Spheroids and Disks

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). 

A number of authors have measured the drag on disks (Jl, Kl, L5, P5, R5, S2, S5, S8, W4, W5). For supported disks with steady motion parallel to the axis, numerical and experimental results at low and intermediate Re are well correlated (P5) by  

Data are scant for spheroids other than disks and spheres. Experimental results for axisymmetric flow outside the Stokes range appear to be limited to 

Hence A — 2.63 for a disk. The results of List and Dussault (L3) are interpolated from wind-tunnel measurements on approximately spheroidal hailstone models (L2) while those of List et al (L4) are for true spheroids in a wind 

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This theorem could have been used to obtain the drag for fluid and solid spheres in Chapter 3. Explicit analytic solutions are available for bodies whose boundaries are easily described in relatively simple coordinate systems. Results for spheroids and disks (Ol, P3, SI) are discussed below. Solutions are also available for lenses and hemispheres (P3), hollow spherical caps (D3, C3, P3), toroids (P4), long spindles or needles (P5), and pairs of identical spheres (S7). 

Fig. 6.2 Wake lengths for spheroids and disks. Numerical predictions for spheroids (M4, P5) flow visualization for disks (K2).

Fig. 6.11 Correlations and numerical calculations for heat transfer to spheroids and disks with Pr = 0.7.

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. 

For axisymmetric flow at higher Re the most reliable data are those of Beg for the sublimation of oblate naphthalene spheroids (B4) (0.25 < E < 1) and disks (B3). His correlations are in terms of the characteristic length L defined... 

The dimensionless shape factor for the right circular cylinder is in very close agreement with the values for the oblate spheroid in the range 0 < Lid < 1 and with the values for the prolate spheroid in the range 1 < Lid < 8. The difference when Ud = 1 is less than 1 percent. This shows that the results for the sphere and a finite circular cylinder of unit aspect ratio are very close. The simple expression obtained from the Smythe solution can be used to estimate the shape factors of circular disks, oblate spheroids, and prolate spheroids in the range 0 < Ltd < 8. For Ud > 8, the prolate spheroid asymptotic result can be used to provide accurate results for long circular cylinders and other equivalent bodies. 

External transient conduction from an isothermal convex body into a surrounding space has been solved numerically (Yovanovich et al. [149]) for several axisymmetric bodies circular disks, oblate and prolate spheroids, and cuboids such as square disks, cubes, and tall square cuboids (Fig. 3.10). The sphere has a complete analytical solution [11] that is applicable for all dimensionless times Fovr = all A. The dimensionless instantaneous heat transfer rate is QVa = Q AI(kAQn), where k is the thermal conductivity of the surrounding space, A is the total area of the convex body, and 0O = T0 - T, is the temperature excess of the body relative to the initial temperature of the surrounding space. The analytical solution for the sphere is given by... 

E. J. Normington and J. H. Blackwell, Transient Heat Flow From Constant Temperature Spheroids and the Thin Circular Disk, Quarterly Journal of Mechanics and Applied Mathematics Vol. 17, pp. 65-72,1964. 

M. M. Yovanovich, General Conduction Resistance for Spheroids, Cavities, Disks, Spheroidal and Cylindrical Shells, AIAA 77-742, AIAA 12th Thermophysics Conference, Albuquerque, New Mexico, June 27-29,1977. 

The most metal-rich stars in dwarf spheroidals (dSph) have been shown to have significantly lower even-Z abundance ratios than stars of similar metallicity in the Milky Way (MW). In addition, the most metal-rich dSph stars are dominated by an s-process abundance pattern in comparison to stars of similar metallicity in the MW. This has been interpreted as excessive contamination by Type la super-novae (SN) and asymptotic giant branch (AGB) stars ( Bonifacio et al. 2000, Shetrone et al. 2001, Smecker-Hane McWilliam 2002). By comparing these results to MW chemical evolution, Lanfranchi Matteucci (2003) conclude that the dSph galaxies have had a slower star formation rate than the MW (Lanfranchi Matteucci 2003). This slow star formation, when combined with an efficient galactic wind, allows the contribution of Type la SN and AGB stars to be incorporated into the ISM before the Type II SN can bring the metallicity up to MW thick disk metallicities. 

Thermotropic liquid crystalline (LC) phases or mesophases are usually formed by rod-like (calamitic) or disk-like (discotic) molecules. Spheroidal dendrimers are therefore incapable of forming mesophases unless they are flexible, because this would allow them to deform and subsequently line up in a common orientation. However, poly(ethyleneimine) dendrimers were reported to exhibit lyotropic liquid crystalline properties as early as 1988 [123],... 

Some simple examples may help to clarify these classes of symmetry. Circular cylinders, disks, and spheroids are axisymmetric and orthotropic cones are axisymmetric but not orthotropic none of these are strictly spherically isotropic. Parallelepipeds are orthotropic, but the cube is the only spherically isotropic parallelepiped. Regular octahedra and tetrahedra are spherically isotropic octahedra are orthotropic whereas tetrahedra are not. 

Spheroidal particles can be treated analytically, and allow study of shapes ranging from slightly deformed spheres to disks and needles. Moreover, a spheroid often provides a useful approximation for the drag on a less regular... 

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. 

Different eorrelations are required for three-dimensional bodies (spheres, disks, and spheroids) than for the two-dimensional shapes (cylinders and wedges). For three-dimensional shapes transfer in the aft region is correlated by... 

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