Cylinder arbitrary shape

Unfortunately, Maxwell s equations can be solved analytically for only a few simple canonical resonator structures, such as spheres (Stratton, 1997) and infinitely long cylinders of circular cross-sections (Jones, 1964). For arbitrary-shape microresonators, numerical solution is required, even in the 2-D formulation. Most 2-D methods and algorithms for the simulation of microresonator properties rely on the Effective Index (El) method to account for the planar microresonator finite thickness (Chin, 1994). The El method enables reducing the original 3-D problem to a pair of 2-D problems for transverse-electric and transverse-magnetic polarized modes and perform numerical calculations in the plane of the resonator. Here, the effective... 

Proof of boundedness of the force of interaction between two charged particles of an arbitrary shape in H3, held at a given distance from each other in an electrolyte solution, upon an infinite increase of the particle s charge. (It was shown in 2.2 that the repulsion force between parallel symmetrically charged cylinders saturates upon an infinite increase of the particle s charge. This is also true for infinite parallel charged plane interaction [9]. The appropriate result is expected to be true for particles of an arbitrary shape.)... 

In this connection, as commented in a footnote given under Table 24 in Section 9.5, it has been reported by J. Isler that the critical temperature for the thermal explosion of nitrocellulose powder (13.4 % N) formed into a cylinder, 1,1 cm ill diameter, 1,3 in specific gravity, and placed in the atmosphere under isothermal, but open, conditions, is 145 "C [75], This temperature value, 145 C, is far higher than 76" 77 °C, i.e., the value of SADT calculated herein for guncotton or collodion cotton, having an arbitrary shape and an arbitrary size, confined in an arbitrary closed container of the corresponding shape and size, and placed in the atmosphere under isothermal conditions (refer to the superscript, given in Table 24). 

The boundary-layer problem for the specific case of a circular cylinder is (10-40), (10 41), (10-43), and (10-47), with ue and 3p/dx given by (10-122) and (10-123). The first point to note is that a similarity solution does not exist for this problem. Furthermore, in view of the qualitative similarity of the pressure distributions for cylinders of arbitrary shape, it is obvious that similarity solutions do not exist for any problems of this general class. The Blasius series solution developed here is nothing more than a power-series approximation of the boundary-layer solution about x = 0. 

Although we are interested in the response of the cylinder to a perturbation of arbitrary shape, such a perturbation can be constructed as a Fourier sine series, and it is sufficient to look at the stability of a single mode for all possible values of k. If the shape-perturbation mode grows for any k, the system is unstable. Now, if / has the form (12-25), we see from the normal-stress condition (12-24) that p must be equal to... 

Cylinder of arbitrary shape. Let us consider mass exchange for cylindrical bodies of arbitrary shape in a uniform translational flow of viscous fluid at small... 

Figure 4.3. Heat exchange between a cylinder of an arbitrary shape and a translational flow (a) the original system of rectangular coordinates (b) the plane of the new variables ip, ip...

FIGURE 4.19 Definition sketch for long horizontal convex cylinder of arbitrary shape. The constants G, C2, and C, are for use in Eq. 4.48. 

H. Nakamura and Y. Asako, Laminar Free Convection From a Horizontal Cylinder With Uniform Cross-Section of Arbitrary Shape, Bull. JSME (21/153) 471-478,1978. 

Sonshine, R. M., Cox, R. G., and Brenner, H., The Stokes translation of a particle of arbitrary shape along the axis of a circular cylinder filled to a finite depth with viscous liquid I and II. Appl. Sci. Res., Ser. A (in press) see also Sonshine, R. M., The Stokes settling of one or more particles along the axis of finite and infinitely long circular cylinders. Ph.D. Dissertation, New York University, New York, 1966. 

In this equation S, and Vp are the external surface area and the volume of the particle and S, is the surface of the equivalent volume sphere = 1 for spheres, 0.874 for cylinders with height equal to the diameter, 0.39 for Raschig-rings, 0.37 for Berl saddles). extends the correlation to particles of arbitrary shape. The product l> dp is sometimes written as a diameter dj, ... 

Expansion work does not require a cylinder-and-piston device. Suppose the system is an isotropie fluid or solid phase, and various portions of its boundary undergo displaeements in different directions. Figure 3.5 shows an example of compression in a system of arbitrary shape. The deformation is eonsidered to be carried out slowly, so that the pressure p of the phase remains uniform. Consider the surface element t of the boundary, with area /4s,T, indicated in the figure by a short thick curve. Beeause the phase is isotropic, the force = pA z exerted by the system pressure on the surroundings is perpendicular to this surface element that is, there is no shearing force. The force exerted by the... 

