Ellipsoid of revolution prolate

The intrinsic viscosity of poly(7-benzyl-L-glutamate) (Mq = 219) shows such a strong molecular weight dependence in dimethyl formamide that the polymer was suspected to exist as a helix which approximates a prolate ellipsoid of revolution in its hydrodynamic behaviorf ... 

R/Ro)soiv(f/fo)ellip = n + (mib/m2)(P2/Pi)] (f/fo)eiiip-Briefly justify this expansion of the (f/fo oiv factor. Assuming these particles were solvated to the extent of 0.26 g water (g protein)", calculate (f/fo)eiHp-For prolate ellipsoids of revolution (b/a < 1), Perrin has derived the following expression ... 

Observed properties of many nuclei have been interpreted as showing that the nuclei are not spherical but are permanently deformed (4). The principal ranges of deformation are neutron numbers 90 to 116 and 140 to 156. Most of the deformed nuclei are described as prolate ellipsoids of revolution, with major radii 20 to 40 percent larger than the minor radii. 

Once a general conformation type or preliminary classification has been established it is possible to use sedimentation data to obtain more detailed information about polysaccharide conformation. For example, the low value of ks/[v 0 25 found for the bacterial polysaccharide xylinan has been considered to be due to asymmetry [115]. If we then assume a rigid structure the approximate theory of Rowe [36,37] can be applied in terms of a prolate ellipsoid of revolution to estimate the aspect ratio p L/d for a rod, where L is the rod length and d is its diameter) 80. 

Perrin s theory (108) for prolate ellipsoids of revolution gives for thin rigid rods of L/d> 10... 

FIG. 4.12 Viscosity of dispersions of some nonspherical particles (a) intrinsic viscosity as a function of the axial ratio a/b for oblate and prolate ellipsoids of revolution according to the Simha theory (redrawn with permission of Hiemenz 1984) (b) experimental values of relative viscosity versus volume fraction for tobacco mosaic virus particles of different a/b ratios (data from M. A. Lauffer, J. Am. Chem. Soc., 66, 1188 (1944)). 

Atoms and ions with noble-gaS electron configurations have usually been described as having spherical symmetiy. For some considerations this description is satisfactory for others, however, it is advantageous to consider the atoms or ions to have a shape other than spherical—the helium atom can be described as deformed to a prolate ellipsoid of revolution, and the neon atom and other noble-gas atoms as deformed to a shape with cubic symmetry. 

Finally, the dotted curve in Fig. 13 traces the relation between v and vs for rigid prolate ellipsoids of revolution [see Peterlin (16) or Frisch and Simha (6 )] with axial ratio proportional to molecular weight. This curve lies very far from those for flexible molecules except for very low values of the axial ratio p. This seems to exhaust the available information of the type represented by Fig. 13. In connection with the behavior of DNA and perhaps other naturally occurring macromolecules, it would be interesting to have calculations for rods with one or two or at most a small number of flexible joints, such as might correspond to almost completely helical structures [see Section III D]. In spite of the absence of theories for this and possibly other relevant molecular models, it is often possible to arrive at useful indications of conformation by comparing the experimental data with Fig. 13. 

If we consider the microstructural changes to the particles, it is clear that the spherical particles will alter their shape with time to that of a prolate ellipsoid of revolution by tins transfer of material finm the spherical surface to that of the neck. In fact, the distance between the centers of the particles is not affected by vapor phase transport and only the shape of the pores is changed. Without a decrease in the center-to-center distance there is no densification or sintering. Therefore, this is one example of coarsening during the initial stage sintering. 

The overall shape of the RAGl zinc-binding dimerization domain is elongated as modelled by a prolate ellipsoid of revolution. Both sedimentation velocity and small-angle x-ray scattering experiments yielded axial ratios consistent with an extended molecule in solution for the ZDD dimer. 

In the remainder of this section, we outline results for two types of problems for which internal distributions of singularities have been used to advantage. The first, originally pursued by Chwang and Wu,13 considers bodies of very simple shape - spheres, prolate ellipsoids of revolution (spheroids), and similar cases for which exact solutions can be obtained for some flows either by a point or line distribution involving only a few singularities. The second class of problems is for very slender bodies for which an approximate solution can be obtained by means of a distribution of stokeslets along the particle centerline.14... 

