Physics

Nonlinear system

Homogenization (climate)

Mathematics

Mechanics

Porosity

Biology

Porous medium

Biodiversity

Mathematics education

Microscale chemistry

Stokes flow

Composite material

Ecology

Materials science

Flow (mathematics)

Quantum mechanics

Fluid dynamics

Journal of Fluid Mechanics

Natural and engineered surfaces are never smooth, but irregular, rough at different scales, compliant, possibly porous, liquid impregnated or superhydrophobic. The correct numerical modelling of fluid flowing through and around them is important but poses problems. For media characterized by a periodic or quasi-periodic microstructure of characteristic dimensions smaller than the relevant scales of the flow, multiscale homogenization can be used to study the effect of the surface, avoiding the numerical resolution of small details. Here, we revisit the homogenization strategy using adjoint variables to model the interaction between a fluid in motion and regularly micro-textured, permeable or impermeable walls. The approach described allows for the easy derivation of auxiliary/adjoint systems of equations which, after averaging, yield macroscopic tensorial properties, such as permeability, elasticity, slip, transpiration, etc. When the fluid in the neighbourhood of the microstructure is in the Stokes regime, classical results are recovered. Adjoint homogenization, however, permits simple extension of the analysis to the case in which the flow displays nonlinear effects. Then, the properties extracted from the auxiliary systems take the name of effective properties and do not depend only on the geometrical details of the medium, but also on the microscopic characteristics of the fluid motion. Examples are shown to demonstrate the usefulness of adjoint homogenization to extract effective tensor properties without the need for ad hoc parameters. In particular, notable results reported herein include: (i) an original formulation to describe filtration in porous media in the presence of inertial effects; (ii) the microscopic and macroscopic equations needed to characterize flows through poroelastic media; (iii) an extended Navier’s condition to be employed at the boundary between a fluid and an impermeable rough wall, with roughness elements which can be either rigid or linearly elastic; (iv) the microscopic problems needed to define the relevant parameters for a Saffman-like condition at the interface between a fluid and a porous substrate; and (v) the macroscopic equations which hold at the dividing surface between a free-fluid region and a fluid-saturated poroelastic domain.

Flow over natural or engineered surfaces: an adjoint homogenization perspective