Using the projection gradient method for the numerical solution of the coupled system of Stokes and Reynolds equations

A numerical coupled model is designed to investigate creeping flow in the computational domain that consists of a thin multi-layered viscous sheet underlaid by a thick viscous layer. The sheet is assumed to have lower density than the layer. The model combines the Reynolds equations describing the flow in the sheet and the Stokes equations describing the flow in the layer. Analytical study of this model with single-layered sheet reveals a significant discrepancy between its short- and long-time evolutions. Using the perturbation method to investigate the long-time behavior of the flow within the sheet, we derive an ordinary differential equation which relates the surface and interface displacements of the sheet with the velocities at the interface between the sheet and the underlaying layer. We use this asymptotic equation as an internal boundary condition to couple the Stokes equations and the Reynolds equations. Numerical implementation is fulfilled by the modified finite element method combined with the projection gradient method. The proposed model reduces the computational costs in comparison with the majority of existing coupled models because it enables us to combine different-type equations of hydrodynamics without applying any iterative improvements. Comparison of the analytical and numerical solutions confirms the good accuracy of the presented model.