Axisymmetric Velocity Fields

(7.5) |

When the fluid velocity is written in this form it becomes obvious that the incompressibility constraint is satisfied [because --see Equations (A.175) and (A.176)]. It is also clear that the Stokes stream function, , is undefined to an arbitrary additive constant.

In fact, the most general expression for an axisymmetric incompressible flow pattern is

where is the angular velocity of flow circulating about the -axis. (This follows because when . See Appendix C.) The previous expression implies that (because when ). In other words, when plotted in the meridian plane, streamlines in a general axisymmetric flow pattern correspond to contours of .

Making use of the vector identities (A.176) and (A.178), we can also write Equation (7.6) in the form

(7.7) |

It follows from the identity (A.177) that

(7.8) |

because , by symmetry. Hence, the vorticity of a general axisymmetric flow pattern is written

(7.9) |

where , and [see Equations (C.52)-(C.54)]

In the following, we shall concentrate on axisymmetric flow patterns in which there is no circulation about the -axis (i.e., ).