Mixing in the presence of non-monotonicity

Non-monotonic velocity profiles are an inherent feature of mixing flows obeying no-slip boundary conditions. Here we consider simple 'stretching and folding' models of laminar fluid mixing, composing orthogonal shears on the two dimensional torus, and study the effect of imposing non-monotonic, piecewise linear shears. We give conditions under which non-mixing regions (elliptic islands) emerge and the factors which determine their size. We further study examples where no islands form, proving (measure theoretic) mixing properties over open parameter windows. Over the variety of systems considered, we encounter both uniformly and non-uniformly hyperbolic examples (with singularities). This is reflected in their mixing rates, exponential and polynomial respectively, which we establish using results from the chaotic billiards literature. We put these systems in the context of similar laminar mixing models, linked twist maps and a map of Cerbelli and Giona. Finally we consider a broader range of mixing protocols, rigorously comparing their efficiency, and discuss the challenges of relaxing piecewise linearity.