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 $z=r\,\cos\theta$ .