The invariant of stagnation streamline for a stationary vortex flow of an ideal incompressible fluid around a body

In this study, using the Euler equations we investigate the stagnation streamline in the general spatial case of a stationary incompressible fluid flow around a body with a smooth convex bow. It is assumed that in some neighborhood of the stagnation point everywhere, except for the stagnation point, the fluid velocity is nonzero; and that all streamlines on the surface of the body in this neighborhood start at the stagnation point. Here we prove the following three statements. 1) If on a certain segment of the vortex line the vorticity does not turn to zero, then the value of the fluid velocity in this segment is either identically equal to zero or nonzero at all points of the segment of the vortex line (velocity alternative). 2) The vorticity at the stagnation point is equal to zero. 3) On the stagnation streamline, the vorticity is collinear to the velocity, and the ratio of the vorticity to the velocity is the same at all points of the stagnation streamline (invariant of the stagnation streamline). On the basis of the obtained results, it is concluded that if in the free stream the velocity and vorticity are not collinear, a stationary flow around the body is impossible. However, the question of vorticity at the stagnation point in plane-parallel flows remains open, because the accepted assumption that the velocity of the fluid differs from zero in some neighborhood of the stagnation point everywhere, except for the stagnation point itself, excludes plane-parallel flows from consideration.

Euler equations, Helmholtz vortex theorems, Zorawski's criterion, stagnation streamline