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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)

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.

View Article: PubMed Central - PubMed

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.

No MeSH data available.


Related in: MedlinePlus

Geometric mixing in the journal bearing flow.(a–c) Poincaré maps demonstrate geometric mixing for the journal–bearing flow for the same cylinder radii and eccentricity as in Fig 1. Chaotic trajectories are marked in red while regular ones appear in blue. 10000 iterations of the parameter loop are shown for: (a) θ = π/2 radians; (b) θ = 2π and (c) θ = 4π. (d–f) The geometric phase across the domain for the same parameters; the color scale denoting the phase at a given point is given by the intensity of red, positive, and blue, negative. (g–i) The evolution of a line segment—initial segments in red; final segments in blue—across the widest gap between the cylinders after one cycle for the same parameters.
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pone.0130735.g002: Geometric mixing in the journal bearing flow.(a–c) Poincaré maps demonstrate geometric mixing for the journal–bearing flow for the same cylinder radii and eccentricity as in Fig 1. Chaotic trajectories are marked in red while regular ones appear in blue. 10000 iterations of the parameter loop are shown for: (a) θ = π/2 radians; (b) θ = 2π and (c) θ = 4π. (d–f) The geometric phase across the domain for the same parameters; the color scale denoting the phase at a given point is given by the intensity of red, positive, and blue, negative. (g–i) The evolution of a line segment—initial segments in red; final segments in blue—across the widest gap between the cylinders after one cycle for the same parameters.

Mentions: Let us now consider the long-term fluid dynamics elicited by a repeated realization of the same contractible non-reciprocal loop that induces a given map. The dynamics is described by the repeated iteration of this map that acts as the stroboscopic map of the time-periodic Hamiltonian system constituted by the incompressible flow periodically driven by the motion of the walls. For small loops, the map is a small perturbation of the identity that can be thought of as the implementation of the Euler algorithm for a putative continuous time dynamical system defined by this perturbation. Therefore, in 2D we expect that the iterations of the map will closely follow the trajectories of this 2D continuous system which is integrable. Therefore, fluid particles will mix very slowly in space: this is, so to speak, mixing by “quasi-static” fluids. This is nicely illustrated in Fig 2(a), where even for a square loop formed with values as large as θ = π/2 the positions of fluid particles after successive loops smoothly shift along the closed curves that are the trajectories of the continuous dynamics. The trajectories are composed of segments that nearly follow the integrable trajectories of a 2D flow (approximated as an Euler map) until it reaches the region of large phase, where chaos and heteroclinic tangles occur. There the particle jumps into another quasi-integrable trajectory, until it again reaches the region of large phase. In typical Hamiltonian chaos (the standard map, for example) the map is not a perturbation of the identity but a perturbation of a linear shear (i.e. with the canonical action-angle dynamical variables (I, φ) following I′ = I, φ′ = φ+I′) for which reason this behavior is not normally seen [17, 18]. 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.


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)

Geometric mixing in the journal bearing flow.(a–c) Poincaré maps demonstrate geometric mixing for the journal–bearing flow for the same cylinder radii and eccentricity as in Fig 1. Chaotic trajectories are marked in red while regular ones appear in blue. 10000 iterations of the parameter loop are shown for: (a) θ = π/2 radians; (b) θ = 2π and (c) θ = 4π. (d–f) The geometric phase across the domain for the same parameters; the color scale denoting the phase at a given point is given by the intensity of red, positive, and blue, negative. (g–i) The evolution of a line segment—initial segments in red; final segments in blue—across the widest gap between the cylinders after one cycle for the same parameters.
© Copyright Policy
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4496066&req=5

pone.0130735.g002: Geometric mixing in the journal bearing flow.(a–c) Poincaré maps demonstrate geometric mixing for the journal–bearing flow for the same cylinder radii and eccentricity as in Fig 1. Chaotic trajectories are marked in red while regular ones appear in blue. 10000 iterations of the parameter loop are shown for: (a) θ = π/2 radians; (b) θ = 2π and (c) θ = 4π. (d–f) The geometric phase across the domain for the same parameters; the color scale denoting the phase at a given point is given by the intensity of red, positive, and blue, negative. (g–i) The evolution of a line segment—initial segments in red; final segments in blue—across the widest gap between the cylinders after one cycle for the same parameters.
Mentions: Let us now consider the long-term fluid dynamics elicited by a repeated realization of the same contractible non-reciprocal loop that induces a given map. The dynamics is described by the repeated iteration of this map that acts as the stroboscopic map of the time-periodic Hamiltonian system constituted by the incompressible flow periodically driven by the motion of the walls. For small loops, the map is a small perturbation of the identity that can be thought of as the implementation of the Euler algorithm for a putative continuous time dynamical system defined by this perturbation. Therefore, in 2D we expect that the iterations of the map will closely follow the trajectories of this 2D continuous system which is integrable. Therefore, fluid particles will mix very slowly in space: this is, so to speak, mixing by “quasi-static” fluids. This is nicely illustrated in Fig 2(a), where even for a square loop formed with values as large as θ = π/2 the positions of fluid particles after successive loops smoothly shift along the closed curves that are the trajectories of the continuous dynamics. The trajectories are composed of segments that nearly follow the integrable trajectories of a 2D flow (approximated as an Euler map) until it reaches the region of large phase, where chaos and heteroclinic tangles occur. There the particle jumps into another quasi-integrable trajectory, until it again reaches the region of large phase. In typical Hamiltonian chaos (the standard map, for example) the map is not a perturbation of the identity but a perturbation of a linear shear (i.e. with the canonical action-angle dynamical variables (I, φ) following I′ = I, φ′ = φ+I′) for which reason this behavior is not normally seen [17, 18]. 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.

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.

View Article: PubMed Central - PubMed

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.

No MeSH data available.


Related in: MedlinePlus