(5.60) |

In cylindrical coordinates, because , this expression implies that (see Section C.3)

Let us search for a separable solution of Equation (5.61) of the form

(5.62) |

It is easily seen that

(5.63) |

which can only be satisfied if

where is an arbitrary (positive) constant. The general solution of Equation (5.65) is a linear combination of and factors. However, assuming that the flow extends over all values, the function must be single-valued in , otherwise --and, hence, --would not be be single-valued (which is unphysical). It follows that can only take integer values (and that must be a positive, rather than a negative, constant). The general solution of Equation (5.64) is a linear combination of and factors, except for the special case , when it is a linear combination of and factors. Thus, the general stream function for steady two-dimensional irrotational flow (that extends over all values of ) takes the form

where , , and are arbitrary constants. We can recognize the first few terms on the right-hand side of the previous expression. The constant term has zero gradient, and, therefore, does not give rise to any flow. The term is the flow pattern generated by a vortex filament of intensity , coincident with the -axis. (See Section 5.6.) The term corresponds to uniform flow of speed whose direction subtends a (counter-clockwise) angle with the minus -axis. (See Section 5.4.) Finally, the term corresponds to a dipole flow pattern. (See Section 5.5.)

The velocity potential associated with the irrotational stream function (5.66) satisfies [see Equations (4.89) and (5.7)]

(5.67) | ||

(5.68) |

It follows that

(5.69) |