Spheroids oblate

Oblate Spheroid (formed by the rotation of an ellipse about its minor axis [ ]) Data as given previously. 

Agrees with cylinder and oblate spheroid results, 15%. Assumes molecular (iffusion and natural convection are negligible. 

Differences based on ends of extraction column 100 measured values 2% deviation. Based on area oblate spheroid. 

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. 

If this equality is valid, then the oblate spheroid is indeed a figure of equilibrium. As follows from Equation (2.342) the frequency is defined as... 

Oblate spheroid with very small eccentricity... 

Suppose that we deal with a slightly oblate spheroid, ei < 1. Then, making use of an expansion... 

Investigations performed by Poincare and Lyapunov have shown that only slightly oblate spheroids are stable and for them... 

Ellipsoid If the base of a vessel is one-half of an oblate spheroid (the cross section fitting to a cylinder is a circle with radius of D/2 and the minor axis is smaller), then use the formulas for one-half of an oblate spheroid. 

A modified version of the TAB model, called dynamic drop breakup (DDB) model, has been used by Ibrahim et aU556l to study droplet distortion and breakup. The DDB model is based on the dynamics of the motion of the center of a half-drop mass. In the DDB model, a liquid droplet is assumed to be deformed by extensional flow from an initial spherical shape to an oblate spheroid of an ellipsoidal cross section. Mass conservation constraints are enforced as the droplet distorts. The model predictions agree well with the experimental results of Krzeczkowski. 311 ... 

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. 

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... 

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). 

As shown in Chapters 3 and 4, creeping flow analyses have little value for Re > 1. A number of workers (M4, M7, Mil, P5, R3) have obtained numerical solutions for intermediate Reynolds numbers with motion parallel to the axis of a spheroid. The most reliable results are those of Masliyah and Epstein (M4, M7) and Fitter et al (P5). Flow visualization has been reported for disks (K2, W5) and oblate spheroids (M5). 

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. 

No data are available for heat and mass transfer to or from disks or spheroids in free fall. When there is no secondary motion the correlations given above should apply to oblate spheroids and disks. For larger Re where secondary motion occurs, the equations given below for particles of arbitrary shape in free fall are recommended. 

Heywood gave drag curves for various values of k (H3), and tabulated the velocity correction factor (H2). Figure 6.15 shows plotted from Heywood s table. There is empirical evidence for the validity of this approach (Dl). As with sphericity, comparison for specific shapes is informative. For oblate spheroids (for which Ja is the equatorial diameter) and Re < 100,... 

In the intermediate regime it is recommended that the particle be treated as an oblate spheroid with major and minor axes determined from the particle... 

The conditions under which fluid particles adopt an ellipsoidal shape are outlined in Chapter 2 (see Fig. 2.5). In most systems, bubbles and drops in the intermediate size range d typically between 1 and 15 mm) lie in this regime. However, bubbles and drops in systems of high Morton number are never ellipsoidal. Ellipsoidal fluid particles can often be approximated as oblate spheroids with vertical axes of symmetry, but this approximation is not always reliable. Bubbles and drops in this regime often lack fore-and-aft symmetry, and show shape oscillations. 

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