On Interface Conditions for Flows in Coupled Free-Porous Media

Abstract

Many processes in nature (e.g., physical and biogeochemical processes in hyporheic zones, and arterial mass transport) occur near the interface of free-porous media. A firm understanding of these processes needs an accurate prescription of flow dynamics near the interface which (in turn) hinges on an appropriate description of interface conditions along the interface of free-porous media. Although the conditions for the flow dynamics at the interface of free-porous media have received considerable attention, many of these studies were empirical and lacked a firm theoretical underpinning. In this paper, we derive a complete and self-consistent set of conditions for flow dynamics at the interface of free-porous media. We first propose a principle of virtual power by incorporating the virtual power expended at the interface of free-porous media. Then by appealing to the calculus of variations, we obtain a complete set of interface conditions for flows in coupled free-porous media. A noteworthy feature of our approach is that the derived interface conditions apply to a wide variety of porous media models. We also show that the two most popular interface conditions—the Beavers–Joseph condition and the Beavers–Joseph–Saffman condition—are special cases of the approach presented in this paper. The proposed principle of virtual power also provides a minimum power theorem for a class of flows in coupled free-porous media, which has a similar mathematical structure as the ones enjoyed by flows in uncoupled free and porous media. The derived interface conditions are summarized along with a pictorial description of the problem, which pertains to the flow of an incompressible fluid in coupled free-porous media. $$\Psi $$ is the power expended density along the interface. $$\mathbf {v}_{\text {free}}$$ and $$\mathbf {v}_{\text {por}}$$ are the velocities in the free and porous regions, respectively. A superposed asterisk on a (vectorial) quantity denotes its tangential component along the interface. $$v_n$$ is the normal component of the velocity at the interface from the free region into the porous region. $$\mathbf {T}_{\text {free}}^{\text {extra}}$$ and $$\mathbf {T}_{\text {por}}^{\text {extra}}$$ , respectively, denote the extra Cauchy stresses in the free and porous regions. $$\mathbf {t}_{\text {free}}$$ and $$\mathbf {t}_{\text {por}}$$ , respectively, denote the tractions on the free and porous sides of the interface with outward normals $$\widehat{\mathbf {n}}_{\text {free}}$$ and $$\widehat{\mathbf {n}}_{\text {por}}$$ . A unit tangential vector along the interface is denoted by $$\widehat{\mathbf {s}}$$ .