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The resulting characteristic direction is the direction of the unstable manifolds of the underlying map. we create a flow by alternating horizontal and vertical shears. The fact that the flow directions are orthogonal means that the streamline crossing is in some sense the ‘most transverse’. 1. The map is known as the Arnold Cat Map (Arnold & Avez (1968)), and forms a fundamental example of a dynamical system for which rigorous mathematical results about mixing can be proved, and also has been used as a prototype for mixing in many applications (see for example, Childress & Gilbert (1995), Thiffeault & Childress (2003) or Boyland et al.

If F is applied to a point (x, y) in P, the y-coordinate is left unchanged, but the x-coordinate is altered by an amount dependent on the value of y. , the component of F defined on P and the component of F defined on R\P) should ‘join up’ at the boundary of P – that is, F should be continuous on the boundary of P, denoted ∂P. This point is crucial mathematically and will be carefully discussed in later chapters. In terms of fluids, it corresponds to systems where initially connected blobs of fluid remain connected blobs, with no tearing or break-up taking place.

Note the highlighted square in panel (a). A fluid line element in the square perpendicular to a blade is deformed as shown in panel (b) as a blade pushes through it. The other blade pushes through the line in a perpendicular direction, as shown in panel (c). Parts of the line element that extend out the top of the square later re-enter through the bottom. If the blades are rotated alternately then the flow can be described by a LTM. However, this is a LTM on the plane rather than a torus. , the one on the left is rotating clockwise and the one on the right is rotating counterclockwise.

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