There is a famous theory due to Thurston and Nielsen that classifies all possible continuous transformations of surfaces up to topological equivalence. When we stir a fluid in two dimensions we are essentially mapping fluid elements from a starting position to a final position, a continuous transformation. Therefore the Thurston-Nielsen theory applies and we can use it to learn a lot about the types of fluid motions that lead to good mixing, that is, those that give rise to complex, chaotic fluid motion.
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