Axisymmetric Bodies

The projected area and perimeter must be determined normal to some specified axis. For axisymmetric bodies, the reference direction is taken parallel or normal... 

For axisymmetric bodies with creeping flow parallel to the axis of symmetry, Bowen and Masliyah (B3) found that the most useful shape parameter was based on the sphere with the same perimeter, P, projected normal to the axis. Their shape factor is given by... 

A useful theorem due to Payne and Pell (P3) enables the drag on an axisymmetric body to be calculated directly from the stream function ij/ for steady... 

Bowen and Masliyah (B4) give a useful discussion of the axial resistance of various axisymmetric bodies. For particles which may be regarded as spheres with axisymmetric deformations, simple estimates for the resistance are available. Suppose that a particle of volume V with principal resistance c is obtained by deforming a sphere of volume and resistance It is convenient to use two factors introduced by O Brien (02) ... 

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 ... 

If exceeds a critical value of order unity, reverse flow occurs near the forward stagnation point for oblate and prolate axisymmetric bodies (M8, M9), leading to formation of an upstream separation bubble. ... 

Consider two-dimensional flow over an axisymmetric body. Write the governing equations in terms of a suitably defined stream function and vorticity. 

Dhir, V.K. and Lienhard, J.H., Laminar Film Condensation on Plane and Axisymmetric Bodies in Non-uniform Gravity, J, Heat Transfer, Vol. 93, p. 97, 1971. 

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

We shall be interested here in flows that involve an axisymmetric body with the origin of coordinates r = 0 inside the body (and at its centre if the body is spherical). 

For any problem with ip in the general form (7-131), the force exerted by the fluid on an arbitrary, axisymmetric body with its center of mass at x = 0 is generally... 

Figure 7-11. A schematic representation of the domain for a uniform flow past an arbitrary, axisymmetric body. For the case of a solid sphere, this is Stokes problem.

Application to Uniform Streaming Flow past an Arbitrary Axisymmetric Body... 

As an example of the application of (7-131), we consider creeping flow past an arbitrary axisymmetric body with a uniform streaming motion at infinity. For the case of a solid sphere, this is known as Stokes problem. In the present case, we begin by allowing the geometry of the body to be arbitrary (and unspecified) except for the requirement that the symmetry axis be parallel to the direction of the uniform flow at infinity so that the velocity field will be axisymmetric. A sketch of the flow configuration is shown in Fig. 7 11. We measure the polar angle 9 from the axis of symmetry on the downstream side of the body. Thus ij = I on this axis, and ij = — 1 on the axis of symmetry upstream of the body. 

The analysis of the preceding section was carried out by use of a spherical coordinate system, but the majority of the results are valid for an axisymmetric body of arbitrary shape. The necessity to specify a particular particle geometry occurs only when we apply boundary conditions on the particle surface (that is, when we evaluate the coefficients Cn and Dn in the spherical coordinate form of the solution). For this purpose, an exact solution requires that the body surface be a coordinate surface in the coordinate system that is used, and this effectively restricts the application of (7-149) to streaming flow past spherical bodies, which may be solid, as subsequently considered, or spherical bubbles or drops, as considered in section H. 

In spite of the fact that there are actually quite a large number of axisymmetric problems, however, there are many important and apparently simple-sounding problems that are not axisymmetric. For example, we could obtain a solution for the sedimentation of any axisymmetric body in the direction parallel to its axis of symmetry, but we could not solve for the translational motion in any other direction (e.g., an ellipsoid of revolution that is oriented so that its axis of rotational symmetry is oriented perpendicular to the direction of motion). Another example is the motion of a sphere in a simple linear shear flow. Although the undisturbed flow is 2D and the body is axisymmetric, the resulting flow field is fully 3D. Clearly, it is extremely important to develop a more general solution procedure that can be applied to fully 3D creeping-flow problems. 

In the case of an axisymmetric body, the cross-sectional shape at any point is a circle (R independent of ), and for the special case of a spheroid, e = a/b, where a and b are the semi-minor and semi-major axis lengths. 

For a 2D or axisymmetric body, the surface corresponds to either qlorq2 = const. For convenience, we assume that it is <71 (though the required changes are completely trivial if it is e/2 instead). If we denote the velocity vector as... 

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... 

Provided a solution of this equation exists such that g remains finite as q2 -> q2o, the similarity transformation is successful. This has been shown to be the case for the case of a sphere, where q =r, q2 = 0, a = -3 sin0/2, and h2 = (sin0) 1[l + <9(Pe 1/3)]. Solutions can also be obtained for other axisymmetric bodies, but to go further, we would... 

F. Streaming Flow Past Axisymmetric Bodies - A Generalization of the Blasius Series... 

F. STREAMING FLOW PAST AXISYMMETRIC BODIES - A GENERALIZATION OF THE BLASIUS SERIES... 

Figure 10-11. A schematic representation for streaming flow past an arbitrary axisymmetric body. The body geometry is specified by the function r(x). which measures the distance from the symmetry axis to the body surface as a function of the position x.

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