AbstractSome fundamental understanding of the blow-off (i.e., the disappearance of wall shear) of a transient, laminar boundary layer by a strong cross-flow is gained by extending the classical Stokes’first problem to include blowing (and suction). It is found, by asymptotic studies as well as detailed numerical display of exact solutions for a variety of situations (including a similarity solution where the cross-flow varies inversely with the square root of time) that there are, in general, three stages clearly discernible when blowing is present: (1) pre-blow-off stage during which the influence of the eross-flow has not shown up yet; (2) blow-off stage during which the boundary layer exhibits a zero slope in its velocity profile, and the blow-off begins; and (3) post-blow-off stage during which a wave front is seen to ride with the cross-flow, carrying with it a rather rapid change of motion of the wall to the rest-state before it, while the blow-off at the wall is sustained. Furthermore, for very large Reynolds numbers, the front in the third stage becomes very sharp, and the accompanying change very sudden. In contrast, in the case of suction, the boundary layer is seen to stick to the wall, in a more exaggerated manner; and the situation approaches a steady limit for large times.