The Oseen’s Spiral Flow

Abstract

The Navier-Stokes equations for a flow associated to a logarithmic spiral in an incompressible, viscous, planar fluid, become a non linear ordinary fourth order differential equation for a Stokes stream function. After having made a first integration, we found that the resulting equation is the same as the one associated to the movement of a particle subjected both to a cubic potential and to a friction force proportional to the velocity. This allows us to find a new integral in the particular case that corresponds to the frictionless particle. Use is made of the fact that the independent coordinate, in the case of a fluid, has boundary conditions that require periodic restrictions. This fact is also used in the general case, that for the particle includes friction, to find the general solution for the fluid and justify its uniqueness.