By Damascelli L., Pacella F.

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This flow lacks the hyperbolic structure necessary for this direction to be an unstable manifold (a key difference is that nearby trajectories are attracted to this direction only algebraically fast, rather than exponentially fast), but nevertheless the horizontal direction can be regarded as the characteristic direction for this shear flow. 3 shows an analogous shear flow in a vertical annulus, with top and bottom edges identified. The shear is stronger at the right edge of the square than the left edge.

The same dichotomy presents itself – that of co-rotating and counter-rotating systems. To be consistent with the majority of the literature on shear flows, we define a co-rotating system to be a system for which the shears either both act clockwise, or both act counterclockwise (that is, both with increasing angle or both with decreasing angle). Note that this definition is the same as for toral linked twist maps. Similarly, if the shears act in opposite senses, the system is a counter-rotating linked twist map.

2(a), where the left and right edges of the square are identified (suspending for the time being concerns about how such a flow might be created). The length of the arrows indicate that in unit time a point in the flow nearer the top of the annulus moves further than a point nearer the bottom. 2(b), is released into the flow (it may seem unlikely to have a perfectly square blob of fluid, but it helps to visualize what happens to vertical and horizontal lines). 2(e). The longer the flow is run (or equivalently, the stronger the effect of the shear), the closer the alignment of the points in the blob gets to the direction of the streamlines.

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Monotonicity and symmetry results for p-Laplace equations and applications by Damascelli L., Pacella F.

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