Publication
Porous media are solid structures perforated by a matrix of pores, which may let a fluid flow across them. Geometrically, we can classify them as "thin" if their thickness is comparable with the size of their pores, and "bulk" otherwise. These structures can be found in various biological and man-made systems, such as cell membranes, sponges, insect feathers, flow control devices for aerodynamics and cooling applications and filtration devices. A common feature of these systems is the presence of a large difference in size between the pores and the full membrane. From the modeling point of view, two approaches are generally considered: direct solutions of the governing equations and average models. The first strategy guarantees high accuracy but may easily become computationally unaffordable, particularly when the pores are very small compared to the membrane. The second approach models the effect of the membrane using mathematical relations between the flow far from the membrane and the pore-scale behavior. These laws often rely on empirical coefficients, thus limiting their predictive capabilities. However, the computational cost associated with them is tiny compared to direct solutions, and they find use in many engineering simulation routines. Thanks to multi-scale techniques such as homogenization, we can derive such average models from first principles and find out that the coefficients appearing inside them stem from the solution of a set of solvability conditions valid at the pore level. Starting from the elementary case of Stokes flow across a thin membrane, developed by Zampogna & Gallaire (2020), we build further pieces of theory into this framework. Inspired by the microstructure heterogeneities found at the cellular level or in defective man-made membranes, in chapter 2, we introduce an efficient adjoint-based methodology to evaluate the effect of many geometric modifications of the microstructure at a glance. The effect of inertia at the pore level, which is relevant for applications in the flow-control domain, is introduced in chapter 3, where we also consider the presence of a solute advected by the fluid flow. These chapters use a simple but effective approximation, consisting of a unique value for the homogenized flow variables on the membrane. This approximation, however, is sub-optimal when we deal with ``contradictory'' membrane properties, found in the so-called "Janus membranes". A new, discontinuous flow description at the membrane level in the presence of solute transport is described in chapter 4. The model is subsequently used in chapter 5 to deal with diffusio-phoresis, i.e. when a difference in local solute concentration, interacting with the membrane wall, causes a fluid flow. However, in many systems such as reverse-osmosis filters and cell membranes, the pore dimensions are so small that a continuous flow description may not be accurate. In such cases, the proper simulation tool is constituted by molecular dynami