On an equation arising in the boundary-layer flow of stretching/shrinking permeable surfaces

In a recent paper, Al-Housseiny and Stone (J Fluid Mech 706:597–606, 2012) considered the dynamics of a stretching surface and how this interacts with the boundary-layer flow it generates. These authors discussed the cases $$c= -3$$ c=-3 for an elastic sheet and $$c= -1$$ c=-1 for the viscous fluid, c being representative for the stretching velocity of the sheet. The aim of the present paper is to extend the analysis of Al-Housseiny and Stone (2012) to the general values of c , to allow for both a stretching and a shrinking sheet and for the surface to be permeable through the parameter S , where $$S>0$$ S>0 for the fluid withdrawal and $$S < 0$$ S<0 for fluid injection. Both the cases $$S=0$$ S=0 (impermeable surface) and S ≠ 0 S≠0 (permeable surface) are considered for both stretching surfaces and shrinking surfaces. In all these cases, asymptotic solutions are presented for large values c and S (both withdrawal and injection).