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GEOMETRIC MIXING, PERISTALSIS, AND THE GEOMETRIC PHASE OF THE STOMACH

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  1. GEOMETRIC MIXING, PERISTALSIS, AND THE GEOMETRIC PHASE OF THE STOMACH Julyan Cartwright CSIC ! With: Jorge Arrieta, Universidad Carlos III de Madrid, Madrid, Spain Emmanuelle Gouillart, UMR 125 CNRS, Aubervilliers, France Nicolas Piro, EPFL, Lausanne, Switzerland Oreste Piro, Department of Physics, Univ. of Balearic Islands, Spain Idan Tuval, IMEDEA CSIC–Univ. of Balearic Islands, Spain

  2. Mixing at low Reynolds Number An interesting problem for dynamics: ! Reversibility of the flows in the Stokes regime ! Hamiltonian and volume preserving dynamics in real space. An interesting problem for industrial applications: ! Mixing of very viscous flows ! Mixing at very small scales (microfluidics) ! An emblematic industrial case of mixing at low Re: ! Food processing

  3. Our natural food processor The stomach is not just a transport device, but a mixer, where food is mixed with gastric juices to produce the chyme that later passes to the intestine.

  4. The stomach at work
 (peristalsis) We can see the stomach as a container whose boundary performs a repeated cycle on the space of shapes to produce transport and mixing.

  5. Re at the stomach - Human stomach volume (≈L3)= 330 mL, - Viscosity of the chyme : 1 Pa s - Density of chyme 103 kg/m3 -Typical flow 2.5–7.5 mm/s ! Re lies in the range 0.2–0.5 ! (deep in the Stokes regime)

  6. Reversibility 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. Borrowed from G. I. Taylor, Low Reynolds Number Flow (Educational Services Incorporated, 1960), (16 mm film).

  7. Reversibility In mathematical terms, we could say that after a reciprocal cycle in the motion of the internal cylinder, the initial configuration of the fluid has mapped onto itself. ! If we call TRL(r) the map giving the position of a given element of the fluid initially located at the position r, after a cycle we trivially have:

  8. Chaotic advection To the help of mixing at low Re comes CHAOTIC ADVECTION ! This uses non-reciprocal cycles to produce chaotic trajectories of the fluid particles inside the container: Paradigm of chaotic advection: The journal bearing flow. ! The cylinders are now eccentric and they rotate alternately with two different (time dependent) angular velocities

  9. In 2D the flow can be derived from a stream function which is a solution of the stationary Stokes equation with no-slip boundary conditions at both cylinders. and are respectively the stream functions corresponding to having only the inner or the outer cylinder rotating while the other one is kept fixed. ! Both functions have an analytical expression which is too large to fit in this margin (Ballal & Rivlin 1976).

  10. Typical stirring protocol Net displacement of one wall with respect to the other

  11. J. M. Ottino, 1988

  12. Typical stirring protocol Net displacement of one wall with respect to the other NOT GOOD TO MIMICK PERISTALSIS

  13. Geometric mixing protocol -θ0 -θ1 θ1 θ0 A loop in the shape space without net displacement A BETTER MIMICK FOR PERISTALSIS

  14. Geometric Mixing Julyan H. E. Cartwright,1Emmanuelle Gouillart,2Nicolas Piro,3Oreste Piro,4and Idan Tuval5 1Instituto Andaluz de Ciencias de la Tierra, CSIC–Universidad de Granada, Campus Fuentenueva, E-18071 Granada, Spain 2Surface du Verre et Interfaces, UMR 125 CNRS/Saint-Gobain, 93303 Aubervilliers, France 3´Ecole Polytechnique F´ ed´ erale de Lausanne, CH-1015 Lausanne, Switzerland 4Departament de F´ ısica, Universitat de les Illes Balears, E-07071 Palma de Mallorca, Spain 5Mediterranean 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. On occasions, however, mixing is attempted by a cyclic deformation of the container walls that does not allow for a net relative displacement of the corresponding surfaces, situations that often occur both in artificial devices and in liv- ing organisms. At the lowest Reynolds numbers, under what is known as creeping flow conditions, fluid inertia is negligi- ble, 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 re- ciprocating motion to stir fluid at low Reynolds numbers; it would appear to lead to perpetual cycles of mixing and un- mixing. The question then arises of how cyclic changes in the shape of the containers could lead to efficient mixing. FIG. 1: The journal bearing flow with cylinder radii R1 = 1.0, R2 = 0.3 and eccentricity " = 0.4, taken around a closed square parameter loop with ✓1 = ✓2 ⌘ ✓ = 2⇡ 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 solution to this conundrum involves the concept of ge- ometric phase. A geometric phase [3] is an example of an- holonomy: 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 introduced by nonreciprocal cycling of 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 the geometrical constraints of cyclic boundary deformations. Taylor [1] and Heller [2] used theCouette 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 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 ✓1 and ✓2from a given starting point, a geometric phase might arise from driving this system around a loop in the parameter space. In Hamiltonian dynamics, when one takes a system through a parameter loop, one obtains as a result three phases: a dy- namic phase due to the time evolution of the dynamical vari- ables, a nonadiabatic phase that is also dynamic but induced

