Abstract 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, e (η and e are two small parameters). In the three-dimensional case the critical behaviour appears for e = η 2 : 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 e tends to a constant; there is a differential relation between the flow through the wall and the jump of pressure.
[1]
R. Finn.
On the exterior stationary problem for the navier-stokes equations, and associated perturbation problems
,
1965
.
[2]
J. G. Heywood.
On uniqueness questions in the theory of viscous flow
,
1976
.
[3]
A. Bensoussan,et al.
Asymptotic analysis for periodic structures
,
1979
.
[4]
J. L. Lions,et al.
Remarks on Some Asymptotic Problems in Composite and in Perforated Materials
,
1980
.
[5]
Enrique Sanchez-Palencia,et al.
Equations and interface conditions for acoustic phenomena in porous media
,
1977
.
[6]
E. O. Tuck,et al.
Matching problems involving flow through small holes
,
1975
.
[7]
R. Temam.
Navier-Stokes Equations
,
1977
.
[8]
E. Sanchez-Palencia.
Non-Homogeneous Media and Vibration Theory
,
1980
.