A Residual Theorem Approach Applied to Stokes' Problems with Generally Periodic Boundary Conditions including a Pressure Gradient Term

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. In the present paper this problem is solved by making use of the residuals theorem; as it will be shown, this strategy allows achieving 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. This approach allows getting closed form solutions in the time domain in a rather fast and simple way. An ad hoc numerical algorithm, based on a finite difference approximation of the differential equation, has been developed to check the correctness of the analytical solutions.