Reduction of dimension for diffusion in a perforated thin plate

Abstract

The steady-state diffusion problem is considered in a thin plate perforated periodically by many cylindrical holes of critical sizes. The plate is scaled to a plate of thickness 1. The asymptotic behaviour of the solution to the resulting rescaled equation is studied when the thickness of the original plate, the holes' size and period converge to 0. The phenomenon of dimension reduction occurs, i.e the limiting equation is posed in the cross-section only. The equation contains a linear term which describes the sink effect of the holes. This term depends on the relationship between the thickness, the period and the size of the perforations.