Complete decomposition of axisymmetric Stokes flow

Abstract

As Stokes has shown, axisymmetric, incompressible, viscous creeping flow can be studied through the use of a stream function Ψ which belongs to the kernel of the fourth-order differential operator E 4, where E 2 Ψ measures the vorticity of the flow. In fact, irrotational flows are described by stream functions that belong to the ker E 2, while rotational flows are described by stream functions that do not belong to the ker E 2. It is shown that a decomposition, of the form Ψ= Ψ 1+ r 2 Ψ 2, for any stream function Ψ is possible, where Ψ 1 and Ψ 2 belong to ker E 2 and r is the radial spherical variable. Consequently, a stream function that describes a rotational flow can always be divided by a stream function that describes an irrotational flow in a way that renders the ratio always equal to the square of the Euclidean distance. If no singularities are observed on the axis of symmetry then the above decomposition is unique.