An analysis is made of the two-dimensional flow under gravity of an inviscid non-diffusive stratified fluid into a line sink, involving a velocity discontinuity in the flow field. The fluid above the discontinuity is stagnant and hence is not drawn into the sink. At sufficiently low values of the modified Froude number, this is the only physically possible mode of flow, and is the cause of flow separation in many industrial and natural processes. A proper mathematical solution for flows with a stagnant zone has so far been lacking. This paper presents such a solution, after posing the problem as one involving a free-streamline, which is the line of velocity discontinuity. The solution to be given here is obtained by an inverse method. It is also found herein that the modified Froude number has a value of 0·345 for all separated flows of the kind in question.