Cylinders and disks

Au nanorods (NRs) are particularly useful for biological applications since their near-field and far-field optical properties can be tuned controllably throughout the NIR water window by varying their geometry [34], Metallic nanorods and nanowires can have enhanced electric fields at the tips compared to spheres, which would lead to increased SERS activity [1, 35-38]. Moreover it has been shown that Ag cylinders or nanorods exhibit extraordinary electrocataljAic activities towards the electro-reduction process, especially in the case of benzyl chloride [29]. 

Nanorods can be modeled as cylinders in Fig. 3.5 the optical response of Ag cylinders excited with a plane-wave polarized 

By moving from nanorods to nanodisks (see Fig. 3.6), for which s 1 being h d, a strong red-shift of the resonances appears in the spectra by increasing the dimensions of the base (d). This is due to retardation effects experienced by radiation inside a scatterer by enlarging its dimensions along the polarization direction. Moreover for a thin disk (h = 10 nm) only the dipolar resonance (/ = 1) seems to be generated by an in-plane polarized incident field. 

Of course, an average of the target orientations with respect to the incident field should produce richer spectra an out-plane polarization should excite also longitudinal modes (along h direction) and this in agreement with the experimental measurements already done on solutions of this kind of nanoparticles [30,39]. 

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Figure 11-2 Drag coefficient for spheres, cylinders, and disks. (From Perry, 1984.) o Eq. (11-5), spheres. Eq. (11-7), cylinders.

Fig. 23. Position dependence of the effective collective diffusion constant normalized by the collective diffusion constant of spherical gels. D0 = (K + 4ji/3)/f. At the boundary, the values for sphere, cylinder, and disk are 1, 2/3, and 1/3, respectively...

Individual diatom frustules are porous. The diatoms are highly variable in shape and size, having particles that range in effective diameter from 0.75 to 1000 pm, but most are 50 to 100 pm in diameter. Diatom shapes can range from simple cylinders and disks to complex, highly variable, but always punclaie, forms. 

In Chapters 1 and 2 we study fluid flows, which underlie numerous processes of chemical engineering science. We present up-to-date results about translational and shear flows past particles, drops, and bubbles of various shapes at a wide range of Reynolds numbers. Single particles and systems of particles are considered. Film and jet flows, fluid flows through tubes and channels of various shapes, and flow past plates, cylinders, and disks are examined. 

DRAG COEFFICIENTS OF TYPICAL SHAPES. In Fig. 7.3, curves of versus JVrc p are shown for spheres, long cylinders, and disks. The axis of the cylinder and the face of the disk are perpendicular to the direction of flow these curves apply only when this orientation is maintained. If, for example, a disk or cylinder is moving by gravity or centrifugal force through a quiescent fluid, it will twist and turn as it moves freely through the fluid. 

Variations in particle shape can be accounted for by obtaining separate curves of Cjj versus p for each shape, as shown in Fig. 7.3 for cylinders and disks. As pointed out earlier, however, the curves for cylinders and disks in Fig. 

Certain macromolecular species are modeled more accurately as a cylinder rather than an ellipsoid. Examples include short DNA fragments and rodlike viruses such as tobacco mosaic virus (TMV). Extensive theoretical studies of the hydrodynamic properties of cylindrical particles have been carried out. These have been recently reviewed by Ortega and de la Torre [2003], who have extended prior calculations using the bead-shell model to short cylinders and disks. Eor long cylinders of length L and cross-sectional diameter d, and hence axial ratio J=Lid, such analysis leads to... 

Ellipsoids, Cylinders, and Disks. Eor ellipsoids, the solution for a sphere... 

B Flow Past Sphere, Long Cylinder, and Disk... 

For each particular shape of object and orientation of the object with the direction of flow, a different relation of Cp versus exists. Correlations of drag coefficient versus Reynolds number are shown in Fig. 3.1-2 for spheres, long cylinders, and disks. The face of the disk and the axis of the cylinder are perpendicular to the direction of flow. These curves have been determined experimentally. However, in the laminar region for low Reynolds numbers less than about 1.0, the experimental drag force for a sphere is the same as the theoretical Stokes law equation as follows. 

Figure 3. F2. Drag coejficients for flow past immersed spheres, long cylinders, and disks. (Reprinted with permission from C. E. Lapple and C. B. Shepherd, Ind. Eng. Chem., 32, 606 (1940). Copyright by the American Chemical Society.)...

The initial porosity of the nonspherical particles can be approximated by the proper use of the packing results of cylinders and disks shown in Eqs. (31) through (34) and expressed as... 

Electrowinning for recovering a metal M, such as Cu, Ni, Zn, Ti, Pb, etc. from their low-grade ores. The metal is recovered by electrodeposition in the form of cathodes, which are normally rectangular shapes (plates). If rotating cylinders and disks are used, the final electrodeposition product can be in the form of ribbons and powders, respectively. 

Other hydrodynamic cases for corrosion studies can be found elsewhere [10]. On the other hand, for a continuous metal removal from solution in electrowinning, rotating cylinders and disks are used as cathodes (Figme 7.6). The classical rotating-disk cathode is known as Weber s disk [25-26] having a diameter of 2a. The corresponding differential equation in cylindrical coordinates is free of the diffusivity and it is given by... 

Zhu J, Zhang S, Zhang K, Wang X, Mays JW, Wooley KL (2013) Disk-cylinder and disk-sphere nanoparticles via a block copolymer blend solution constiuction. Nat Commun 4 2297... 

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