THE BOUNDARY CONDITIONS

The boundary condition for Eq. (9.1) can directly specify a zero normal velocity component ЭФ*/дп = 0 on the surface SB, in which case this “direct” formulation is called the Neumann problem. It is possible to specify Ф* on the boundary, so that indirectly the zero normal flow condition will be met, and this “indirect” formulation is called the Dirichlet problem. Of course, a combination of the above boundary conditions is possible, too, and this is called a mixed boundary condition problem.

An additional approach would be to search for a singularity distribution that creates enclosed streamlines, equivalent to the geometry of the surface SB. This method is useful in two dimensions, where the stream function Ф is well defined (and hence the streamlines Ф = const, can be easily derived as in Sections 3.10, 3.11), but for complex, three-dimensional geometries the implementation of this method is difficult and will not be dealt with here.