Axisymmetric Helical Flows of Viscous Fluid

Abstract

In this paper, helical flows are flows in which the velocity vector is collinear to the vorticity vector. For an ideal fluid, examples of stationary helical flows are known (Gromeka-Beltrami flows, ABC-flows, etc.) and it has long been proven that the existence of unsteady helical flows is impossible (Beltrami, 1889). For a viscous fluid, examples of unsteady helical flows are known (Trkal, 1919). But it is still unknown whether there can exist a stationary helical flow of incompressible fluid. In the present paper 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 stationary helical flows of viscous incompressible fluid are impossible.