  15. Observe that the geometric mixing protocol induces a map that is a relatively small perturbation of the identity. ! A small perturbation of the identity map produces by iteration a trajectory that resembles a continuous time dynamical system (flow). ! (The map can be seen as the Euler discretization of a flow)

  16. The geometric phase:
 a measure of geometric chaos The geometric phase (a.k.a Berry phase, Hannay angle, etc…) is a measure of the failure of a system to return to the original position in the phase space as a parameter performs an adiabatic closed cycle. Pedagogical example: Foucault’s pendulum: ! ! ! ! ! ! ! ! ! Proportional to the solid angle spanned by pole or the area encircled by the suspension point

  17. The geometric phase:
 a measure of geometric chaos The geometric phase (a.k.a Berry phase, Hannay angle, etc…) is a measure of the failure of a system to return to the original position in the phase space as a parameter performs an adiabatic closed cycle. Pedagogical example: Foucault’s pendulum: ! ! ! ! ! ! ! ! !

  18. The geometric phase:
 a measure of falling cats

  19. The geometric phase:
 a measure of falling rhinos?

  20. The geometric phase:
 a measure of geometric chaos An intuitive way to understand the properties of the phase is given by the parallel transport of a vector on a sphere. The angle α by which the transported vector twists is proportional to the area inside the loop.

  21. The geometric phase:
 a measure of geometric chaos Since in general, in dynamics this geometric effect on the phase comes together with the dynamical evolution, one should separate the two contributions. Since the latter changes sign under time reversal and the geometric one does not, we can define the geometric phase as where φ is a suitably defined angle in the phase space and φ+ and φ− are respectively the time evolved angle as the cycle proceeds in one direction and in the opposite direction, respectively.

  22. In low Re fluid flow the Lagrangian trajectory of the parcel evolves just due to the geometric effect and does not carry any further dynamics. But we maintain the same definition.

  23. Poincaré map Geometric phase Evolution of a material line after one cycle Area of the loop FIG. 2: (a–c) Poincar´ e 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) ✓ = ⇡/2 radians; (b) ✓ = 2⇡ and (c) ✓ = 4⇡. (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. by the finite rate of change of the parameters, and a geo- metric phase, originated only by the geometry of the system, which would be present even when the parameters are var- ied infinitely slowly. If one then traverses the same loop in the opposite direction, the dynamic phase accumulates as be- fore, while the geometric phase is reversed in sign. Therefore, subtracting the total phase accumulated when traveling very slowly (in order to make the nonadiabatic effects negligible) in opposite directions around the loop, one can compute the geometric phase. However, in a fluid system in the Stokes regime, like ours, the motion is by definition always adiabatic and only induced by the change in the parameters: the posi- tions of the cylinders. Therefore, any resulting phase is a geo- metric phase. In the Heller–Taylor demonstration the parame- ter loop is very simple: ✓1first increases a certain amount and then decreases the same amount while ✓2remains fixed. This loop encloses no area, and reversibility ensures that the phase is zero. More complex zero-area loops can be constructed by combining in succession arbitrary pairs of reciprocal rota- tions of both cylinders, and they also lead to a null phase. We shall call these constructs reciprocal cycles. In order to con- sider less trivial loops, we may first note that the parameter space is homotopic to a 2-torus. Loops on such a space can be classified according to the number of complete turns that both parameters accumulate along the loop. Note also that a rela- tive rotation of 2⇡ between the walls brings the container to the original configuration except for a global rotation. Since we are interested in shape loops that can be achieved without a net cumulative displacement of the surfaces of the containers, we need to consider only the class of type-0 or contractible (to a point) loops. 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 ✓, a tracer particle will move along a cir- cle an angle fr(✓) that only depends on ✓. Then it is ob- vious that the cumulative effect of moving one cylinder ✓1, then the other ✓2, then the first ?✓1, and the second ?✓2, is to return the particle to its original position: there is no ge- ometric phase, and unmixing still occurs. But if we modify the Heller–Taylor setup and offset the inner cylinder, we ar- rive at what is known as journal–bearing flow. On introducing an eccentricity " between the cylinders, this flow has a ra- dial component. In the creeping-flow limit, the Navier–Stokes equationsforthejournal–bearingflowreducetoalinearbihar-