The thermal boundary-layer equation, (9-257), also apphes for axisymmetric bodies. One example that we have already considered is a sphere. However, we can consider the thermal boundary layer on any body of revolution. A number of orthogonal coordinate systems have been developed that have the surface of a body of revolution as a coordinate surface. Among these are prolate spheroidal (for a prolate ellipsoid of revolution), oblate spheroidal (for an oblate ellipsoid of revolution), bipolar, toroidal, paraboloidal, and others.22 These are all characterized by having h2 = h2(qx, q2), and either h2/hx = 1 or h2/hx = 1 + 0(Pe 1/3) (assuming that the surface of the body corresponds to q2 = 1). Hence the thermal boundary-layer equation takes the form... 

Figure 2.5. Flow past oblate and prolate ellipsoids of revolution...

Prolate ellipsoid of revolution. To solve the corresponding problem about an ellipsoidal particle (on the right in Figure 2.5) in a translational Stokes flow, we use the reference frame cr, r, fixed to the prolate ellipsoid of revolution. The transformation to the coordinates (o, r, ip) is determined by the formulas... 

In the problem about a prolate ellipsoid of revolution moving with velocity Ui in a stagnant fluid, the corresponding stream function has the form... 

Figure 4.1. Shape factor ratio against perimeter-equivalent factor for particles of various shape in a stagnant medium 1, circular cylinder 2, oblate ellipsoid of revolution 3, prolate ellipsoid of revolution 4, cube...

Rotational Brownian motion results in the disordering of anisometric particles previously oriented in some particular way owing to the flow of the dispersion medium (see Chapter IX) or the application of an electric field. From the time of the disordering one can determine the rotational diffusion coefficient, and, for known particle size and shape, also Avogadro s number. If the particles are able to undergo co-orientation, they usually are of substantially anisometric shape, and their translational and rotational diffusion coefficients differ from those obtained for spherical particles. For example, for prolate ellipsoids of revolution with a ratio of their main diameters d] d2-= 1 10, the diffusion coefficient, D, is about 2/3 of the value obtained for spherical particles of the same volume. 

Figure 3. The muscle calcium binding protein molecule has the general shape of a prolate ellipsoid of revolution (59). The shell, 2.7-A thick, contains those atoms, exclusive of hydrogen, that would be exposed to the solvent were there no surface indentations. The oblate ellipsoid hydrocarbon core consists of side chains of phenylalanine, leucine, isoleucine, and valine.

It is useful to visualize how the three correlation times depend on the axial ratia This dependence is shown in Figure 12.7 for a prolate ellipsoid of revolution. The dio-... 

The disk can be regarded as a flattened sphere oblate ellipsoid of revolution) and the needle as a stretched sphere (prolate ellipsoid of revolution). A spheroid is the same as an ellipsoid of revolution. 

Simha provided an equation for the viscosities of ellipsoids of revolution. The prolate ellipsoids of revolution are cigar-shaped while the oblate ellipsoids of revolution are disc-shaped (see Figure 5.3). According to derivations by Einstein and later by R. Simha,... 

For the motion of a prolate ellipsoid of revolution transverse to the polar axis, as shown in Fig. 7b... 

Several authors have extended the calculations of EiksteiK to non-spherical parti-cies The models that have been used are oblate or prolate ellipsoids of revolution long cylinders or a stiff row of spheres K The calculations are more difficult than in the case of spherical particles, not only in the mathematical sense, but also on account of two new physical aspects of the problem. It will be evident that the contribution to the viscosity of a long or flat particle depends upon its orientation. This orientation is constantly modified by the toppling over of the particles in the field of shear and by the rotational BROWNian motion of the particles. The viscosity is therefore dependent upon the rate of shear. At low rates of shear, the BROWNian motion prevails and the orientation of the particles is completely at random. At high shear, however, or for large particles, the BROWNian motion is negligible and the orientation completely determined by hydrodynamics. 

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