Irrotational Stokes flow in three system of coordinates

Abstract

Irrotational flow is described with the second order elliptic partial differential equation $E^2\psi=0,$ where $\psi$ is the function to be derived and $E^2$ is the Stokes operator. In the present paper we derive the solution of $E^2\psi=0$ in three axisymmetric system of coordinates: the parabolic, the tangent sphere and the cardioid. We prove that Stokes equation separates variables in the parabolic coordinate system, but it R-separates variables in the other two. At the end of this manuscript we summarize all the known results for the 0-eigenspace of $E^2$ in axisymmetric system of coordinates.