  24. induce larger deviations. 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 thestroboscopicmapofthetime-periodicHamiltoniansystem 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 im- plementation of the Euler algorithm for a putative continuous time dynamical system defined by this perturbation. There- fore, in2Dweexpectthattheiterationsofthemapwillclosely follow the trajectories of this 2D continuous system which is integrable. Therefore, fluid particles will mix very slowly in space. This is nicely illustrated in Fig. 2(a), where even for a square loop formed with values as large as ✓ = ⇡/2 the po- sitions of fluid particles after successive loops smoothly shift along the closed curves that are the trajectories of the continu- ous dynamics. The trajectories are composed of segments that nearly follow the integrable trajectories of a 2D flow (approx- imated 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 Hamilto- nian chaos (the standard map, for example) the map is not a perturbation of the identity but a perturbation of a linear shear (I0= I, '0= ' + I0) for which reason this behavior is not normally seen. As the geometric phase and the corresponding perturbation from the identity map increase, the former argument begins to fail [11]. A more chaotic 2D-area preserving map emerges and with it the corresponding space-filling fully chaotic tra- jectories. The KAM islands typically become smaller and smaller as the characteristic values of geometric phase in- crease. As we see in Fig. 2(b) for ✓ = 2⇡ radians, and even more so in 2(c) for ✓ = 4⇡ radians, after 10000 cycles the fluid particle has covered most of the area available to it be- tween the two cylinders. This is fluid mixing induced entirely by a geometric phase; we may call it geometric mixing. Geo- metric mixing therefore creates chaotic advection [9], as does the classical journal–bearing protocol. In Fig. 2(d–f) we show the corresponding distributions of the geometric phase over the domain. The value of the geo- metric phase at a given initial position, obtained in terms of the final angle minus the initial angle in bipolar coordinates after one iteration, ? = ⇠f? ⇠i, is plotted on a color scale of intensities of red (positive) and blue (negative). Note that the phase goes to zero at the walls, as it must, but varies strongly across the domain. In particular, for parameters of ✓ = 2⇡ radians (Fig. 2(e)), we see the development of a tongue of high values of the geometric phase in one sense interpene- trating a region of high values of the phase in the opposite sense. The trajectory plotted in Fig. 1 shows the origin of the tongue; fluid particles that are advected to the vicinity of the inner cylinder by the first ✓1step are then advected to a signif- icantly different value of r by the inner cylinder. As a result, FIG. 3: (a) The L2-norm of the ? field grows quadratically with ✓ for loops with small area. Two distinct loops with equal area are shown. (b) The final length of a line segment as shown in Figure 2 (g–i) plotted after one cycle for flows with different rotation angles ✓. monic one, r4= 0, and we may model this system utilizing an analytical solution of the stream function in bipolar coor- dinates [4–6]. 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 [7–10], but never with contractible loops so that this geometric effect was never em- phasized. A fluid particle that at the beginning of the loop is in a position (x,y), reaches, at the end of the same loop, a unique corresponding point (x0,y0) which is a one-to-one function (x0,y0) = T[(x,y)] of the initial one. For homo- geneous fluids, T must also be continuous and differentiable, whereas incompressibility implies that T preserves the area of any domain of points. In other words, incompressible flow implies Hamiltonian dynamics for the fluid particles, and the map that this dynamics induces in one loop is area preserving. Forcontractiblezero-arealoopsthemapissimplytheidentity; each particle ends in the position in which it started. Hence, a finite-area loop induces, in general, a finite deviation from the identity map and a characteristic value of the geometrical phase gives an estimate for the extent of this deviation. Since generically the geometric phase increases with the area of the loop (see Fig. 3(a)), for small loops the map is a small pertur- bation away from the identity whereas loops of greater area

