Stokes flow in a double-lid-driven cavity with free surface side walls

Abstract

Stokes flow is considered in a rectangular double-lid-driven cavity with free surface side walls, aspect ratio A and speed ratio S= U1/ U2, where U1 and U2 are lid velocities, as an idealized model for two-dimensional viscous flow in the ‘bead’ of a two-roll, meniscus coating system. For S ≤ 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. In the case of asymmetric flow, S ≠ -1, S < 0, the above pattern of behaviour is repeated as A is reduced from A = 2.5, apart from an initial saddle-node bifurcation at which a saddle and a centre annihilate each other. When the lids move in the same direction ( S > 0) bifurcations may arise separately or simultaneously in each of the two large eddies. Again as A → 0 with S constant, nested separatrices appear in both eddies. The key results are displayed via bifurcation curves in an ( S, A) control space diagram.