Abstract Considering steady Hiemenz-Birman flows only, a study is made of flows between porous walls, on the assumption that fluid is injected and extracted at identical rates. It is shown that wherever fluid is being extracted a boundary layer forms at the wall. A class of unsteady two-dimensional flows, more general than Hiemenz-Birman flow, is investigated. In a class of flows generalized Jeffrey-Hamel flow, attention is devoted to flows in a dihedral angle between porous walls when fluid is injected and extracted. A class of steady (unsteady) two-dimensional flows is found, in which flow between coaxial porous cylinders, with fluid injected and extracted at arbitrary rates, is considered. Some exact solutions of the steady- and unsteady-state Navier-Stokes equations are found.