  25. induce larger deviations. 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 thestroboscopicmapofthetime-periodicHamiltoniansystem 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 im- plementation of the Euler algorithm for a putative continuous time dynamical system defined by this perturbation. There- fore, in2Dweexpectthattheiterationsofthemapwillclosely follow the trajectories of this 2D continuous system which is integrable. Therefore, fluid particles will mix very slowly in space. This is nicely illustrated in Fig. 2(a), where even for a square loop formed with values as large as ✓ = ⇡/2 the po- sitions of fluid particles after successive loops smoothly shift along the closed curves that are the trajectories of the continu- ous dynamics. The trajectories are composed of segments that nearly follow the integrable trajectories of a 2D flow (approx- imated 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 Hamilto- nian chaos (the standard map, for example) the map is not a perturbation of the identity but a perturbation of a linear shear (I0= I, '0= ' + I0) for which reason this behavior is not normally seen. As the geometric phase and the corresponding perturbation from the identity map increase, the former argument begins to fail [11]. A more chaotic 2D-area preserving map emerges and with it the corresponding space-filling fully chaotic tra- jectories. The KAM islands typically become smaller and smaller as the characteristic values of geometric phase in- crease. As we see in Fig. 2(b) for ✓ = 2⇡ radians, and even more so in 2(c) for ✓ = 4⇡ radians, after 10000 cycles the fluid particle has covered most of the area available to it be- tween the two cylinders. This is fluid mixing induced entirely by a geometric phase; we may call it geometric mixing. Geo- metric mixing therefore creates chaotic advection [9], as does the classical journal–bearing protocol. In Fig. 2(d–f) we show the corresponding distributions of the geometric phase over the domain. The value of the geo- metric phase at a given initial position, obtained in terms of the final angle minus the initial angle in bipolar coordinates after one iteration, ? = ⇠f? ⇠i, is plotted on a color scale of intensities of red (positive) and blue (negative). Note that the phase goes to zero at the walls, as it must, but varies strongly across the domain. In particular, for parameters of ✓ = 2⇡ radians (Fig. 2(e)), we see the development of a tongue of high values of the geometric phase in one sense interpene- trating a region of high values of the phase in the opposite sense. The trajectory plotted in Fig. 1 shows the origin of the tongue; fluid particles that are advected to the vicinity of the inner cylinder by the first ✓1step are then advected to a signif- icantly different value of r by the inner cylinder. As a result, FIG. 3: (a) The L2-norm of the ? field grows quadratically with ✓ for loops with small area. Two distinct loops with equal area are shown. (b) The final length of a line segment as shown in Figure 2 (g–i) plotted after one cycle for flows with different rotation angles ✓. monic one, r4= 0, and we may model this system utilizing an analytical solution of the stream function in bipolar coor- dinates [4–6]. 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 [7–10], but never with contractible loops so that this geometric effect was never em- phasized. A fluid particle that at the beginning of the loop is in a position (x,y), reaches, at the end of the same loop, a unique corresponding point (x0,y0) which is a one-to-one function (x0,y0) = T[(x,y)] of the initial one. For homo- geneous fluids, T must also be continuous and differentiable, whereas incompressibility implies that T preserves the area of any domain of points. In other words, incompressible flow implies Hamiltonian dynamics for the fluid particles, and the map that this dynamics induces in one loop is area preserving. Forcontractiblezero-arealoopsthemapissimplytheidentity; each particle ends in the position in which it started. Hence, a finite-area loop induces, in general, a finite deviation from the identity map and a characteristic value of the geometrical phase gives an estimate for the extent of this deviation. Since generically the geometric phase increases with the area of the loop (see Fig. 3(a)), for small loops the map is a small pertur- bation away from the identity whereas loops of greater area