The book by Clift et al. (1978) contains an extensive review on this subject. The treatment is, however, mostly for the axisymmetric particles such as spheroids and cylinders and orthotropic particles such as rectangular parallelepipeds. For particles of arbitrary shape. 

The radiation and temperature dependent mechanical properties of viscoelastic materials (modulus and loss) are of great interest throughout the plastics, polymer, and rubber from initial design to routine production. There are a number of laboratory research instruments are available to determine these properties. All these hardness tests conducted on polymeric materials involve the penetration of the sample under consideration by loaded spheres or other geometric shapes [1]. Most of these tests are to some extent arbitrary because the penetration of an indenter into viscoelastic material increases with time. For example, standard durometer test (the "Shore A") is widely used to measure the static "hardness" or resistance to indentation. However, it does not measure basic material properties, and its results depend on the specimen geometry (it is difficult to make available the identity of the initial position of the devices on cylinder or spherical surfaces while measuring) and test conditions, and some arbitrary time must be selected to compare different materials. 

The boundary layer equations for an axisymmetric body, Eqs. (1-55), (10-17), and (10-18) have been solved approximately for arbitrary Sc (L4). For Sc oo the mean value of Sh can be computed from Eq. (10-20). Solutions have also been obtained for Sc oo for some shapes without axial symmetry, e.g., inclined cylinders (S34). Data for nonspherical shapes are shown in Fig. 10.3 for large Rayleigh number. The characteristic length in Sh and Ra is analogous to that used in Chapters 4 and 6 ... 

The classical method of solving scattering problems, separation of variables, has been applied previously in this book to a homogeneous sphere, a coated sphere (a simple example of an inhomogeneous particle), and an infinite right circular cylinder. It is applicable to particles with boundaries coinciding with coordinate surfaces of coordinate systems in which the wave equation is separable. By this method Asano and Yamamoto (1975) obtained an exact solution to the problem of scattering by an arbitrary spheroid (prolate or oblate) and numerical results have been obtained for spheroids of various shape, orientation, and refractive index (Asano, 1979 Asano and Sato, 1980). 

For a dielectric of the shape of a plane-parallel plate or arbitrary cylinder in a longitudinally applied field (L, = 0), equations (343) and (344) yield ... 

Similarly, once the shape of a powdery chemical of the TD type is approximated by an infinite cylinder, a value of rf of 2.0 is fixed automatically for the chemical having an arbitrary value of radius. It then becomes possible to calculate the for the imaginary infinite cylinder, having an arbitrary value of radius, placed in the atmosphere under isothermal conditions, provided the value of of the chemical is known and so forth. 

Once the plot of 7) versus r is made beforehand for a series of similar bodies of a powdery chemical of the TD type, having each a definite shape as well as an arbitrary value of r, confined each in a closed container of the corresponding shape and size, and placed each in the atmosphere under isothermal conditions, it enables us to read approximately but readily the value of for a similar body of the chemical, having the definite shape as well as an arbitrary value of r, confined in a closed container of the corresponding shape and size, and placed in the atmosphere under isothermal conditions, on the plot. Such a plot made for a series of similar cylinders of powdery AIBN, having each a definite shape as well as an arbitrary value of r, confined each in a fiber drum of the corresponding shape and size, and placed each in the atmosphere under isothermal conditions, is shown in Figs. 72 and 73 as an example. 

Figure 72. The plot of TV versus r made for a series of similar cylinders of powdery AI BN, having each a definite shape as well as an arbitrary value of r, confined each in a fiber drum of the corresponding shape and size, and placed each in the atmosphere under isothermal conditions.

Alternatively, we could choose to express the shape function in the more general form / = C expiimz + 3t). Because the cylinder is considered to be infinitely long, the use of exp(imz) instead of sin mz (or cos mz) is completely arbitrary and will not change the results of the analysis. 

Thus our immediate problem is to determine the thermal-flux distribution 4>(r,0 in an arbitrary geometry, given the distribution 4>q t) at i = 0. For simplicity, both analytical and experimental, we assume that the specimen is a bare homogeneous system and of some elementary shape such as a parallelepiped, cylinder, or sphere. The differential equation for such a system which contains fissionable material is given by [cf. Eq. (9.10)1... 

The two discs being placed at an arbitrary distance from one another, at a distance equal to their diameter, for example, one forms between them a cylinder, and one lowers the upper disc then gradually the shape, we know, becomes then an unduloid, and it bulges more and more, until it constitutes a portion of sphere (fig. 45). 

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