Suppose, however, that, in addition to being irrotational, the flow pattern is also incompressible: that is, . In this case, Equation (7.11) yields

In spherical coordinates, assuming that the flow pattern is axisymmetric, so that , the previous equation leads to (see Section C.4)

Let us search for a separable solution of Equation (7.13) of the form

(7.14) |

It is easily seen that

(7.15) |

which can only be satisfied provided

where , and is a constant. The solutions to Equation (7.17) that are well behaved for in the range to are known as the

(7.18) |

Hence,

(7.19) | ||

(7.20) | ||

(7.21) | ||

(7.22) |

et cetera. The general solution of Equation (7.16) is a linear combination of and factors. Thus, the general axisymmetric solution of Equation (7.12) is written

where the and are arbitrary coefficients. It follows from Equations (7.4) that the corresponding expression for the Stokes stream function is

where .