Stretching and folding

The key to effective mixing lies in producing stretching and folding, an operation that is referred to in the mathematics literature as a horseshoe map. Horseshoe maps, in turn, imply chaos. The 2D case is the simplest. The equations of motion for a two-dimensional area preserving flow can be written as... 

Dye structures of passive tracers placed in time-periodic chaotic flows evolve in an iterative fashion an entire structure is mapped into a new structure with persistent large-scale features, but finer and finer scale features are revealed at each period of the flow. After a few periods, strategically placed blobs of passive tracer reveal patterns that serve as templates for subsequent stretching and folding. Repeated action by the flow generates a lamellar structure consisting of stretched and folded striations, with thicknesses s(r), characterized by a probability density function, f(s,t), whose... 

Figure 8.1 Distributive mixing mechanism for the baker s fold process a) stretch and fold method, and b) stretch, cut, and stack method...

Electric fields may interact with flows fed by hydrostatic or pumping action [91]. Flows driven by electroosmotic means may be mixed as well by the action of fluctuating electric fields, which creates oscillating electroosmotic flows and has been termed electrokinetic instability (EKI) [25, 93], In this way, rapid stretching and folding of material lines are induced, not unlike the effect of stirring. In one realized example, comparatively low frequencies, below -100 Hz, and electric field strengths in excess of 100 V mm1 are applied for channels with dimensions of about 50 pm [25],... 

M 3] [P 3] Fluorescence images at various times were taken in the main channel, i.e. along the direction of the electric field, of the first-generation micro mixer [25, 93]. After a period of 2 s, the flow becomes unstable and transverse velocities stretch and fold material lines in the flow. The initial seeded/unseeded interface becomes rapidly deformed. Finally after about 13 s, a random distribution of the tracer transverse to the applied AC field is achieved. EKI action is visible throughout the whole channel length of 7 mm. Thus, feasibility of EKI action for micro mixing has been demonstrated. 

A number of micro mixers use secondary or rotational flows, which are, e.g., created by in-channel flow structures, to stretch and fold fluids. The mixing approach here superposes the rotation by a break-up step, which basically is a splitting step [146], This was done based on the analysis of elementary mixing steps and their corresponding transfer to low Reynolds number mixing. 

Another paper describes the stretching and folding of material lines yielded by simulation and experimental imaging, induced by time-pulsing mixing via unsteady cross-flow injection in a steady-flow main channel [48],... 

Figure 1.170 Schematic of the working principle of the mixer stretching and folding of material lines are achieved by periodic perturbation of the main flow with adjacent flows [48] (by courtesy of Springer-Verlag).

Figure E7.2 compares a stepwise increase in interfacial area in simple shear flow with optimal initial orientation, and simple shear flow where, at the beginning of each step, the interfacial area element is placed 45° to the direction of shear. The figure shows that, whereas in the former case the area ratio after four shear units is 4.1, in the latter case the ratio is 6.1, with a theoretical value of 7.3 when the 45° between the plane and direction of shear is maintained at all times. We note, however, that it is quite difficult to generate steady extensional flows for times sufficiently long to attain the required total elongational strain. This is why a mixing protocol of stepwise stretching and folding (bakers transformation) is so efficient. Not only does it impose elongational stretching, but it also distributes the surface area elements over the volume.

Time-dependent, periodic, two-dimensional flows can result in streamlines that in one flow pattern cross the streamlines in another pattern, and this may lead to the stretching-and-folding mechanism that we discussed earlier, which results in very efficient mixing. In such flow situations, the outflow associated with a hyperbolic point can cross the region of inflow of the same or another hyperbolic point, leading, respectively, to homoclinic or heteroclinic intersections these are the fingerprints of chaos. 

Fig. 11.33 Dispersed PS 685 streaks of ligaments and droplets in the equiviscous Blend 2. PS is flowing/deforming in an affine fashion in the expected stretching and folding pattern. Experiments were conducted at 180°C and 120rpm. [Reprinted with permission from Proceedings of the Tenth Semiannual Meeting of the Polymer Mixing Study, Polymer Processing Institute, Hoboken, NJ (1995).]...

With these properties, if locally separates phase space, as illustrated in the scheme (Figure 1). It is very important to note that even if if has codimension 1 and is locally a separatrix, it does mean in n DOFs that if neither has a simple geometry, because it is subject to stretching and folding because of chaos [24-26], nor separates globally (see Ref. 27). Let us now make a summary of the... 

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