Spheroids prolate

Small micelles in dilute solution close to the CMC are generally beheved to be spherical. Under other conditions, micellar materials can assume stmctures such as oblate and prolate spheroids, vesicles (double layers), rods, and lamellae (36,37). AH of these stmctures have been demonstrated under certain conditions, and a single surfactant can assume a number of stmctures, depending on surfactant, salt concentration, and temperature. In mixed surfactant solutions, micelles of each species may coexist, but usually mixed micelles are formed. Anionic-nonionic mixtures are of technical importance and their properties have been studied (38,39). 

Prolate Spheroid (formed by rotating an ellipse about its major... 

Symmetrical tops are of two types. A prolate spheroid (football shape) in which... 

The electrochemical behavior of the C70 solvent-cast films was similar to that of the C60 films, in that four reduction waves were observed, but some significant differences were also evident. The peak splitting for the first reduction/oxidation cycle was larger, and only abont 25% of the C70 was rednced on the first cycle. The prolate spheroidal shape of C70 is manifested in the II-A isotherm of C70 monolayers. Two transitions were observed that gave limiting radii consistent with a transition upon compression from a state with the long molecnlar axes parallel to the water snrface to a state with the long molecnlar axes per-pendicnlar to the water surface. 

The formula for the surface area of a prolate spheroid such as a cell is given by ... 

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... 

Let us also consider the application of these transformations to diatomic molecules. We note that the density transformations carry the initial prolate spheroidal coordinates (C, t], (j)) into the transformed coordinates (C, j/p, 4>)- The equations that define these transformations are [119] ... 

D, more spherical than myoglobin 3D, prolate spheroid... 

Axisymmetric shapes are conveniently described by the aspect ratio E, defined as the ratio of the length projected on the axis of symmetry to the maximum diameter normal to the axis. Thus, E is the ratio of semiaxes for a spheroid, with < 1 for an oblate spheroid and > 1 for a prolate spheroid. 

Figure 4.5 shows the variation of A with E for flow parallel and normal to the axis, and averaged over random orientations. Except for disk-like particles, the dependence of A on aspect ratio is rather weak. In axial motion, a somewhat prolate spheroid experiences less drag than the volume-equivalent sphere A passes through a minimum of 0.9555 for E = 1.955. For motion normal to the axis of symmetry, A 2 takes a minimum of 0.9883 at = 0.702. However, the average resistance A is a minimum for a sphere. 

For prolate spheroids, Eq. (4-37) with k — 0.5 again agrees with the limiting exact result for -> oo. The validity of these equations for cylinders is demonstrated in Figs. 4.7 and 4.8. Comparison of Eqs. (4-36) and (4-37) shows that the ratio of C2 to cq tends to 2 as -> 00. This result holds for any axisymmetric particle, while cq < 2c for finite aspect ratios (W2). Consequently a needlelike particle falls twice as fast when oriented vertically at low Re than when its axis is horizontal. 

Cox (C5) and Tchen (Tl) also obtained expressions for the drag on slender cylinders and ellipsoids which are curved to form rings or half circles. The advantages of prolate spheroidal coordinates in dealing with slender bodies have been demonstrated by Tuck (T2). Batchelor (Bl) has generalized the slender body approach to particles which are not axisymmetric and Clarke (C2) has applied it to twisted particles by considering a surface distribution rather than a line distribution. 

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

With K-values from Fig. 4.14 and Sho derived from Table 4.2, Eq. (4-69) predicts Sh within 10% of the numerical values of Masliyah and Epstein (Ml) for Pe < 70 and E = 0.2 for oblate spheroids and E = 5 for prolate spheroids. The analogous correlation with L as the characteristic length is... 

The time variation of concentration at the center and at the foci of prolate spheroids has been calculated for negligible external resistance, Bi oo (H2). These appear to be the only calculations for shapes other than those mentioned above. 

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. 

For large bubbles where inertia effects are dominant, enclosed vertical tubes lead to bubble elongation and increased terminal velocities (G7). The bubble shape tends towards that of a prolate spheroid and the terminal velocity may be predicted using the Davies and Taylor assumptions discussed in Chapter 8, but with the shape at the nose ellipsoidal rather than spherical. The maximum increase in terminal velocity is about 16% for the case where 2 is small (G6) and 25% for a bubble confined between parallel plates (G6, G7) and occurs for the enclosed tube relatively close to the bubble axis. 

If T is based on volume-equivalent radius, rather than equatorial radius as used here, E has almost no effect on the trajectory for prolate spheroids (LI). However, this definition of t obscures the effect of shape for oblate particles. 

I or the prolate spheroid a similar, but opposite, effect is evident the first extinction maximum decreases and shifts to larger values of x with increasing... 

Microwave ( = 3 cm) extinction measurements for beams incident parallel ( = 0°) and perpendicular (f = 90°) to the symmetry axis of prolate spheroids... 

Figure 11.23 Measured extinction of microwave radiation by prolate spheroids. From Greenberg et al. (1961).

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