Three-dimensional coupled numerical model of creeping flow of viscous fluid

A three-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 sheet is assumed to have the lower density than the layer. The model couples the Stokes equations describing the flow in the layer and the Reynolds equations describing the flow in the sheet. We investigate the long-time behavior of the flow in the sheet using the asymptotic method 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. Numerical implementation is fulfilled by the modified finite element method combined with the projection gradient method. The examined domain is discretized with hexahedrons. Piecewise square basic functions are used. 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 experiments confirm a good accuracy of the developed three-dimensional coupled model.