Numerical study of the long-time evolution for inhomogeneous creeping flow

A two-dimensional numerical coupled model is developed to describe creeping flow in a computational domain that consists of thick viscous layer overlaid by a thin multi-layered viscous sheet. The model couples the Stokes equations describing the flow in the layer and the Reynolds equations describing the flow in the sheet. We obtain an analytical solution and study the short- and long-time evolution of the surface and interfaces between layers. We investigate the long-time behavior of the flow in the sheet using the method of asymptotic expansions and derive an ordinary differential equation with respect to sheet boundary displacements and velocities at the interface between the sheet and layer. Applying the obtained equation as an internal boundary condition, we couple the Stokes and Reynolds equations. Based on this condition, the system of quasi-linear parabolic equations describing the long-time evolution of sheet boundaries has been developed. Numerical implementation is fulfilled by the modified finite element method combined with the projection gradient method. The proposed model enables different-type hydrodynamic equations to be coupled without any iterative improvements. This reduces significantly computational costs in comparison with the available coupled models. Numerical simulation of the velocity field and the boundary topography at different stages of evolution is fulfilled. Comparison between the analytical and numerical results confirms that the developed coupled model enables simulation of the inhomogeneous flow with a good accuracy at low computational cost. We investigate the evolution of the flow with large displacement of layer boundaries. Some possible applications in tectonics and geophysics of these model results are outlined.