in the first section. 

Oblate spheroid with very small eccentricity... 

Comparison of these equations shows that the area-free Sherwood number is only slightly affected by eccentricity e.g. Sh/Pe for a spheroid with E = 0.4 is only 8.5% larger than that for the equivalent sphere while the area ratio A/A is 17% larger. Therefore, we expect little effect of deformation on the area-free Sherwood number for bubbles and drops at high Re. This is borne out by the agreement of the data in Fig. 7.14 with Eq. (5-39), derived for fluid spheres. 

Ham further shows that the free-boundary problem, starting with a precipitate particle of negligible dimensions, is not unique, since an arbitrary spheroid will grow at constant eccentricity, its dimensions being... 

L. Oblate spheroid, forced convection , d0h up j total surface area i perimeter normal to flow e.g., for cube with side length a, d = 1.27a. k ch [E] Used with arithmetic concentration difference. 120 < NRe 6000 standard deviation 2.1%. Eccentricities between 1 1 (spheres) and 3 1. Oblate spheroid is often approximated by drops. [141] p. 284 [142]... 

Not all bodies are capable of attaining steady motion with unsymmetrical bodies spiraling and wobbling may occur. For an oblate spheroid [12 p 144] of eccentricity and equatorial diameter a, the maximum drag force, for tending to 0, is ... 

The complex polarizability, oc, measured along the axis of symmetry of a prolate spheroid, of volume V and unit eccentricity, may be expressed in terms of e and of by tbe equations... 

For this purpose we view a craze as a lenticular region with its principal plane normal to the tensile axis z. Although actual shapes of isolated crazes differ significantly from regular shapes as we will discuss below, a craze can be considered as a very eccentric oblate spheroid with a radius a in its principal plane and a half thickness b parallel to the z axis. Furthermore, since the main microstructural feature of a craze is extended fibrils with a relatively constant extension ratio we can view the... 

Njig 6000 standard deviation 2.1%. Eccentricities between 1 1 (spheres) and 3 1. Oblate spheroid is often approximated by drops. 

Fig. 6 Eccentricity dependence of the F function. The eccentricity e of a spheroid is related to the aspect ration (c/a) by e =1-(c/a). Reproduced with permission from Ref. 16, Fig. 4. Copyright 1984, Electrochemical Society.

Indeed, the quadratic term in / representing the energy vanishes, the separation constant A becomes infinitely large, the confining spheroids become very eccentric, so that u becomes almost one. This suggests using the transformation complementary to the one introduced in equation (11) of [9] for the hyperboloidal boundary ... 

It is of interest to note that, by judicious definition of the characteristic diameter of nonspherical bodies, good agreement with the equations for spherical solids was obtained. A diameter defined by the total surface area of the body, divided by the perimeter normal to flow, was successfully used for spheres, hemispheres, cubes, prisms, and cylinders (PI), yielding a = 0 b - 0.692 m = 0.514 and n = [Eq. (4)]. Similar results were obtained for spheroids (S14), namely a = 0 6 = 0.74 w = 0.5 and n =. The commonly used equivalent diameter of a sphere of the same volume as the body yields transfer coefficients increasing with eccentricity (SI4). 

For comparison with the experiments, the special case of spheroids has been found suitable. They are characterized by coincidence of two of the three elementary axes. One distinguishes elongated or prolate spheroids with B = C and distinguishes flattened or oblate spheroids with A = B. In the following, A is always the major axis and B the minor axis of the spheroid. Accordingly, only two factors Gi(e) and G2(e) are relevant, depending on the eccentricity e, defined for a spheroid as... 

In Figure 6.4, Gi(e) is plotted versus the eccentricity e for prolate and oblate spheroids. It starts with Gi(0) = which is the case of a spherical particle (A... 

B = C) and decreases with increasing eccentricity. Vice versa, following from equation (5), G2(e) also starts with GaCO) =4, but it increases with increasing eccentricity. It follows now from equation (4) that for a spheroidal silver particle, two surface plasma resonances are induced, namely, when the optical constants of silver satisfy the conditions... 

Janz et al. (1980) developed a mathematical model for the finite deformation of a prolate spheroidal membrane subjected to a hydrostatic pressure and unconstrained with respect to eccentricity. Figure 1 displays the deformed and undeformed geometry of this membrane model. This deformed geometry is determined by solving a system of differential equations for the stretch ratios and... 

Here, the sum over j considers the three dimensions of the particle. P. includes, Pg, and Pj, termed depolarization factors, for each axis of the particle, where A > B = C for a prolate spheroid. The depolarization factors anisotropically alter the values of and u and the resulting LSPR peak frequencies. Here e. and are the same as described in Equation 13.5. Eccentricity is denoted by e which has a direct relationship with major and minor axes. Explicitly, they are expressed by the following equations. 

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