Bodies of revolution

Many problems of practical interest are, indeed, two dimensional in nature. Impact and penetration problems are examples of these, where bodies of revolution impact and penetrate slabs, plates, or shells at normal incidence. Such problems are clearly axisymmetric and, therefore, accurately modeled with a two-dimensional simulation employing cylindrical coordinates. 

Equation (24) is originally derived for a conical indenter. Pharr et al. showed that Eq (24) holds equally well to any indenter, which can be described as a body of revolution of a smooth function [67]. Equation (24) also works well for many important indenter geometries, which cannot be described as bodies of revolution. 

For an orthotropic particle in steady translation through an unbounded viscous fluid, the total drag is given by Eq. (4-5). In principle, it is possible to follow a development similar to that given in Section IT.B.l for axisymmetric particles, to deduce the general behavior of orthotropic bodies in free fall. This is of limited interest, since no analytic results are available for the principal resistances of orthotropic particles which are not bodies of revolution. General conclusions from the analysis were given in TLA. 

Mason and co-workers (B8, F3, Gll, M5, T15) have shown that Eqs. (10-32) to (10-35) can also be applied to disks and cylinders provided that one uses an apparent value of , calculated from Eq. (10-36) and the observed Bretherton (B15) considered more general shapes and proved that most bodies of revolution, except for some extreme shapes, show periodic rotation with no lateral migration (i.e., no lift) provided that inertia terms are neglected. In reality all these particles migrate in the direction of positive lift (see Chapter 9). For a useful extended review on particle motion in shear fields, see Goldsmith and Mason (G12). 

As indicaated in Fig. 3, consider the simplest case of such a field-containing inductor, a body of revolution as a toroid. The cross section of the toroid need not be circular. It lies on ST and contains the volume VT- With surface current density Js(rs, t) on ST as indicated, we have the following for zero frequency ... 

To analyze the properties of the toroidal antenna, consider it first as a receiver. As in Fig. 6, let the antenna be a body of revolution with respect to the z axis with the usual cordinates. With the incident electric field Ei,nc) taken initially parallel to the z axis, let the antenna be electrically small. Neglect the field distortion due to the antenna conductors, or equivalently consider the antenna (as in Section VII) as a set of distributed sources in space specified by a surface current density Js with... 

F>g. 2.21 A body of revolution for which the surrounding velocity field is described as an analytic function. 

Chang, S. S.-H. (1975). Nonequilibrium Phenomena in Dusty Supersonic Flow Past Blunt Bodies of Revolution. Phys. Fluids, 18,446. 

It is possible to wind integrally most of the bodies of revolution, such as spheres, oblate spheres, and torroids. Each application, however, requires a study to insure that the winding geometry satisfies the membrane forces induced by the configuration being wound. 

Homogeneous symmetrical particles can take up any orientation as they settle slowly in a fluid of infinite extent. Spin-free terminal states are attainable in all orientations for ellipsoids of uniform density and bodies of revolution with fore and aft symmetry, but the terminal velocities will depend on their orientation. A set of identical particles will, therefore, have a range of settling velocities according to their orientation. This... 

MVFN methods have been used with some success in compressible flows. Figure 9 shows a prediction of Herring and Mellor (H5) of the Mach number correction to the skin friction factor for a flat-plate boundary layer. Figure 10 shows their prediction for the boundary layer on a waisted body of revolution. We note that, while the momentum thickness is quite accurately predicted, the velocity-profile details are in considerable error. 

Fig. 10. Herring and Mellor MVFN calculation for a compressible boundary layer on a waisted body of revolution (a) integral parameters, (b) profiles.

For this purpose, a worm-like coil is represented by an extended body of revolution for which the central radius of gyration is related to the longitudinal, H, and transverse, Q, body dimensions by the general equation... 

Since 1970, two generic types of viscometer have received the greatest attention the first makes use of the torsional oscillations of bodies of revolution and the second is based on the rather simpler concept of laminar flow through capillaries. Both reduce the measurement of viscosity to measurements of mass, length and time. 

Lochiel, a. C. Calderbank, P. H. 1964 Mass transfer in the continuous phase around axisymmetric bodies of revolution. Chemical Engineering Science 19,471 84. 

Problem 7-3. Linearity of Creeping Flow. In the laboratory you are looking at the sedimentation of rods under creeping-flow conditions (zero Reynolds number). These may be regarded as bodies of revolution characterized by an orientation vector (director) pL. In a simple experiment you measure the sedimentation velocity when the director is parallel to gravity (e.g., point down) to be 0.03 cm/s and when it is perpendicular to gravity to... 

The thermal boundary-layer equation, (9-257), also apphes for axisymmetric bodies. One example that we have already considered is a sphere. However, we can consider the thermal boundary layer on any body of revolution. A number of orthogonal coordinate systems have been developed that have the surface of a body of revolution as a coordinate surface. Among these are prolate spheroidal (for a prolate ellipsoid of revolution), oblate spheroidal (for an oblate ellipsoid of revolution), bipolar, toroidal, paraboloidal, and others.22 These are all characterized by having h2 = h2(qx, q2), and either h2/hx = 1 or h2/hx = 1 + 0(Pe 1/3) (assuming that the surface of the body corresponds to q2 = 1). Hence the thermal boundary-layer equation takes the form... 

N. Farahat, W. Yu, and R. Mittra, A fast near-to-far-field transformation in body of revolution finite-difference time-domain method, IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2534-2540, Sep. 2003.doi 10.1109/TAP.2003.816360... 

The similar relation for the wake behind a body of revolution has the form... 

Translational Stokes Flow Past Bodies of Revolution... 

It follows from (2.6.14) and (2.6.15) that to calculate the drag force of a body of revolution of any shape with arbitrary orientation in a Stokes flow, it suffices to know the value of this force only for two special positions of the body in space. The axial (Fy) and transversal (Fl) drags can be obtained both theoretically... 

Figure 2.6. Body of revolution in translational flow (arbitrary orientation)...

One can interpret the table data as follows. Let us project a body of revolution on the plane perpendicular to the symmetry axis. The projection is a disk of... 

For a body of revolution whose axis is inclined at an angle ui to the incoming flow direction, the following formula [166] holds in the Stokes approximation (as Re —> 0) ... 

At low Peclet numbers, for the translational Stokes flow past an arbitrarily shaped body of revolution, formula (4.10.8) coincides with the exact asymptotic expression in the first three terms of the expansion [358], Since (4.10.8) holds identically for a spherical particle at all Peclet numbers, one can expect that for particles whose shape is nearly spherical, the approximate formula (4.10.8) will give good results for low as well as moderate or high Peclet numbers. 

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 ... 

For surfaces in unbounded convection, such as plates, tubes, bodies of revolution, etc., immersed in a large body of fluid, it is customary to define h in Eq. (1.12) with 7 as the temperature of the fluid far away from the surface, often identified as T. (Fig. 1.2). For bounded convection, including such cases as fluids flowing in tubes or channels, across tubes in bundles, etc., Tf is usually taken as the enthalpy-mixed-mean temperature, customarily identified as T . 

Sibulkin, M. "Heat Transfer Near the Forward Stagnation Point of a Body of Revolution." Journal of Aeronautical Science and Technologies 19 (1952) 570-71. 

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. 

Solutions for the transfer coefficients around axisymmetric bodies of revolution (oblate and prolate spheroids and bubbles with spherical cups shapes) in potential flow were also reported (LI8) and related to experiment (Cl a). 

To illustrate the application of these relations, consider the force experienced by the ovoid body of revolution... 

Fig. 1. Simple shear flow parallel to the axis of an ovoid body of revolution.

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