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

The unsteady dynamics of the Stokes flows, where ∇⃗ 2(𝑝𝜌)=0, is shown to verify the vector potential–vorticity ( 𝜓⃗ ,𝜔⃗ ) correlation ∂𝜓⃗ ∂𝑡+𝜈𝜔⃗ +Π⃗ =0, where the field Π⃗ is the pressure-gradient vector potential defined by ∇⃗ (𝑝𝜌)=∇⃗ ×Π⃗ . This correlation is analyzed for the Stokes eigenmodes, ∂𝜓⃗ ∂𝑡=𝜆𝜓⃗ , 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, 𝜈(𝜔⃗ −𝜔⃗ 0)=−𝜆𝜓⃗ , where 𝜔⃗ 0 is a constant offset field, possibly zero.