Next: Inviscid Flow Past a Up: Incompressible Inviscid Fluid Dynamics Previous: Two-Dimensional Vortex Filaments

Two-Dimensional Irrotational Flow in Cylindrical Coordinates

As we have seen, in a two-dimensional flow pattern, we can automatically satisfy the incompressibility constraint, , by expressing the pattern in terms of a stream function. Suppose, however, that, in addition to being incompressible, the flow pattern is also irrotational. In this case, Equation (477) yields
 (516)

In cylindrical coordinates, since , this expression implies that (see Section C.3)
 (517)

Let us search for a separable steady-state solution of Equation (517) of the form

 (518)

It is easily seen that
 (519)

which can only be satisfied if
 (520) (521)

where is an arbitrary (positive) constant. The general solution of Equation (521) 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). Now, the general solution of Equation (520) 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
 (522)

where , , and are arbitrary constants. We can recognize the first few terms on the right-hand side of the above 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 5.11.) The term corresponds to uniform flow of speed whose direction subtends a (counter-clockwise) angle with the minus -axis. (See Section 5.9.) Finally, the term corresponds to a dipole flow pattern. (See Section 5.10.)

The velocity potential associated with the irrotational stream function (522) satisfies [see Equations (459) and (474)]

 (523) (524)

It follows that
 (525)

Next: Inviscid Flow Past a Up: Incompressible Inviscid Fluid Dynamics Previous: Two-Dimensional Vortex Filaments
Richard Fitzpatrick 2012-04-27