Modelling of Poro-Acoustic Media

This paper deals with the homogenization of a binary three-dimensional structure made of damped Helmholtz resonators embedded in a rigid porous matrix when both parts of each resonator, namely the duct and the chamber, are filled with the same Stokes fluid and when the Darcy and Stokes flows are coupled by a Beavers–Joseph type condition. As the small period of the distribution shrinks to zero, we study the asymptotic behaviour of the flow when the permeability and transfer coefficients on the interface are of unity order in contrast with the periodic distribution of resonators characterized by highly conductive parallel thin ducts. To that aim the energetic procedure of homogenization is associated to the control-zone method taking into account the geometry of the microstructure and heterogeneities of the conductivity coefficients. The main result lies in the asymptotic behaviour of the periodic distribution in the binary medium characterized by a significant increase of conductivity in thin ducts whose everlasting action contrasts with their local vanishing volume. The dependence with respect to the relative thickness of the ducts is studied.