Stokes flow is considered in a rectangular double-lid-driven cavity with free surface side walls, aspect ratio A and speed ratio S = U-1/U-2, where U-1 and U-2 are lid velocities, as an idealized model for two-dimensional viscous flow in the 'bead' of a two-roll, meniscus coating system. For S less than or equal to 0 (> 0) there is one (two) large eddy (eddies) within the cavity with one (two) stagnation points on the centre-line between the free surfaces. Various transformations of flow structure arising as a result of flow bifurcations at these stagnation points are identified as the two control parameters A and S are continuously varied. In the case of symmetric flow, S = -1, reducing A from A = 2.5 produces a sequence of pitchfork bifurcations at which a stagnation-saddle point changes to a centre (or vice versa) with the generation of two additional stagnation points. As A-->0 and the number of stagnation points increases, it is shown that the flow structure consists of a series of nested separatrices, each contained within the central section of the previous one.