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Axisymmetric Flow

A flow pattern is said to be axisymmetric when it is identical in every plane that passes through a certain straight-line. The straight-line in question is referred to as the symmetry axis. Let us set up a Cartesian coordinate system in which the symmetry axis corresponds to the $ z$ -axis. The flow is most conveniently described in terms of the cylindrical coordinates ($ \varpi$ , $ \varphi$ , $ z$ ), or the spherical coordinates ($ r$ , $ \theta $ , $ \varphi$ ). Here, $ \varpi=(x^{\,2}+y^{\,2})^{1/2}$ , $ \varphi= \tan^{-1}(y/x)$ , $ r=(x^{\,2}+y^{\,2}+z^{\,2})^{1/2}$ , and $ \theta=\cos^{-1}(z/r)$ . (See Appendix C.) In particular, $ \varpi= r\,\sin\theta$ and $ z=r\,\cos\theta$ .

Richard Fitzpatrick 2016-03-31