Vector potential–vorticity relationship for the Stokes flows: application to the Stokes eigenmodes in 2D/3D closed domain

Abstract

The unsteady dynamics of the Stokes flows, where $\vec{\nabla}^{2} \left(\frac{p}{\rho}\right) =0$, is shown to verify the vector potential–vorticity ( $\vec{\psi},\,\vec{\omega}$) correlation $\frac{\partial\vec{\psi}}{\partial t}+\nu\,\vec{\omega}+\vec{\Pi}=0$, where the field $\vec{\Pi}$ is the pressure-gradient vector potential defined by $\vec{\nabla} \left(\frac{p}{\rho}\right)=\vec{\nabla}\times\vec{\Pi}$. This correlation is analyzed for the Stokes eigenmodes, $\frac{\partial\vec{\psi}}{\partial t}=\lambda\,\vec{\psi}$, subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity, $\nu\,\left(\vec{\omega}-\vec{\omega}_0\right)=-\lambda\,\vec{\psi}$, where $\vec{\omega}_0$ is a constant offset field, possibly zero.