Application to Uniform Streaming Flow past an Arbitrary Axisymmetric Body

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 asymptotic form of the streamfunction for r oo can be obtained from the uniform 

The constant that appears in (7-145) is completely arbitrary. Notice from (7-102) that we can always add an arbitrary constant to xjr without changing the velocities at all. Because the constant is arbitrary, we set it equal to zero, so that 

Note that the choice const = 0 here also requires that the constant in (7-103) be zero, so that 

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