General Axisymmetric Bodies

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... 

Thus, regardless of the details of the body geometry, or the form of the boundary conditions at the body surface, the streamfunction must exhibit the asymptotic form (7 147) if there is a uniform streaming flow at large distances from the body. Comparing (7 147) and (7-131), we see that many of the coefficients A and B in the general axisymmetric solution must be zero for this case. In particular, the asymptotic condition (7-147) requires that... 

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. 

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

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

Cone in Supersonic Flow. The preceding solutions for a flat plate may be applied to a cone in supersonic flow through the Mangier transformation [39], which in its most general form relates the boundary layer flow over an arbitrary axisymmetric body to an equivalent flow over a two-dimensional body. This transformation is contained in Eq. 6.89, which results in transformed axisymmetric momentum and energy equations equivalent to the two-dimensional equations (Eqs. 6.95 and 6.97). Hence, solutions of these equations are applicable to either a two-dimensional or an axisymmetric flow, the differences being contained solely in the coordinate transformations. 

Cox (C5) and Tchen (Tl) also obtained expressions for the drag on slender cylinders and ellipsoids which are curved to form rings or half circles. The advantages of prolate spheroidal coordinates in dealing with slender bodies have been demonstrated by Tuck (T2). Batchelor (Bl) has generalized the slender body approach to particles which are not axisymmetric and Clarke (C2) has applied it to twisted particles by considering a surface distribution rather than a line distribution. 

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. 

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

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