  26. FIG. 4: Peristalsis mixing is generated by a geometric phase. (a) The minimal geometry of a model for the stomach. A peristaltic wave propagates along the upper and lower boundaries of a closed cavity zw(x,t) = 1 + bsin(kx ? ?t) of aspect ratio ⇡. (b) Contours of concentration of a passive scalar whose initial spatial distribution at T = 0 is given by a blurred step (i.e., tanhz). The temporal evolution of the spatial concentration is obtained integrating the advection–diffusion equation for Pe = 15 ⇥ 103. (c) The spatial concentration after 20 peristaltic cycles. (d) The geometric phase of the system. Pink solid lines show some representative trajectories with initial conditions marked by thick black dots. importance, and in any smaller animal it will be inappreciable. We note that previous work on gastric mixing have mostly considered the case of inertial contributions [16, 18, 19] for which the dynamical constraints discussed herein do not ap- ply. The gastric mixing is brought about by peristaltic waves — transverse traveling waves of contraction — that propa- gate along the stomach walls at some 2.5 mms?1. They are initiated approximately every 20 s, and take some 60 s to pass the length of the stomach, so 2–3 waves are present at one time, while on average the stomach width as the wave passes is 0.6 times its normal width [16, 17]. We thus have their velocity c = 2.5 mms?1, frequency ! = 0.05 Hz, and thence wavelength ? = c/! = 5 cm, and their amplitude b = 1/2 ⇥ 0.6L ⇡ 2 cm. These waves force the stomach through a nonreciprocal loop in the space of shapes, as a re- sult of which geometric mixing is expected. One can give a rough estimate of the size of the expected geometric phase by taking advantage of results obtained for another geomet- ric phase problem: that of low-Reynolds-number microorgan- isms swimming. Many bacteria swim by deforming their bod- ies in the same way as the peristaltic waves of the stomach and their speed has been well estimated by modeling such de- formations as plane waves [20]. Similar calculations for the stomach render the flow velocity induced by the peristaltic waves V = ⇡c(b/?)2, which comes out at approximately 1 mms?1, from where a displacement of about 6 cm per peri- staltic cycle is expected or, considering a circular stomach of radius L, a geometric phase of the order of 2 radians. To show the effects of this phase, we have constructed a minimal model of the stomach undergoing peristalsis, as sketched in figure 4(a). We have intentionally reduced the ge- ometric, dynamic, and functional complexity of the stomach and model a stomach of uniform radius, with sealed pyloric and esophageal valves, to focus on the role peristaltic con- tractions may play in mixing within the enclosed inertialess cavity. In our model a peristaltic wave deforms the upper and lower boundaries of a symmetric cavity of aspect ratio ⇡ ac- cording to zw(x,t) = 1+bsin(kx ? !t). The flow within the cavity is obtained by integrating the Stokes equations with the corresponding boundary conditions for the peristaltic wave, u = 0 and w = @zw/@t at z = zw(x,t), and symmetry boundary conditions at z = 0. Lateral walls deform vertically to match the vertical velocity of the peristaltic wave at x = 0 and x = 2⇡. In figure 4(a) black solid lines represent the streamlines of the induced fluid motion within the cavity due to the peristaltic wave. The contour plot corresponds to the time-averaged velocity over one full peristaltic cycle. Areas of maximum average velocity are close to the axis of symme- try, whereas near the wall the average velocity is zero and no average motion is produced. We consider the mixing of a passive scalar ? whose ini- tial spatial distribution at T = 0 is given by the blurred step (?t=0 = 1 + tanh[(z/zw? 1/2)/?]), as represented in the contour map of figure 4(b). The temporal evolution of this spatial concentration is obtained integrating the advection-

