Geometric Mixing Julyan H. E. Cartwright, 1 Emmanuelle Gouillart, 2 Nicolas Piro, 3 Oreste Piro, 4 and Idan Tuval 5 1 Instituto Andaluz de Ciencias de la Tierra, CSIC–Universidad de Granada, Campus Fuentenueva, E-18071 Granada, Spain 2 Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 93303 Aubervilliers, France 3 ´ Ecole Polytechnique F´ ed´ erale de Lausanne, CH-1015 Lausanne, Switzerland 4 Departament de F´ ısica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain 5 Department of Global Change Research, Mediterranean Institute for Advanced Studies (CSIC-UIB), E-07190 Esporles, Spain Mixing fluid in a container at low Reynolds number — in an inertialess environment — is not a trivial task. Reciprocating motions merely lead to cycles of mixing and unmixing, so continuous rotation, as used in many technological applications, would appear to be necessary. However, there is another solution: movement of the walls in a cyclical fashion to introduce a geometric phase that avoids unmixing. We show using journal–bearing flow as a model that such geometric mixing is a general tool for using deformable boundaries that return to the same position to mix fluid at low Reynolds number. PACS numbers: 05.45.-a, 47.51.+a, 47.52.+j 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 con- tainer 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 [1] and Heller [2] demonstrated. This would seem to preclude the use of recipro- cating motion to stir fluid at low Reynolds numbers; it would appear to lead to perpetual cycles of mixing and unmixing. The solution to this conundrum involves a geometric phase induced by a cyclic variation of the boundary shape. A geo- metric phase [3] is an example of anholonomy: the failure of system variables to return to their original values after a closed circuit in the parameters. In this Letter we propose what we term geometric mixing: the use of the geometric phase intro- duced by the deformable boundaries of a container as a tool for fluid mixing at low Reynolds number. We note that since a flow produced by a reciprocal cycle of the boundaries induces an identity map for the positions of each fluid element at suc- cessive cycles, the problem of mixing by nonreciprocal ones is closely related to the class of dynamical systems consti- tuted by perturbations of the identity. The structure of chaos in this class of dynamics has been greatly overlooked in the literature, and the present research opens a new avenue to the understanding of this associated problem. 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 geometrical constraints. We lastly propose that such a geo- metric phase — the “belly phase” [4] — may be found in the stomach. Taylor [1] and Heller [2] used the Couette flow of an incom- pressible fluid contained between two concentric cylinders to demonstrate fluid unmixing due to the time reversibility of the Stokes regime. They showed that after rotating the cylinders FIG. 1: The journal bearing flow with cylinder radii R1 =1.0, R2 = 0.3 and eccentricity ε =0.4, taken around a square closed parameter loop with θ1 = θ2 ≡ θ =2π radians. The four segments of the loop are plotted in different colours (red, yellow, green, blue) to enable their contributions to the particle motion to be seen. A trajectory initially at (0.0, -0.8) is shown. through a certain angle, it is possible to arrive back at the ini- tial state — to unmix the flow — by reversing this rotation through the same angle with the opposite sign, even when the angle is large enough that a blob of dye placed in the fluid has been apparently well mixed. Considering as parameters in this device the positions of the outer and inner cylindrical walls of the container specified respectively with the angles θ 1 and θ 2 from a given starting point, a geometric phase might arise from driving this system around a loop in the parameter space. In general, if one takes a system through a parameter loop, one obtains as a result three phases: a dynamic phase, a nonadiabatic phase, and a geometric phase. If one then tra- verses the same loop in the opposite direction, the dynamic phase accumulates as before, while the geometric phase is re- versed in sign. To get rid of the nonadiabatic phase too one must travel slowly around the loop. However, in a fluid sys- tem in the Stokes regime, like ours, the motion is by defini- tion always adiabatic and only associated with the change in arXiv:1206.6894v1 [nlin.CD] 28 Jun 2012