Finite aspect ratio

For prolate spheroids, Eq. (4-37) with k — 0.5 again agrees with the limiting exact result for -> oo. The validity of these equations for cylinders is demonstrated in Figs. 4.7 and 4.8. Comparison of Eqs. (4-36) and (4-37) shows that the ratio of C2 to cq tends to 2 as -> 00. This result holds for any axisymmetric particle, while cq < 2c for finite aspect ratios (W2). Consequently a needlelike particle falls twice as fast when oriented vertically at low Re than when its axis is horizontal. 

Thus, P oo for either p —> oo, which is the limit of an infinitely thin prolate ellipsoid, or p — 0, which is the limit of an infinitely flat oblate one. In these two limits, the particle rotates until a long axis is parallel to the flow direction, and then rotation slows to a halt. For large, but finite, aspect ratios, the particle rotates slowly when its long axis is nearly parallel to the flow direction, and rapidly otherwise. The Jeffery orbits of rod-like and disklike particles have been observed directly (Anczurowski and Mason 1967a, 1967b) and indirectly by optical dichroism (Frattini and Fuller 1986). 

In fact, the fiber contribution to the shear viscosity of a fiber suspension at steady state is modest, at most. The reason is that, without Brownian motion, the fibers quickly rotate in a shear flow until they come to the flow direction in this orientation they contribute little to the viscosity. Of course, the finite aspect ratio of a fiber causes it to occasionally flip through an angle of n in its Jeffery orbit, during which it dissipates energy and contributes more substantially to the viscosity. The contribution of these rotations to the shear viscosity is proportional to the ensemble- or time-averaged quantity (u u ), where is the component of fiber orientation in the flow direction and Uy is the component in the shear gradient direction. Figure 6-21 shows as a function of vL for rods of aspect... 

Therefore, the unified model employs two empirical parameters a and P) and two known parameters b = -I and cTo- Although this empiricism is not desired, the a value in varies from zero in the slip flow regime to an order-one value of o as oo. Finally, the model is adapted to the finite aspect ratio rectangular ducts using a standard aspect ratio correction given in Eq. (7). 

These equations are suitable for single calculation and were employed previously for the single ply and angle ply properties. The short fiber composite properties are also given by the Halpin-Tsai equations where the moduli in the fiber orientation direction is a sensitive function of aspect ratio (1/d) at small aspect ratios and has the same properties of a continuous fiber composite at large but finite aspect ratios. 

The first term in this expression, which corresponds to the crystalline bridge sequences, is next treated as an array of short fibres, so introducing the shear lag (efficiency) factor d>, which is a function of the finite aspect ratio of the crystalline bridges. The analogous equation to Equation (8.20) is... 

The flat film process, in which an extruded sheet is drawn up on a roll, is a two-dimensional analog of melt spinning, and the dynamical equations and instabilities are essentially the same. The finite aspect ratio of the film die and the presence of an edge on the extruded sheet introduce stresses that cause necking in of the sheet. This is a steady-state phenomenon that does not seem to have major dynamical implications. 

I>, which is a function of the finite aspect ratio of the crystalline bridges. The analogous equation to Equation (9.22) is... 

Choice of Aspect Ratio. Because of the finite aspect ratio of the capillary there are velocity gradients in the a and b directions. To determine a certain viscosity coefficient, the influence of the gradient in the b direction should be negligible. For a given aspect ratio this influence is large in the case of 773, where the gradient in the b direction affects the measurement via the usually large viscosity coefficient Pj. There is no influence of other viscosity coefficients in the case of ri2- The apparent viscosity coefficients can be extrapolated to an infinite aspect ratio [10]. For example, in the case of 771 the correction amounts to 4% for b/a = 8 and 77,/t73=4. 

White et al recently conducted simulation and experimental studies to investigate the L/D dependence of 4>c in polymer composites with finite-aspect ratio silver nanowire fillers. Their simulation program determines the percolation threshold for very large systems of isotropic soft-core cylinders (comprising 10 -10 rods) with aspect ratio ranging between 5 and 80. The authors then compared the simulated thresholds to analytical results from the excluded volume theory for inter-penetrable cylinders (eqn [7]), as well as experimental results from their silver nanowire/polystyrene composites (Figure 5j. 

Of much greater relevance in micro reactors are rectangular channels, which were the subject of a study by Cheng et al. [110], among others. They solved the Navier-Stokes equation for channel cross-sections with an aspect ratio between 0.5 and 5 and Dean numbers between 5 and 715 using a finite-difference method. The vortex patterns obtained as a result of their computations are depicted in Figure 2.20 for two different Dean numbers. 

