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# Flow Around a Submerged Prolate Spheroid

Consider the conformal map

 (7.142)

where is real and positive. It follows that

 (7.143) (7.144)

Let

 (7.145) (7.146)

Thus, in the meridian plane, the curve corresponds to the ellipse

 (7.147)

We conclude that the surface is an prolate spheroid (i.e., the three-dimensional surface obtained by rotating an ellipse about a major axis) of major radius and minor radius . The constraints (7.87) and (7.90) yield

 (7.148) (7.149)

respectively. Setting , and substituting into the governing equation, (7.86), we obtain

 (7.150)

The solution that satisfied the constraint (7.148) is

 (7.151)

Let be the eccentricity of the spheroid. Thus, , , , and . The constraint (7.149) yields

 (7.152)

Hence,

 (7.153)

Finally, from Equation (7.84),

 (7.154)

which can be integrated to give

 (7.155)

It is easily demonstrated that

 (7.156)

and

 (7.157)

Thus,

 (7.158)

It follows that the added mass of the spheroid is

 (7.159)

where

 (7.160)

is a monotonic function that takes the value when , and asymptotes to as .

Next: Exercises Up: Axisymmetric Incompressible Inviscid Flow Previous: Flow Around a Submerged
Richard Fitzpatrick 2016-03-31