Geometric mixing

Abstract

Mixing fluids often involves a periodic action, like stirring one's tea. But reciprocating motions in fluids at low Reynolds number, in Stokes flows where inertia is negligible, lead to periodic cycles of mixing and unmixing, because the physics, molecular diffusion excepted, is time reversible. So how can fluid be mixed in such circumstances? The answer involves a geometric phase. Geometric phases are found everywhere in physics as anholonomies, where after a closed circuit in the parameters, some system variables do not return to their original values. We discuss the geometric phase in fluid mixing: geometric mixing. This article is part of the theme issue 'Stokes at 200 (part 2)'.

Keywords: belly phase; fluid mixing; geometric phase; journal-bearing flow.

Figure 1.

The geometric phase for the journal-bearing flow with cylinder radii R₁ = 1.0, R₂ = 0.3 and eccentricity ε = 0.4, taken around a closed loop with θ₁ = θ₂ = 6 radians. The four segments of the loop are plotted in different colours (red, green, blue, magenta) to enable their contributions to the particle motion to be seen. The trajectories of two passive scalars initially at x = 0.5, y = 0.0 and x = 0.6, y = 0.0 are shown. (Online version in colour.)

Figure 2.

(a–c) Poincaré maps demonstrate geometric mixing for the journal-bearing flow for the same cylinder radii and eccentricity as figure 1. Several initial conditions, marked with different colours, are shown for 10 000 iterations of the parameter loop: (a) θ₁ = θ₂ = π/2 radians; (b) θ₁ = θ₂ = 2π radians; (c) θ₁ = θ₂ = 4π radians. (d–f ) The geometric phase across the domain for the same parameters; the colour 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 across the widest gap between the cylinders after one cycle for the same parameters. (Online version in colour.)

Figure 3.

The final length of a line segment as shown in figure 2g–i after one cycle plotted against the rotation angle θ₁ = θ₂ = θ. (Online version in colour.)

Figure 4.

The value of the commutator is shown for three different values of the eccentricity ε, 0.01(a), 0.1(b) and 0.5(c). The lower panel shows different global measures of the commutator, the spatial mean of the L²-norm, as a function of ε for a prescribed rotation protocol. (Online version in colour.)

Figure 5.

A typical Lagrangian trajectory shows how it is winding around the inner cylinder and the period-1 orbit is located in the ‘hole’ on the right. Such an orbit is an example of a so-called ghost rod. (Online version in colour.)

Figure 6.

Poincaré section obtained for a small value of θ₁ (θ₁ = θ₂ = π/8), by plotting the positions of 20 trajectories (coloured lines) strobed every period. The fluid domain is divided into two regions separated by the stable and unstable manifolds of two 1-periodic points (red squares). No exchange of fluid is observed between these two regions during the 20 000 periods of the simulation. It is likely that there is a weak heteroclinic tangle resulting from the manifolds of the two periodic points, yet chaos, if present, is very weak. (Online version in colour.)

Figure 7.

Material line advected by the geometric-mixing journal-bearing flow, for different values of θ₁ = θ₂ (see text). Red squares represent the position of periodic points of period 1. (Online version in colour.)

Figure 8.

θ₁ = π and 2π: topological entropy of the braid formed by a single Lagrangian trajectory winding around the two 1-periodic points (red squares in figure 7b,c). Much more braiding occurs for θ₁ = 2π than for θ₁ = π.

Figure 9.

(a) The trajectory of a rod looping around two elliptic islands on a figure-eight path forms a topologically entangled braid when plotted in a space–time diagram. (b) Mixing pattern created by integrating the trajectories of 160 000 particles located initially in a small square. (c) Dye pattern created by a stirring rod moving on a figure-eight path, for a small rod diameter. (d) Same as (c), for a larger rod. Note the characteristic heart-shaped pattern that resembles the pattern in (b) [39]. (Online version in colour.)

Figure 10.

Growth of a material line, plotted versus the positive angular displacement θ = θ₁ × t, where t is the number of stirring periods, for different values of the parameter θ₁ (θ₁ = π/2,  π,  2π,  3π,  4π). An exponential growth of the length is observed, as expected for chaotic advection. However, in the case of strong stretching inhomogeneities (θ₁ = π/2), it may take a long time before the asymptotic exponential regime is reached, because fluid particles wait a long time before visiting some parts of the domain. The topological entropy, that is the asymptotic rate of length growth, increases with θ₁. For θ₁ = 2π, the topological entropy corresponds well to the topological entropy of a figure-eight braid (see text). Inset: the length of a line is not growing monotonically during one stirring period. Alternating the sense of rotation of the cylinders leads to the unwinding of filaments that have been sheared by one cylinder, and have hardly been displaced by the rotation of the other cylinder—as for fluid particles located on the right side of fluid domain when the inner cylinder starts moving. Efficient stretching is nevertheless experienced by fluid particles that are displaced significantly by both cylinders successively. (Online version in colour.)

Figure 11.

Sketch of the evolution of a blob of dye stretched and folded by chaotic advection. The concentration level of the filaments stays close to the original concentration until filaments are compressed down to the Batchelor scale, at which the effects of stretching and diffusion balance. Afterwards, filaments diffuse and are merged with other fluctuations, resulting in efficient mixing. (Online version in colour.)

Figure 12.

(a) θ₁ = 2π, histograms of total stretching experienced by fluid particles at different times (from blue n = 2 to red n = 17), plotted in semilog coordinates. (b) Evolution of the unmixed fraction $f_{{\lambda }_c}$ for two protocols (θ₁ = 2π or 3π), and two values of the critical stretching factor λ_c. Solid lines correspond to $f_{{\lambda }_c}$ measured inside the whole chaotic region (including the vicinity of the outer cylinder), while for dashed lines only particles inside a central disc of radius 0.6 have been taken into account. (c) and (d) θ₁ = 2π, map of stretching values (from blue to red) at t = 10 (c)) and t = 15 periods (d)). (Online version in colour.)