On a new type of non-stationary helical flows for incompressible 3D Navier-Stokes equations

Abstract

In fluid mechanics, a lot of authors have been executing their researches to obtain the analytical solutions of Navier-Stokes equations. But there is an essential deficiency of non-stationary solutions indeed. In our presentation, we proceed exploring the case of non-stationary helical flows of the Navier-Stokes equations for incompressible fluids, with variable (spatially dependent) coefficient of proportionality α between velocity and the curl field of flow. The main motivation of the current research is the exploring the case when velocity field u is supposed to be perpendicular to the vector ∇α. Conditions for the existence of the exact solution for the aforementioned type of flows are obtained, for which non-stationary helical flow with invariant Bernoulli-function is considered. The spatial part of the pressure field of the fluid flow should be determined via Bernoulli-function, if components of the velocity of the flow are already obtained.