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Consider the conformal map
![$\displaystyle z+{\rm i}\,\varpi = c\,\sinh(\xi+{\rm i}\,\eta),$](img2752.png) |
(7.115) |
where
is real and positive. It follows that
Let
Thus, in the meridian plane, the curve
corresponds to the ellipse
![$\displaystyle \left(\frac{\varpi}{a}\right)^2+\left(\frac{z}{b}\right)^2=1.$](img2757.png) |
(7.120) |
We conclude that the surface
is an oblate spheroid (i.e., the three-dimensional surface obtained by
rotating an ellipse about a minor axis) of major radius
and minor radius
.
The constraints (7.87) and (7.90) yield
respectively. Setting
, and substituting into the governing equation, (7.86), we obtain
![$\displaystyle \frac{d}{d\xi}\left(\frac{1}{\cosh\xi}\,\frac{dF}{d\xi}\right) - \frac{2}{\cosh\xi}\,F=0,$](img2760.png) |
(7.123) |
which can be rearranged to give
![$\displaystyle \frac{d}{d\xi}\left(\cosh\xi\,\frac{dF}{d\xi}-2\,\sinh\xi\,F\right)=0.$](img2761.png) |
(7.124) |
On integration, we get
![$\displaystyle \cosh\xi\,\frac{dF}{d\xi}-2\,\sinh\xi\,F = B,$](img2762.png) |
(7.125) |
which can be rearranged to give
![$\displaystyle \frac{d}{d\xi}\left(\frac{F}{\cosh^2\xi}\right)= \frac{B}{\cosh^3\xi}.$](img2763.png) |
(7.126) |
It follows that
where use has been made of the constraint (7.121).
Let
be the eccentricity of the spheroid. Thus,
,
,
, and
. The constraint (7.122) yields
![$\displaystyle -\frac{1}{2}\,V\,a^{\,2} = F(\xi_0)= \frac{1}{2}\,B\left[\frac{(1-e^{\,2})^{1/2}}{e}-\frac{\sin^{-1}e}{e^{\,2}}\right],$](img2773.png) |
(7.128) |
or
![$\displaystyle B = - \left[\frac{V\,a^{\,2}\,e^{\,2}}{e\,(1-e^{\,2})^{1/2}-\sin^{-1}e}\right].$](img2774.png) |
(7.129) |
Hence,
![$\displaystyle \psi(\xi,\eta) =-\frac{1}{2}\,V\,a^{\,2}\,e^{\,2}\left[\frac{\sin...
...\xi\,\tan^{-1}(1/\sinh\xi)} {e\,(1-e^{\,2})^{1/2}-\sin^{-1}e}\right]\sin^2\eta.$](img2775.png) |
(7.130) |
Finally, from Equation (7.84),
![$\displaystyle \frac{\partial\phi}{\partial\eta} = -\frac{1}{c\,\cosh\xi\,\sin\eta}\,\frac{\partial\psi}{\partial\xi},$](img2776.png) |
(7.131) |
which can be integrated to give
![$\displaystyle \phi(\xi,\eta) =-V\,a\,e\left[\frac{1 -\sinh\xi\,\tan^{-1}(1/\sinh\xi)} {e\,(1-e^{\,2})^{1/2}-\sin^{-1}e}\right]\cos\eta.$](img2777.png) |
(7.132) |
It is easily demonstrated that
![$\displaystyle \psi(\xi_0,\eta)= -\frac{1}{2}\,V\,a^{\,2}\,\sin^2\eta,$](img2778.png) |
(7.133) |
and
![$\displaystyle \phi(\xi_0,\eta) = V\,a\left[\frac{e-(1-e^{\,2})^{1/2}\,\sin^{-1} e}{\sin^{-1} e-e\,(1-e^{\,2})^{1/2}}\right]\cos\eta.$](img2779.png) |
(7.134) |
Thus,
![$\displaystyle K = -\pi\,\rho\left.\int_{\eta=0}^{\eta=\pi} \phi\,d\psi\right\ve...
...rac{e-(1-e^{\,2})^{1/2}\,\sin^{-1} e}{\sin^{-1} e-e\,(1-e^{\,2})^{1/2}}\right].$](img2780.png) |
(7.135) |
It follows that the added mass of the spheroid is
![$\displaystyle m_{\rm added} = \frac{4}{3}\,\pi\,\rho\,a^{\,3}\,G(e),$](img2781.png) |
(7.136) |
where
![$\displaystyle G(e)= \frac{e-(1-e^{\,2})^{1/2}\,\sin^{-1} e}{\sin^{-1} e-e\,(1-e^{\,2})^{1/2}}$](img2782.png) |
(7.137) |
is a monotonic function that varies between
when
and
when
.
Figure:
Contours of the Stokes stream function for the case of a disk of radius
, lying in the
-
plane, and placed in a uniform, incompressible, irrotational flow directed parallel to the
-axis.
![\begin{figure}
\epsfysize =3in
\centerline{\epsffile{Chapter07/disk.eps}}
\end{figure}](img2786.png) |
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
and
![$\displaystyle m_{\rm added} = \frac{8}{3}\,\rho\,a^{\,3},$](img2791.png) |
(7.140) |
respectively. Here,
and
. It
follows that, in the instantaneous rest frame of the disk,
![$\displaystyle \psi(\varpi,z)=- \frac{1}{4}\,V\left[a^{\,2}+z^{\,2}+\varpi^{\,2}...
...rpi^{\,2})^2-4\,a^{\,2}\,\varpi^{\,2}}\,\right] + \frac{1}{2}\,V\,\varpi^{\,2}.$](img2794.png) |
(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.
Next: Flow Around a Submerged
Up: Axisymmetric Incompressible Inviscid Flow
Previous: Conformal Maps
Richard Fitzpatrick
2016-01-22