Creeping flow axisymmetric body

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... 

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. 

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. 

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