Nonspherical body

The solution of Stokes flow problems by internal distributions of singularities for nonspherical bodies has been discussed in a series of papers by Chwang and Wu13 and more recently by Kim.14 Here, as an example to demonstrate basic principles, we consider the relatively simple problem of uniform flow past a prolate spheroid... 

The principal drags of some nonspherical bodies are given in [179] and below. 

Substituting the coefficients Ki, Ki, and from (2.6.24)-(2.6.27) into (2.6.36) and using (2.6.35), one obtains the average settling rate for the above-mentioned nonspherical bodies. 

Following the method of asymptotic analogies, we shall use formula (4.2.7) for the calculation of bulk temperature for nonspherical bodies. To this end, for a body of a given shape, we must first calculate the asymptotics of bulk temperature for small and large f and then substitute these asymptotics into (4.2.7). 

Let us compare the approximate dependence (4.2.15) with some well-known exact results on heat exchange for nonspherical bodies. 

The work of Hassani and Hollands [128] on conduction across layers of uniform thickness applied to 3D bodies of various shape has shown that Eq. 4.18 applies approximately to other 3D body shapes as well. (Note that the actual Nucond for the body at hand must be used.) The work of Hassani and Hollands also shows that slightly better results for such nonspherical bodies would be obtained if Eq. 4.18 is modified as follows ... 

It is of interest to note that, by judicious definition of the characteristic diameter of nonspherical bodies, good agreement with the equations for spherical solids was obtained. A diameter defined by the total surface area of the body, divided by the perimeter normal to flow, was successfully used for spheres, hemispheres, cubes, prisms, and cylinders (PI), yielding a = 0 b - 0.692 m = 0.514 and n = [Eq. (4)]. Similar results were obtained for spheroids (S14), namely a = 0 6 = 0.74 w = 0.5 and n =. The commonly used equivalent diameter of a sphere of the same volume as the body yields transfer coefficients increasing with eccentricity (SI4). 

FIGURE 4. (A) Torque induced in nonspherical body by electric field. (B) Nonspherical body aligning parallel to low and high frequencies, but perpendicular at intermediate frequencies. 

With the considerable difficulties still remaining in the analysis of the isothermal drag force for spherical ultrafine particles, it is not surprising that the analysis of this problem for nonspherical particles is incomplete. This subject is reviewed by FUCHS [2.7]. Recent kinetic-theory analyses of drag on nonspherical bodies in the free-molecular regime can be found in CERCIGNANI [2.84]. The excellent series of experimental studies by STDBER and co-workers [2.80,81,116] as well as more recent work on this problem [2.82,117-119] may be consulted for further information. 

Dahneke, B.E. (1973). Slip correction factors for nonspherical bodies The form of the general law. J Aerosol Sci 4 163-170. 

Any study of colloidal crystals requires the preparation of monodisperse colloidal particles that are uniform in size, shape, composition, and surface properties. Monodisperse spherical colloids of various sizes, composition, and surface properties have been prepared via numerous synthetic strategies [67]. However, the direct preparation of crystal phases from spherical particles usually leads to a rather limited set of close-packed structures (hexagonal close packed, face-centered cubic, or body-centered cubic structures). Relatively few studies exist on the preparation of monodisperse nonspherical colloids. In general, direct synthetic methods are restricted to particles with simple shapes such as rods, spheroids, or plates [68]. An alternative route for the preparation of uniform particles with a more complex structure might consist of the formation of discrete uniform aggregates of self-organized spherical particles. The use of colloidal clusters with a given number of particles, with controlled shape and dimension, could lead to colloidal crystals with unusual symmetries [69]. 

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 ... 

For nonspherical particles, values for the slip correction factor are available in slip flow (MU) and free-molecule flow (Dl). To cover the whole range of Kn and arbitrary body shapes, it is common practice to apply Eq. (10-58) for nonspherical particles. The familiar problem then arises of selecting a dimension to characterize the particle. Some workers [e.g. (H2, P14)] have used the diameter of the volume-equivalent sphere this procedure may give reasonable estimates for particles only slightly removed from spherical, or in near-con-tinuum flow, but gives the wrong limit at high Kn. An alternative approach... 

This rotational diffusion coefficient will be used in Chapter 13 to aid in determining if nonspherical particles will orient in shear flow as they are cast to make a ceramic green body. 

For nonspherical particles, the analysis employs the diameter of a sphere of equivalent volume. A correction factor, which depends upon the shape of the body and its orientation in the fluid, must be applied. 

It must be pointed out that this expression implicitly contains terms that depend on the orientation of the molecule with respect to the surface and the orientation of a given molecule with respect to its neighbors when the molecules are nonspherical. Equation (3.19) assumes that the potentials are additive and pairwise since it does not include three-body or higher terms, this must be considered as an effective potential [8]. 

I. THERMAL BOUNDARY-LAYER THEORY FOR SOLID BODIES OF NONSPHERICAL SHAPE IN UNIFORM STREAMING FLOW... 

I. Thermal Boundary-Layer Theory for Solid Bodies of Nonspherical Shape... 

Numerous workers (B16, F2, F4, H17, J4, K4, K7, L8, M2, S5, and others) estimated the external heat-transfer coefficient in the continuous phase by assuming a velocity profile in the boundary layer and ambient fluid. Except for very low Reynolds numbers, the exact boundary layer solutions only apply to the front part of the drop, up to the separation point. Fortunately, simple assumptions sometimes suffice for extending the derivation to the entire drop, and the relationships obtained are in agreement with experimental data. The limitations of the analytical solutions, as well as their application to nonspherical drops, is concisely demonstrated in Lochiel and Calderbank s (LI8) recent study on mass transfer around axisymmetric bodies of revolutions. 

Fig. 1.11 (a) A typical coarse-graining method replaces groups of atoms by ellipsoidal rigid bodies, (b) The Gay-Beme potential models the anisotropic interaction between nonspherical coarse-grained molecules potentials are described by the orientation of the molecules with respect to a fixed frame and the separation between their centers of mass... 

Whenever a body is subject to an electric field and the body is non-spherical, there exists the possibility of creating a torque. See Figure 4A. Even for spherical bodies, if the polarization is a tensor not parallel to the field there exists the possibility for a torque to arise. For example, early experiments by Griffin and StowelF showed that Euglena (nonspherical) would align parallel to the applied field at low frequencies, or at very high... 

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