Abstract

Many physical, biological and engineering processes involve the coupling of free flow with flows in porous media. Well-known examples include all filtration processes, flows in karstic geometry, hyporheic flow, and PEM fuel cell among many others. 

In the laminar flow regime, we derive the interface boundary conditions that couples the Stokes system for free flow with the Darcy system for the porous media flow via Helmhotz minimal dissipation principle. When the Reynolds number of the free-flow is relatively high, the Navier-Stokes system is adopted instead of the Stokes system. However, there are different choices of interface boundary conditions that couples the Navier-Stokes system with the Darcy system.

The coupled system is highly nonlinear. After discussing the well-posedness of the system, we present a linear, uniquely solvable and long-time energy stable scheme that decouples the Navier-Stokes equations from the Darcy equation if appropriate interface boundary condition is adopted. Such kind of long-time stable algorithms are highly desirable for the physically significant  small  Darcy number regime if we are interested in the important transport behavior in porous media. Next, we investigate the behavior of the Navier-Stokes-Darcy system at small Darcy number. We discover that the leading order dynamics are completely decoupled, and different choices of interface boundary conditions lead to the same non-trivial effective dynamics.