Irrotational Stokes flow in three system of coordinates

TL;DR

This paper derives solutions to the irrotational Stokes flow equation in three axisymmetric coordinate systems, revealing separation of variables properties and summarizing known results for the 0-eigenspace.

Contribution

It provides explicit solutions and separation properties of the Stokes equation in parabolic, tangent sphere, and cardioid coordinates, expanding understanding of axisymmetric irrotational flows.

Findings

Stokes equation separates variables in parabolic coordinates.

R-separation occurs in tangent sphere and cardioid coordinates.

Summarizes known results for the 0-eigenspace in axisymmetric systems.

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.