Axisymmetric helical flow of viscous fluid

Helical flow is called a flow in which the velocity vector is collinear to the vorticity vector. For ideal fluid examples are known of stationary helical flows (Gromeka–Beltrami flows, ABC-flows, etc.) and it is proved that the existence of unsteady helical flows is impossible (Beltrami, 1889). For a viscous fluid examples are known of unsteady helical flows (Trkal, 1919). But it is still unknown whether there can exist a stationary helical flow of a viscous fluid. In the present article this question is investigated using the Navier–Stokes equations in the axisymmetric case. It was assumed that the coefficient of proportionality between the vorticity and velocity may depend on the spatial coordinates. It is shown that in the axisymmetric case, the existance of such flows is impossible. This study solved the problem of the existence of axisymmetric helical flows of an incompressible fluid.