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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  =  -  3for an elastic sheet and    $$c= -1$$    c  =  -  1for 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    $$S&gt;0$$    S  &gt;  0for the fluid withdrawal and    $$S &lt; 0$$    S  &lt;  0for fluid injection. Both the cases    $$S=0$$    S  =  0(impermeable surface) and    $$S \ne 0$$    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).

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