Analytical solutions of one-dimensional Stokes'problems for infinite and finite domains with generally periodic boundary conditions

The differential problem given by a parabolic equation describing the purely viscous flow generated by a constant or an oscillating motion of a boundary is the well-known Stokes'problem. The one dimensional equation is generally solved for unbounded or bounded domains; for the latter, either free slip (i.e. zero normal gradient) or no-slip (i.e. zero velocity) conditions are enforced on one boundary. Generally, the analytical strategy to solve these problems is based on finding the solutions of the Laplace-transformed (in time) equation and on inverting these solutions. The inversion is not an easy task. In the present paper this problem is solved by making use of the residuals theorem; as it will be shown, this strategy allows to achieve the solutions of first and second Stokes' problems in both infinite and finite depth. The extension to generally periodic boundaries with the presence of a periodic pressure gradient is also presented. An ad-hoc numerical algorithm, based on a finite difference approximation of the differential equation, has been also developed to check the correctness of the analytical solutions.