Particles oblate

We designate the length of the ellipsoid along the axis of rotation as 2a and the equatorial diameter as 2b to define the axial ratio a/b which characterizes the ellipticity of the particle. By this definition, a/b > 1 corresponds to prolate ellipsoids, and a/b < 1 to oblate ellipsoids. 

A system of particles interacting in this way was studied using a microcanonical ensemble at a scaled density of 3.0 which is close to the transitional density for hard oblate ellipsoids with the same ellipticity (see Fig. 3). At a scaled density of 3.0 the system is found to exhibit isotropic. 

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. 

Uniaxial deformations give prolate (needle-shaped) ellipsoids, and biaxial deformations give oblate (disc-shaped) ellipsoids [220,221], Prolate particles can be thought of as a conceptual bridge between the roughly spherical particles used to reinforce elastomers and the long fibers frequently used for this purpose in thermoplastics and thermosets. Similarly, oblate particles can be considered as analogues of the much-studied clay platelets used to reinforce a variety of materials [70-73], but with dimensions that are controllable. In the case of non-spherical particles, their orientations are also of considerable importance. One interest here is the anisotropic reinforcements such particles provide, and there have been simulations to better understand the mechanical properties of such composites [86,222],... 

AV the Fermi sea has a prolate shape for the majority spin particles, while an oblate shape for the minority spin particles. 

Equation (3-33) shows how the inertia term changes the pressure distribution at the surface of a rigid particle. The same general conclusion applies for fluid spheres, so that the normal stress boundary condition, Eq. (3-6), is no longer satisfied. As a result, increasing Re causes a fluid particle to distort towards an oblate ellipsoidal shape (Tl). The onset of deformation of fluid particles is discussed in Chapter 7. 

As a general guide, is usually less than C2 for a prolate particle, so that 6 < cj) and the direction of motion is between the axis and the vertical. On the other hand, an oblate body usually has > C2 so that the direction of fall is between the vertical and the equator. The settling velocity follows from Eq. (4-9) ... 

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). 

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. 

Since most irregular particles of practical concern tend to be oblate, lenticular, or rod-like with moderate aspect ratio, these comparisons generally support Heywood s approach. Combining this observation with the fact that the volumetric shape factor is more readily determined than sphericity, we conclude that Heywood s approach is preferred for the intermediate range. For convenience in estimating Uj, Table 6.4 gives correlations, fitted to Heywood s values, for 0.1 < k < 0.4 at specific values of Since is relatively insensitive to interpolation for at other values of is straightforward. In common with Heywood s tabulated values, the correlations in Table 6.4 do not extrapolate to = 1 for a sphere k = 0.524). 

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. 

For nonspherical particles the only theoretical treatment available is for potential flow around a spheroid (LIO). For an oblate spheroid the area-free... 

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. 

The biggest difference between biological particles and ceramic particles in the application of Eq. (4.20) is that while most ceramic particles are spherical ( Ch = 2.5), most biological particles can be modeled as either prolate ellipsoids or oblate spheroids (or ellipsoids). Ellipsoids are characterized according to their shape factor, ajb, for which a and b are the dimensions of the semimajor and semiminor axes, respectively (see Eigure 4.17). In a prolate ellipsoid, a > b, whereas in an oblate ellipsoid, b > a.ln the extremes, b approximates a cylinder, and b a approximates a disk, or platelet. 

The hydrodynamic shape factor and axial ratio are related (see Eigure 4.18), but are not generally used interchangeably in the literature. The axial ratio is used almost exclusively to characterize the shape of biological particles, so this is what we will utilize here. As the ellipsoidal particle becomes less and less spherical, the viscosity deviates further and further from the Einstein equation (see Eigure 4.19). Note that in the limit of a = b, both the prolate and oblate ellipsoid give an intrinsic viscosity of 2.5, as predicted for spheres by the Einstein equation. 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

Figure 11.15 shows Asano s calculations of extinction by nonabsorbing spheroids for an incident beam parallel to the symmetry axis, which is the major axis for prolate and the minor axis for oblate spheroids. Because of axial symmetry extinction in this instance is independent of polarization. Calculations of the scattering efficiency Qsca, defined as the scattering cross section divided by the particle s cross-sectional area projected onto a plane normal to the incident beam, are shown for various degrees of elongation specified by the ratio of the major to minor axes (a/b) the size parameter x = 2ira/ is determined by the semimajor axis a. 

Several examples of scattering by spherical and by nonspherical particles are collected in Fig. 13.8 calculations for randomly oriented prolate and oblate spheroids measured scattering of microwave radiation by a polydispersion of nonspherical particles and measured scattering of visible light by irregular... 

Calculated and measured values of P = —Sn/Su, the degree of linear polarization, for several nonspherical particles are shown in Fig. 13.9. The prolate and oblate spheroids, cubes, and irregular quartz particles have made their appearance already (Fig. 13.8) a new addition is NaCl cubes. Also shown are calculations for equivalent spheres. 

Noctilucent cloud particles are now generally believed to be ice, although more by default—no serious competitor is still in the running—than because of direct evidence. The degree of linear polarization of visible light scattered by Rayleigh ellipsoids of ice is nearly independent of shape. This follows from (5.52) and (5.54) if the refractive index is 1.305, then P(90°) is 1.0 for spheres, 0.97 for prolate spheroids, and 0.94 for oblate spheroids. 

It is not surprising, however, that Mie theory is inadequate in this instance the indium particles are not spheres, they are more nearly oblate spheroids with (average) major and minor diameters of about 1390 and 368 A. 

The absorption spectrum of isolated indium spheres differs from that of closely packed oblate spheroids in that the peak shifts from 2230 A to 4100 A about half of this shift is attributable to particle shape and half to particle interaction. Indium particles on immunological slides are not identical, however, but are distributed in size and shape about some mean this tends to broaden the spectrum. 

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