Ellipsoidal bodies

Understandably, it is much more common to see analyses of problems based on Eq. (32) since for simple geometries the solution can be written down in closed form, expressed in terms of simple functions. For plane surfaces, for example, the solutions are elementary hyperbolic functions while for an isolated spherical surface the Debye-Huckel potential expression prevails. For two charged spherical surfaces the general solution can be written down as a convergent infinite series of Legendre polynomials [16-19]. The series is normally truncated for calculation purposes [16] K For an ellipsoidal body ellipsoidal harmonics are the natural choice for a series representation [20]. (The nonlinear Poisson Boltzmann equation has been solved numerically for a ellipsoidal body... 

The smaller spheroidal and ellipsoidal bodies of type E, 4-8 jttm in size and preserved due to impregnation with carbonaceous material, fine-grained hematite, or greenalite, differ from the structures that have been described. These structures are found in black cherts, red jasper bands, and granule-containing greenalite cherts (see Fig. 34). The algal structures at the base of the Biwabik formation also contain unique filiform structures of type F their diameter is 1-2, sometimes up to 6 jam. 

It can be generally shown that usually, for any particle, one clearly defined direction of orientation with regard to the applied field exists (leaving aside completely symmetrical partides). This direction depends on the frequency of the field. More detailed quantitative evaluation requires explidt knowledge of E. The problem has been generally solved for ellipsoidal bodies. On the basis of the results obtained, most partides of practical interest can at least be discussed in first approximation. 

The original Gay-Beme potential forms a nematic phase and a Smectic B phase, which is more solid like than liquid like. Ellipsoidal bodies do usually not form smectic A phases because they can easily diffuse from one layer to another layer. However, if one increases the side by side attraction it becomes possible to form smectic A phases [6]. When one calculates transport coefficients very long simulation runs are required. Therefore one sometimes re-places the Lennard-Jones core by a purely repulsive 1/r core in order to decrease the range of the potential. Thereby one decreases the number of interactions, so that the simulations become faster. The Gay-Beme potential can be generalised to biaxial bodies by forming a string of oblate ellipsoids the axes of which are parallel to each other and perpendicular to the line joining their centres of mass [35]. One can also introduce an ellipsoidal core where the three axis are different [38]. 

The overall shape of the Gaussian chain is thus approximated by an ellipsoidal body with the lengths of its principal axes in the ratio of... 

If the ellipsoidal bodies are pressed together by a normal load (F), then an elliptical contact zone Is developed In the vicinity of the point of contact with semi-major and semi-minor axes denoted by (a,b) respectively. The principal. axes of this contact ellipse may be determined, according to the theory of Hertz (24), and the relevant expressions may be found In Hamrock and Dowson (25). 

Pokrovskii [112] demonstrated theoretically that concentrated suspensions of solid ellipsoidal bodies in a Newtonian fluid give rise to a viscoelastic behavior. He showed that for such suspensions it is possible to use the concept of transverse viscosity which expresses the effect of normal stresses and found that the transverse viscosity increases with velocity gradient. 

Ellipsoidal bodies, unique structures found only in the hyphae of lichen fungi, are especially prevalent in lichens from harsh, dry habitats such as deserts and alpine regions. These bodies have not been found in the hyphae of mycobionts grown in isolated culture except in one instance where the fungus was induced, by drying, to form reproductive structures. Ellipsoidal bodies were not found in Hydrothyria venosa, a freshwater lichen that is always inundated (Jacobs and Ahmadjian, 1973) (see Chapter 5). 

The totally symmetrical sphere is characterized by a single size parameter its radius. Ellipsoids of revolution are used to approximate the shape of unsymmetrical bodies. Ellipsoids of revolution are characterized by two size parameters. 

Let us represent the potential Uq in terms of the angular frequency, geometrical parameters of a body, and its mass M. As we know, at large distances from the ellipsoid the potential of the attraction field is described as... 

This group comprises bodies generated by rotating a closed curve around an axis. Spheroidal particles (also called ellipsoids of revolution) are of particular interest, since they correspond closely to the shapes adopted by many drops and... 

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. 

According to this authority, in our contemporary and already Cubist world, the differential equations of dynamics are characterized by varying transformations, so we must admit that all bodies become deformed, and that a sphere, for example, is transformed into an ellipsoid in which the minor axis is parallel to the translation of the axes. Time itself must be profoundly modified. This disturbing perception leads Poincare to a further conclusion ... 

Another complication is the demagnetization correction due to the geometry of the specimen. Demagnetization (or the equivalent depolarization problem for dielectric bodies in an electric field) can only be solved analytically for an ellipsoid of revolution (27X28). When He is applied parallel to one of the three axes of revolution, the magnetization is parallel to He, but the internal field H is given by (29) ... 

Perrin [223] extended Debye s theory of rotational relaxation to consider spheroids and ellipsoids. Using Edwards analysis [224] of the torque on such bodies, Perrin found two or three rotational relaxation times, respectively. However, except for bodies very far from spherical, these times are within a factor of two of the Debye rotational times [eqn. (108)] for stick boundary conditions. 

Stokes-Einstein Relationship. As was pointed out in the last section, diffusion coefficients may be related to the effective radius of a spherical particle through the translational frictional coefficient in the Stokes-Einstein equation. If the molecular density is also known, then a simple calculation will yield the molecular weight. Thus this method is in effect limited to hard body systems. This method has been extended for example by the work of Perrin (63) and Herzog, Illig, and Kudar (64) to include ellipsoids of revolution of semiaxes a, b, b, for prolate shapes and a, a, b for oblate shapes, where the frictional coefficient is expressed as a ratio with the frictional coefficient observed for a sphere of the same volume. 

Eq. (3.21) discussed in Section 3.3.2 is only valid if the motion of the molecules under study has no preferential orientation, i.e. is not anisotropic. Strictly speaking, this applies only for approximately spherical bodies such as adamantane. Even an ellipsoidal molecule like trans-decalin performs anisotropic motion in solution it will preferentially undergo rotation and translation such that it displaces as few as possible of the other molecules present. This anisotropic rotation during translation is described by the three diagonal components Rlt R2, and R3 of the rotational diffusion tensor. If the principal axes of this tensor coincide with those of the moment of inertia - as can frequently be assumed in practice - then Rl, R2, and R3 indicate the speed at which the molecule rotates about its three principal axes. 

Chapter 7. THE STRUCTURE OF THE NUCLEUS OF THE ATOM What exclaimed Roger, as Karen rolled over on the bed and rested her warm body against his. I know some nuclei are spherical and some are ellipsoidal, but where did you find out that some fluctuate in between ... 

Instead of an assumed contraction of fast moving objects, I have introduced the idea that the travelers spheres of observation by the velocity are transformed into ellipsoids of observation. One advantage is that this new concept is easier to visualize and that it makes possible a simple graphic derivation of distortions of time and space caused by relativistic velocities. Another advantage is that it is mentally easier to accept a deformation of spheres of observation than a real deformation of rigid bodies depending on the velocity of the observer. 

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