(7.115) |

where is real and positive. It follows that

(7.116) | ||

(7.117) |

Let

(7.118) | ||

(7.119) |

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

(7.120) |

We conclude that the surface is an

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

(7.123) |

which can be rearranged to give

(7.124) |

On integration, we get

(7.125) |

which can be rearranged to give

(7.126) |

It follows that

(7.127) |

where use has been made of the constraint (7.121). Let be the

(7.128) |

or

(7.129) |

Hence,

Finally, from Equation (7.84),

(7.131) |

which can be integrated to give

It is easily demonstrated that

(7.133) |

and

(7.134) |

Thus,

(7.135) |

It follows that the added mass of the spheroid is

where

(7.137) |

is a monotonic function that varies between when and when .

In the limit , our spheroid asymptotes to a thin disk of radius moving through the fluid in the direction perpendicular to its plane. Expressions (7.130), (7.132), and (7.136) yield

(7.138) | ||

(7.139) |

and

(7.140) |

respectively. Here, and . It follows that, in the instantaneous rest frame of the disk,

(7.141) |

This flow pattern, which corresponds to that of a thin disk placed in a uniform flow perpendicular to its plane, is visualized in Figure 7.6.