Hydrodynamic Green functions. Paradoxes in unsteady Stokes conditions and infinite propagation velocity in incompressible viscous models

We present a simple representation of the hydrodynamic Green functions grounded on the free propagation of a vector field without any constraints (such as incompressibility) coupled with a gradient gauge in order to enforce these constraints. This approach involves the solution of two scalar problems: a couple of Poisson equations in the case of the Stokes regime, and a system of diffusion/Poisson equations for unsteady Stokes flows. The explicit and closed-form expression of the Green function for unsteady Stokes flow is developed. The relevance of this approach resides in its conceptual simplicity and it enables us to focus on the intrinsic singularities (Stokesian paradoxes) associated with the propagation of the stresses in incompressible flows under unsteady Stokes conditions, determining the occurrence of power-law tails in the velocity profile arbitrarily far away from the location of the impulsive force.