Convex bodies

For any matrix A it is convenient to let AK represent the set of all points Ax for which xeK, and to define aK — (al)K for any scalar a. Then a converse to the above theorem, which also holds, can be stated as follows if K is a bounded, dosed, equilibrated, convex body, then the function... 

Since only bounded, closed, and equilibrated convex bodies come into consideration, these qualifications will be assumed hereafter. Clearly the origin is interior to any such convex body, flence it can be shown that if H and K are both such, then there exists a scalar k such that... 

Note The term domain refers to the smallest convex body that contains the molecule or particle in its average shape. 

Compressive measurements provide a means to determine specimen stiffness, Young s modulus of elasticity, strength at failure, stress at yield, and strain at yield. These measurements can be performed on samples such as soy milk gels (Kampf and Nussi-novitch, 1997) and apples (Lurie and Nussi-novitch, 1996). In the case of convex bodies, where Poisson s ratio is known, the Hertz model should be applied to the data in order to determine Young s modulus of elasticity (Mohsenin, 1970). It should also be noted that for biological materials, Young s modulus or the apparent elastic modulus is dependent on the rate at which a specimen is deformed. 

The mixing rules for the hard convex body part of the GvdW-EOS are given by theory [7]. The most significant parameter for the attractive part of the GvdW-EOS is the critical temperature. For the mixture, this is calculated for an equivalent substance as in the previous paper [2] ... 

Chen and Kreglewski [11] developed a perturbation equation based on Boublik s [12] hard-convex-body equation as the reference fluid equation. The repulsive pressure is Alder et al. s double-series polynomial with modified coefficients fitted to experimental argon data. The Boublik-Alder-Chen-Kreglewski (BACK) eos is... 

Let us consider a class of problems concerning transient heat exchange between convex bodies of various shape and the environment. At the initial time t - 0 the temperature is the same throughout the body and is equal to 2], and for t > 0 the temperature on the surface T of the body is maintained constant and is equal to Ts. The temperature distribution inside the body is described by the heat equation... 

For a translational Stokes flow past a convex body of revolution of sufficiently smooth shape with symmetry axis parallel to the flow, the error (in percent) in formula (4.10.9) for the mean Sherwood number can be approximately estimated as follows ... 

Shape factors of isothermal, three-dimensional convex bodies having complex shapes and small to large aspect ratios are of considerable interest for applications in the nuclear, aerospace, microelectronic, and telecommunication industries. The shape factor S also has applications in such diverse areas as antenna design, electron optics, electrostatics, fluid mechanics, and plasma dynamics [27]. 

Analytical solutions are available for a small number of geometries such as the family of geometries related to the ellipsoid (e.g., sphere, oblate and prolate spheroids, elliptical and circular disks). Precise numerical values of S for other axisymmetric convex bodies have been obtained by various numerical methods such as that proposed by Greenspan [27] and that proposed by Wang and Yovanovich [123]. 

The dimensionless geometric parameter VA/A is proposed as an alternate parameter for determination of shape factors of complex convex bodies. 

The three solutions corresponding to the three boundary conditions can be used to obtain approximate solutions for other convex bodies, such as a cube, for which there are no analytical solutions available. The dimensionless parameters Bi and Fo are defined with respect to the equivalent sphere radius, which is obtained by setting the surface area of the sphere equal to the surface area of the given body, i.e., a = Vi4/(4tt). This will be considered in the following section, which covers transient external conduction from isothermal convex bodies. 

Transient External Conduction From Isothermal Convex 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... 

M. M. Yovanovich, P. Teertstra, and J. R. Culham, Modeling Transient Conduction From Isothermal Convex Bodies of Arbitrary Shape, Journal of Thermophysics and Heat Transfer (9/3) 385-390,1995. 

A simplex in n dimensions is the convex body determined by n+1 vertices. Thus, in a 2-dimensional plane a simplex is a triangle determined by its three vertices. In 3D space a simplex is a tetrahedron determined by its 4 vertices, etc. The idea of Nelder and Mead, for an n-dimensional problem, is to start with n+1 points x, and geometrically transform and move the simplex they determine until a minimum in fix) has been reached. The main steps are outlined, in 2-dimensions, in Figure 8.1. 

The first two terms were derived from Boublik s (9) relation for the compressibility factor of hard convex bodies. Here a is a constant depending on the shape of the molecules ( = 1 for spherical molecules), for mixtures assumed to be am = ... 

Pq cannot be calculated exactly except in very special cases or by Monte Carlo methods. In order to find useful approximations, we follow the procedure of Rothenstein [17]. The escape probability for a single convex body can be written as (see 10.1 of [11])... 

Molecules are not hard spheres and other equations of the same general form as eq 5.179 have been proposed for molecules of various geometrical shapes. For example, hard convex bodies Boublik gives... 

In 1950 Isihara proved a geometrical identity for convex bodies and pointed out the fact that the identity is useful for treatment of molecular interactions in uniform gases and liquid solutions. His work was improved and generalized by the present author. - For spherically symmetric molecules the intermolecular potential is a function of the distance between the molecular centers. The above-mentioned generalization was based on the supposition that the intermolecular potential was a function of the shortest distance between convex cores assumed within the molecules. On this supposition a useful expression for the second virial... 

The works mentioned were based on molecular models to which the geometry of convex bodies could be applied. Besides these geometrical treatments thermodynamic and transport properties of nonspherical molecules were investigated by Comer, by Pople, by Castle, Jansen and Dawson, by Kihara, Midzuno and Kaneko, by Balescu, and by Curtiss. Comer said in his paper, "I have tried to find an intermolecular potential which satisfies the three conditions of accuracy, generality, and integra-bility. It is the purpose of the present article to show that the three conditions of accuracy, generality, and integrability are satisfied in a harmonious manner by the convex-core model of intermolecular potential. 

Let us choose a coordinate origin 0 inside a (fixed) convex body and take coordinates of polar angles 6 and 95 (0 6 tt, 0 99 2tt). For any 0, contact with the convex body and whose normal from the origin is in the direction This plane is named the supporting plane in the... 

We shall later see that M is the mean curvature integrated over the whole surface of the convex body. 

Let us first assume that the convex body has a smooth surface and that each supporting plane has a contact of first order with the convex body. Let T 0,(p) be the radius vector from the origin to the contact point of the body with the supporting plane in the direction 0,(p) compare Fig. 1 (a). Then by use of the unit vector u(0,99) in the direction (6,99) w e have... 

The surface area S and the volume V of the convex body are given by... 

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