Homogenization Techniques for Lower Dimensional Structures

This thesis is concerned with extensions and applications of the theory of periodic unfolding in the field of (mathematical) homogenization. The first part extends the applicability of homogenization in domains with evolving microstructure to the case of evolving hypersurfaces: We consider a diffusion-reaction equation inside a perforated domain, where also surface diffusion and reaction takes place. Upon a transformation to a referential geometry, we (formally) obtain a transformed set of equations. We show that homogenization techniques can be applied to this transformed formulation. Special emphasis is placed on possible nonlinear reaction rates on the surface, a fact which requires special results for estimation and convergence results. In the limit, we obtain a macroscopic system, where each point of the domain is coupled to a system posed in the reference (micro-)geometry. Additionally, this reference geometry is evolving. In a second part, we are concerned with an extension of the notion of periodic unfolding to some Riemannian manifolds: We develop a notion of periodicity on nonflat structures in a local fashion with the help of a special atlas. If this atlas satisfies a compatibility condition, unfolding operators can be defined which operate on the manifold. We show that continuity and compactness theorems hold, generalizing the well-known results from the established theory. As an application of this newly developed results, we apply the unfolding operators to a strongly elliptic model problem. Again, we obtain a generalization of results well-known in homogenization. Moreover, we are also able to show some additional smoothness-properties of the solution of the cell problem, and we construct an equivalence relation for different atlases. With respect to this relation, the limit problem is independent of the parametrization of the manifold

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

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

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

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

Werkzeuge und Techniken für die Homogenisierung niederdimensionaler Strukturen

This thesis is concerned with extensions and applications of the theory of periodic unfolding in the field of (mathematical) homogenization. The first part extends the applicability of homogenization in domains with evolving microstructure to the case of evolving hypersurfaces: We consider a diffusion-reaction equation inside a perforated domain, where also surface diffusion and reaction takes place. Upon a transformation to a referential geometry, we (formally) obtain a transformed set of equations. We show that homogenization techniques can be applied to this transformed formulation. Special emphasis is placed on possible nonlinear reaction rates on the surface, a fact which requires special results for estimation and convergence results. In the limit, we obtain a macroscopic system, where each point of the domain is coupled to a system posed in the reference (micro-)geometry. Additionally, this reference geometry is evolving. In a second part, we are concerned with an extension of the notion of periodic unfolding to some Riemannian manifolds: We develop a notion of periodicity on nonflat structures in a local fashion with the help of a special atlas. If this atlas satisfies a compatibility condition, unfolding operators can be defined which operate on the manifold. We show that continuity and compactness theorems hold, generalizing the well-known results from the established theory. As an application of this newly developed results, we apply the unfolding operators to a strongly elliptic model problem. Again, we obtain a generalization of results well-known in homogenization. Moreover, we are also able to show some additional smoothness-properties of the solution of the cell problem, and we construct an equivalence relation for different atlases. With respect to this relation, the limit problem is independent of the parametrization of the manifold