Sharp Material Interface Limit of the Darcy-Boussinesq System
Hongjie Dong, Xiaoming Wang
TL;DR
The paper addresses the problem of validating the widely used sharp material interface description for Darcy–Boussinesq convection in layered porous media by starting from a physically realistic diffuse-interface formulation with thin transition layers. It develops both sharp and diffuse interface descriptions, establishes weak formulations and a priori estimates, and conducts a convergence analysis that yields quantitative rates for the concentration, pressure, and velocity as the transition width $\varepsilon$ vanishes; a boundary-layer expansion for the velocity is derived to capture the singular limit. The main findings show that $\phi^\varepsilon \to \phi$ and $P^\varepsilon \to P$ in appropriate norms with rate $O(\varepsilon^{1/2})$, while the velocity exhibits a boundary-layer structure and a potential tangential-velocity jump across interfaces in the sharp limit. This work rigorously justifies the sharp-interface model from a more physical diffuse-interface framework on finite times and lays the groundwork for extensions to curved interfaces and more complex layering, while highlighting open questions about long-time behavior and attractors.
Abstract
We investigate the sharp material interface limit of the Darcy-Boussinesq model for convection in layered porous media with diffused material interfaces, which allow a gradual transition of material parameters between different layers. We demonstrate that as the thickness of these transition layers approaches zero, the conventional sharp interface model with interfacial boundary conditions, commonly adopted by the fluids community, is recovered under the assumption of constant porosity. Our results validate the widely used sharp interface model by bridging it with the more physically realistic case of diffused material interfaces. This limiting process is singular and involves a boundary layer in the velocity field. Our analysis requires del