  27. FIG. 5: Mixing quality depends on the accumulated geometric phase. (a) The time evolution of the degree of mixing quantified by the standard deviation of the concentration field in the domain. The black dotted line corresponds to the peristaltic wave; the red solid line to the standing wave and the green dotted line to the random wave. (b) and (c) show contours of concentration of ? after the same integration time, equivalent to 20 peristaltic cycles, shown in fig. 4 (c) for the case of a stationary and random wave, respectively. diffusion equation for a characteristic P´ eclet number, Pe = c?/Dchymerepresentative of the mixing process within the stomach. As the characteristic diffusivity of the chyme is, at most, of order of the molecular diffusion of large macro- molecules Dchyme  10?6cm2/s, Pe ? 1 and advective contributions dominate the mixing process. Figure 4 (c) rep- resents the spatial concentration of the passive scalar ? after 20 peristaltic cycles for Pe = 15 ⇥ 103. The flow induced by peristalsis accumulates a finite geometric phase after each cycle, fluid elements are stretched and folded and, as a conse- quence, thin filaments are formed that facilitate mixing within the cavity. We obtain the geometric phase by integrating the trajec- tory of passive scalars over one full cycle, with uniformly dis- tributed initial conditions in the domain [0,2⇡]⇥[0,zw(x,0)]. The Euclidean distance between the initial and final position after one cycle gives an estimate of the geometric phase. Con- tours in figure 4 (d) represent the geometric phase of the sys- tem. It can be seen that maximum displacements are observed in the central region of the cavity where filaments are created. Note that regions in figure 4 (d) with small displacements cor- respond to regions that remain unmixed in figure 4 (c). Thus, and despite the uniform radius of the cavity in our minimal geometric model, mixing is not spatially uniform. Regions in the central part of the cavity form thin filaments that enhance mixing, whereas regions close to the lateral and to the upper walls remain almost unmixed after 20 cycles. Even further in- homogeneities are expected for more faithful geometries, with changing average wall diameter [21], specific timing of the opening and closing of the pylorus with peristalsis [22] and interactions between the fundic/cardiac region of the stomach [23], all of which are known feature for mixing within the stomach [24]. Stomach contractions that correspond to a standing wave are akin to a zero-area reciprocal loop. As we anticipated for the journal-bearing case, reciprocal loops induce flow which does not generate any mixing. This is shown in figure 5 (b) where the concentration field after 20 cycles of the boundaries deforming as a standing wave is depicted. Since the induced geometric phase is null, mixing is only controlled by (slow) diffusion. The importance of the geometric mixing in the stomach may be appreciated by reference to instances in which it is disrupted. The stomach is like the heart, with electrical activ- ity from a pacemaker region stimulating oscillations; in this case being traveling waves of peristalsis. If this system is not functioning correctly, there can be gastroparesis or gastric fib- rillation [25, 26], in which the peristaltic waves become dis- ordered. We have generated such disordered deformations by interspersing peristaltic waves whose propagation velocities c are chosen randomly from a uniform distribution of zero mean. The scalar field ? remains almost unmixed compared to the peristaltic case after an equivalent integration time, with mixing mostly controlled again by slow diffusion (figure 5 (c)). Thus, in our terms, there is poor mixing or no mixing in gastroparesis because there is not a loop around the space of forms, so no average geometric phase, and instead random peristaltic waves induce only mixing and unmixing. To compare the degree of mixing in the three cases consid- ered herein (peristalsis (pw), stationary (sw) and random (rw) waves), we calculate for each cycle the variance of the spatial concentration field [27], ? =⌦(? ? h?i)2↵1/2, where hi de- of ? with the number of cycles. It reveals the higher mixing efficiency realized in peristalsis by geometric mixing. notes the spatial average. Figure 5 (a) represents the evolution DISCUSSION In summary, we have introduced the concept of geometric mixing in which mixing arises as a consequence of a geomet- ric phase induced by a contractible non-reciprocal cycle in the

  28. The “belly” phase
 Does the stomach mix by means of geometric mixing? ! We do not know, yet, but we do know (experimentally) that if a container is deformed in a symmetric and periodic peristaltic fashion, it does not mix. The cycle is now reciprocal. ! If, on the contrary, we introduce asymmetry in this container or in the peristaltic motion, the mixing efficiency increases in the way predicted by the geometric approach. ! Might this explain some gastric anomalies?

  29. The “belly” phase
 Might this explain some gastric anomalies? ! It very well might! ! For one, gastroparesis, a condition usually associated with a form of stomach fibrillation, a disorder of the rhythm of the peristalsis, is likely to reduce the geometric mixing and the transport function of the organ.

  30. The End