Derivation of boundary conditions at a curved contact interface between a free fluid and a porous medium via homogenisation theory

Abstract

In soil chemistry or marine microbiology (for example when dealing with marine aggregates), one often encounters situations where porous bodies are suspended in a fluid. In this context, the question of boundary conditions for the fluid velocity and pressure at the porous-liquid interface arises. Up to the present, only results for straight interfaces are known. In this work, the behaviour of a free fluid above a porous medium is investigated, where the interface between the two flow regions is assumed to be curved. By carrying out a coordinate transformation, we obtain the description of the flow in a domain with a straight boundary. We assume the geometry in this domain to be epsilon-periodic. Using periodic homogenisation, the effective behaviour of the solution of the transformed partial differential equations in the porous part is obtained, yielding a Darcy law with a non-constant permeability matrix. The boundary layer approach of Jäger and Mikelic is then generalized to construct corrections at the interface. Finally, this allows us to obtain the fluid behaviour at the porous-liquid interface: Whereas the velocity in normal direction is continuous over the interface, a jump appears in tangential direction. The magnitude of this jump can explicitely be calculated and seems to be related to the slope of the interface. Therefore the results indicate a generalized law of Beavers and Joseph.