Surfaces covered by bristles, hairs, polymers and other filamentous structures arise in
a variety of natural settings in science such as the active lining of many biological
organs, e.g. lungs, reproductive tracts, etc., and have increasingly begun to be used in
technological applications. We derive an effective field theory for the elastohydrodynamics
of ordered brushes and disordered carpets that are made of a large number of elastic
filaments grafted on to a substrate and interspersed in a fluid. Our formulation for the
elastohydrodynamic response of these materials leads naturally to a set of constitutive
equations coupling bed deformation to fluid flow, accounts for the anisotropic properties
of the medium, and generalizes the theory of poroelasticity to these systems. We use the
effective medium equations to study three canonical problems—the normal settling of
a rigid sphere onto a carpet, the squeeze flow in a carpet and the tangential shearing
motion of a rigid sphere over the carpet, all problems of relevance in mechanosensation
in biology with implications for biomimetic devices.