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Abstract: How may fluid be mixed at low Reynolds number? Such mixing is normally performed with a stirrer, a rotating device within the container that produces a complex, chaotic flow. Alternatively, in the absence of a stirrer, rotation of the container walls themselves can perform the mixing, as occurs in a cement mixer. At the lowest Reynolds numbers, under what is known as creeping flow conditions, fluid inertia is negligible, fluid flow is reversible, and an inversion of the movement of the stirrer or the walls leads — up to perturbations owing to particle diffusion — to unmixing, as Taylor and Heller demonstrated. This would seem to preclude the use of reciprocating motion to stir fluid at low Reynolds numbers; it would appear to lead to perpetual cycles of mixing and unmixing. Consider a biological case of cavity flow: the stomach. In the stomach food and drink are mixed to form a homogeneous fluid termed chyme, which is then digested by the intestines. Gastric mixing is produced by what is called peristalsis: by the stomach walls moving in a rhythmic fashion. In mathematical terms, the shape of the stomach walls undergoes a closed cycle in the space of shapes during each peristalsis cycle. Obviously only shape cycles that do not require a cumulative net displacement between any two sections of the stomach can be considered. How then is this peristaltic movement of the stomach walls able to produce mixing, especially in animals in which the stomach dimensions are such that fluid inertia of the stomach contents is negligible? The solution to this conundrum involves a geometric phase induced by a cyclic variation of the boundary shape. A geometric phase is an example of anholonomy: the failure of system variables to return to their original values after a closed circuit in the parameters. We propose what we term geometric mixing: the use of the geometric phase introduced by the deformable boundaries of a container as a tool for fluid mixing at low Reynolds number. To exemplify how this process leads to efficient mixing, we use the well-known two-dimensional mixer based on the journal bearing flow but subject to a much-less-studied rotation protocol that satisfies the geometrical constraints required by the stomach example. We lastly show that peristaltic mixing and digestion may operate thanks to a geometric phase in the stomach.