Acoustic fluid flow through holes and permeability of perforated walls

We first study the unsteady incompressible fluid flow through a hole in a wall in the two- and three-dimensional cases. In the first case, a convolution relation is obtained between the fluid flux through the hole and the difference of pressure between the far regions on the two sides of the wall. In the two-dimensional case, the pressure increases logarithmically with distance from the wall. In a second part, we study acoustic flow in a domain containing a wall with many small holes. The distance between two contiguous holes is of order η and the size of each hole, ε (η and ε are two small parameters). In the three-dimensional case the critical behaviour appears for ε = η²: it is described by a convolution relation between the flow through the wall and the jump of pressure. In the two-dimensional case, the critical behaviour appears if η log ε tends to a constant; there is a differential relation between the flow through the wall and the jump of pressure.