Fig. 5a-c Principal stress values for rectangular bars loaded along their length in tension by forces applied at the four corners. (Finite element calculations in plane strain), a Isotropic material aspect ratio 5 b Isotropic material aspect ratio 10 c Anisotropic material (E/G),n = 15.1, aspect ratio 10 (Note the figures are not drawn to scale). (From Ref. 16)... 

Fig. 6. Tensile strain along direction of loading at surface (top) and at centre (bottom) in a rectangular bar loaded along its length by forces at its four comers. (Finite element calculations in plane strain). Anisotropic material (E/G)1 2 = 15.1. Aspect ratio 40/3...

Fig. 6. Sherwood numbers computed for the transport of finite particles to a spherical collector under the combined action of convective-diffusion and London forces. Values of the aspect ratio are (a) ft - 104, (b) ft — 105, (c) ft =- 10s, and (d) ft = 10. For each aspect ratio, the value of A/kT was taken (upper curves to lower curves) as ID2, 1, 10 2 and 10 4. Dashed lines represent the Levich-Lighthill equation (19), while the dotted curves represent Sherwood numbers deduced from Figure 4 which ignores the transport from diffusion.

No attempt has been made to discuss, in a comprehensive manner, models which are based on finite element calculations or other numerical analyses. Only some results of Schmauder and McMeeking10 for transverse creep of power-law materials were discussed. The main reason that such analyses were, in general, omitted, is that they tend to be in the literature for a small number of specific problems and little has been done to provide comprehensive results for the range of parameters which would be technologically interesting, i.e., volume fractions of reinforcements from zero to 60%, reinforcement aspect ratios from 1 to 106, etc. Attention in this chapter was restricted to cases where comprehensive results could be stated. In almost all cases, this means that only approximate models were available for use. 

Further examples of recent applications of the finite element method can be found in Targett et al. (1995) who studied flow through curved rectangular channels of large aspect ratio using the FEM-based FIDAP code. They also compared their computational results with experimental data obtained from visualization experiments and found good agreement between theory and experiment. 

Fig. 16. Finite element results for the velocity vector flow field within rectangular cavities, (a) Aspect ratio 1 1. (b) Aspect ratio 4 1. The arrow length is proportional to the velocity. (Figure and caption reprinted from Alkire and Deligianni [55] by permission of the publisher. The Electrochemical Society, Inc.).

In this paper we shall explain how it is possible to determine theoretically y, ) giving a few technical details concerning a recently introduced method which exploits Monte Carlo within a Finite Element Methods approach (MC-FEM). After applying such MC-FEM method to intermixed Ge pyramids of different aspect ratios, we shall show how the steepness of the island facets influences the SiGe distribution. 

Ge distribution. With this respect, the method is fully self-consistent. Further details on MC-FEM can be found in Refs. [9,10]. In Ref. [10] the method was also extended to treat entropic contributions at finite temperatures. Here, however, we shall focus on elastic-energy minimization only. Let us now apply the method to islands of different height-to-base aspect ratios, quantifying the effect of non-uniform concentration profiles on the elastic energy stored in 3D islands. 

Combination of several properties is becoming increasingly important in modem industry. One example may be taken from electronics, where in addition to mechanical properties and electric resistance, themial stability and conductivity are important requirements. It was estimated that the increase of temperature by 10°C reduces time to failure by the factor of two." A finite analysis model was developed which accounts for the following properties of filled composites microstructure, effect of particle shape, formation of conductive chains, effect of filler aspect ratio, and interfacial thermal resistance. The predictions of the model indicate the most... 

Nanotubes in the laboratory often exhibit an aspect ratio of 10 000, i.e., length L approximately in microns. From the fundamental perspective, an important stimulus of this research is the realization that such a linear geometry provided by small R nanotubes yields one-dimensional (ID) phases of matter that description is certainly true from the phase transition perspective (since only one dimension approaches infinity in the thermodynamic limit). The subject of ID matter has been studied as an academic problem for many years [17, 18]. An intriguing aspect of the subject is that no phase transitions occur in a strictly ID system at finite temperature (T). In the nanotube environment, however, ID lines of adsorbed molecules can interact with neighboring lines of molecules, resulting in a 3D transition at finite T. To this date, in fact, predictions have been made of ID, 2D, 3D, and even 4D phases of matter in this novel environment [19, 20]. All such regimes will be discussed, to some extent, in this chapter and Chapter 15. The rich variety of phenomena has made theoretical study both enjoyable and rewarding. 

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