Geometric Mixing, Peristalsis, and the Geometric Phase of the Stomach.

Arrieta J, Cartwright JH, Gouillart E, Piro N, Piro O, Tuval I - PLoS ONE (2015)

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pone.0130735.g001: A finite-area non-reciprocal contractible loop.The journal bearing flow with cylinder radii R1 = 1.0, R2 = 0.3 and eccentricity ɛ = 0.4, taken around a closed square 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 beginning at (0.0, −0.8) is shown. The inset shows the performed loop in parameter space.
Mentions: All zero-area reciprocal loops are contractible, but there are many more enclosing a finite area. To obtain a finite-area non-reciprocal contractible loop we can, for instance, rotate first one cylinder, then the other, then reverse the first, and finally reverse the other. However, for concentric cylinders the streamlines are concentric circles; if we move one of the cylinders by angle θ, a tracer particle will move along a circle an angle that only depends on θ. Then it is obvious that the cumulative effect of moving one cylinder θ1, then the other θ2, then the first −θ1, and the second −θ2, is to return the particle to its original position: there is no geometric phase, and unmixing still occurs. But if we modify the Heller–Taylor setup and offset the inner cylinder, we arrive at what is known as journal–bearing flow. On introducing an eccentricity ɛ between the cylinders, this flow has a radial component. In the creeping-flow limit, the Navier–Stokes equations for the journal–bearing flow reduce to a linear biharmonic one, ∇4ψ = 0, for the stream function, ψ, and we may model this system utilizing an analytical solution (see [10–12] and the Materials and Methods section for further details). If we now perform a parameter loop by the sequence of rotations detailed above, we arrive back at our starting point from the point of view of the positions of the two cylinders, so it is, perhaps, surprising that the fluid inside does not return to its initial state. We illustrate the presence of this geometric phase in Fig 1 in which an example of the trajectory of a fluid particle is shown as the walls are driven through a nonreciprocal contractible loop. Journal–bearing flow has been much studied in the past [13–16], but never with contractible loops so that this geometric effect was never emphasized. This minor protocolary modification in a well established flow has, nonetheless, a substantial effect on the fluid dynamics as we describe below.

Bottom Line: However, there is another solution: movement of the walls in a cyclical fashion to introduce a geometric phase.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.We then simulate a biological example: we show that mixing in the stomach functions because of the "belly phase," peristaltic movement of the walls in a cyclical fashion introduces a geometric phase that avoids unmixing.

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Affiliation: Mediterranean Institute for Advanced Studies (CSIC-UIB), E-07190 Esporles, Spain; Área de Mecánica de Fluidos, Universidad Carlos III de Madrid, E-28911, Leganés, Spain.

ABSTRACT
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. 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. We then simulate a biological example: we show that mixing in the stomach functions because of the "belly phase," peristaltic movement of the walls in a cyclical fashion introduces a geometric phase that avoids unmixing.

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