Vortex flow over a flat surface with suction

TLDR: In this paper, the boundary-layer equations under the boundary condition of distributed suction were solved, and a formal similarity solution was obtained for suction parameter greater than a certain value.

Abstract: Introduction B the use of momentum integral methods, Taylor and Cooke calculated the laminar boundary-layer development on the interior surface of a frustum of a cone, with a potential vortex as the outer flow. Taylor's swirl atomizer problem has also been examined by means of similarity transformations of the boundary-layer equations by Moore, Mager, and Rott and Lewellen, and in Ref. 3 it is concluded that no valid solution can be found. However, in Ref. 4 it is noted that the valid solution may be found if the velocity profile of secondary flow changes sign somewhere in the boundary layer. Contrary to this, in Ref. 5 the existence of similarity solution is denied by a simple and clear consideration. Here, for simplicity we restrict our attention to the vortex flow over an infinite flat surface, i.e., to the case when the vertex angle of a cone is equal to 180°. For this problem, even the complete Navier-Stokes equations have a similarity solution only for a limited range of the Reynolds number." In this Note, we attempt to solve the boundary-layer equations under the boundary condition of distributed suction, and show without any speculation to the singularity on the vortex axis that a formal similarity solution can be obtained for suction parameter greater than a certain value.