Fluid mixing enhancement through chaotic advection in mini micro channel 3

Chapter Analysis of Three-dimensional and Spatial-periodic Chaotic Mixer Chapter Analysis of Three-dimensional and Spatialperiodic Chaotic Mixer 5.1 Projection of a spatial-periodic mixer into a 3D torus The passive micromixer design usually consists of periodic structures. At low Reynolds numbers, the involved flow also exhibits periodicity. If the small influence of the inlet conditions can be neglected, the flow field in each mixer unit remains the same. The outlet of one mixer unit can be considered as the inlet of the next one, and we can bend the mixer unit into a continuous circular structure. Thus the flow is projected onto a three dimensional torus T3 as shown in Fig. 5.1. The mixer length corresponds to the circular axial length of the torus and the cross-section corresponds to the Poincaré section (see Doherty & Ottino, 1988). P1 P2 Fig. 5.1 The flow in a spatial-periodic mixer can be projected onto a 3D torus T3. In this way, the flow and mixing in a spatial-periodic micromixer can be analyzed with one single mixer unit. Though the flow is regular, the trajectory of an individual particle could be either periodic or chaotic depending on the local dynamical properties. Its behavior is determined by the Poincaré mapping function. — 89 — Chapter Analysis of Three-dimensional and Spatial-periodic Chaotic Mixer 5.2 A characterization method with one single mixer unit1 5.2.1 Lyapunov exponent Lyapunov exponent ( λ ) is an important parameter for distinguishing various chaotic systems. It measures the exponential rate of divergence between two initially neighboring orbits. Positive λ is one signature of chaos, where the nearby trajectories diverge exponentially fast. Zero λ indicates stable systems which exhibit Lyapunov stability. The negative λ is the characteristic of dissipative systems which exhibit asymptotic stability. In a flow system, the value of λ can be measured from the separation between nearby fluid particles. Its magnitude reflects the stretching rate or how fast a concentrated solute/dye is transported into the fluid domain. In Niu and Lee’s (2003) study, Lyapunov exponent was used as a mixing index to evaluate the chaotic behavior of their mixer. In Aubin et al.’s (2003) study, they also discussed the application of this parameter as a criterion to quantify the mixing. The Lyapunov exponent is usually calculated as  δ (t )   λ = lim ln t →∞ t  δ ( 0)  (5.1) where δ (0) is the initial separation of two close trajectories in a phase plane and δ (t ) is the separation at time t. For a passive mixer consisting of periodic structures, it involves a timeindependent open flow. To calculate λ , the separation between particle trajectories is projected onto one mixer cross-section, i.e. Poincaré section (x-y plane) at each mixer The characterization method is included in our paper: Xia HM, Shu C, Wan S, Chew YT. 2006. Influence of Reynolds number on chaotic mixing in spatially periodic micromixer and its characterization using dynamical system techniques. J. Micromech. Microeng., 16, 53. — 90 — Chapter Analysis of Three-dimensional and Spatial-periodic Chaotic Mixer unit. Suppose that two initially close inert particles are released at the inlet. As they move downstream, their trajectories (streamlines) successively intersect with the Poincaré sections. Their separations δ (z ) can be recorded (as shown in Fig. 5.2). Then the Lyapunov exponent λ is calculated by N →∞ N λ = lim  δ i +1 ( z )   i  ln ∑ i =0  δ ( z)  N −1 (5.2) where δ i (z ) is the separation between the particle trajectories at the sampled crosssection of the i th mixer unit. It is a function of the mixer length z (or the number of the mixer unit i ). Fig. 5.2 The definition of Poincaré section and the separation ( δ ) between two particle trajectories in a spatial-periodic mixer. Usually, λ is calculated after iteration of many mixer units. However, in numerical studies, it involves new problems. Since the evolution of the particle trajectories is based on the discrete flow field solution, it is reasonable to set the initial separation δ o (0) comparable with the computational mesh size. In such a situation, according to our studies, the saturation length may soon be reached. That is, the particle separation δ N ( z ) has increased to a scale comparable with the dimension of the mixer cross section. Afterwards it only exhibits some kind of oscillation but will — 91 — Chapter Analysis of Three-dimensional and Spatial-periodic Chaotic Mixer no longer increase. One example is given in Fig. 5.3. The results are for mixer TLCCM-A at Re 0.2. Two inert particles are released at Pa (100, 135) and Pb (100, 150) ( µm ) in section A0 (the reference coordinates are shown in Fig. 5.2). The initial separation is δ (0) = 15µm . The 3D view of the trajectories (expressed as Φ( Pa ) and Φ( Pb )) is illustrated in Figure 5.3(a). Figure 5.3(b) plots the separation as a function of the mixer length, which is translated into the number of mixer units (per mixer unit length is ≈ 2.546 mm). It shows that within 1.6 mixer-unit length, the trajectories diverge roughly exponentially by δ N ≈ δ o ⋅ e1.72 N . Positive λ indicates the occurrence of chaotic advection. After that, the distance in the plane between the particles has increased to δ s ≈ δ ⋅ e1.72×1.6 = 235 ( µm) (δ = 15µm) . This value is comparable with the dimension of the Poincaré section (cross section of the mixer) which is 424.3× 300 ( µm ). Afterwards, δ shows no further increase. The Lyapunov exponent λ should be calculated within the saturation length, which is hard to predict as it depends on different mixer designs and the local dynamical properties of the flow. — 92 — Chapter Analysis of Three-dimensional and Spatial-periodic Chaotic Mixer (a) Ln (δ(z)/ δ(0)) (1.6, 2.75) -1 mixer length (N) (b) Fig. 5.3 Particle trajectories Φ(100, 135) and Φ(100, 150) ( µm ) in TLCCM-A at Re = 0.2. (a) 3D view. (b) The separation δ (z ) versus the mixer length. The saturation length is ≈ 1.6 mixer units (4.07 mm). 5.2.2 Averaged dispersion rate of the mixer As mentioned in Section 5.1, for a spatially periodic mixer, its flow field solution in each mixer unit stays invariant. It contains all the information of the flow in the whole mixer system. It might not be necessary to calculate λ after many mixer units as defined by Eq. (5.2) to analyze the dynamical properties of the flow. Here we propose a method to characterize mixers consisting of spatial-periodic structure. Firstly, the divergence rate between neighboring particles is calculated within one single mixer cycle. Suppose that a pair of particles is released at ( xi ,1 , yi ,1 ) — 93 — Chapter Analysis of Three-dimensional and Spatial-periodic Chaotic Mixer N and ( xi , , y i , ) in a mixer cross-section AN . The initial separation is δ i . After one mixer cycle, their separation becomes δ i N +1 on section AN +1 . Then, the divergence rate within one mixer cycle at λi′ is calculated as λi′ ( x, y ) = ln δ i N +1 . δiN (5.3) For fair comparison among different mixers of various unit-lengths, λ ′ can be calculated over a same distance L. L is not limited to one mixer cycle, but it should be within the saturation length. Another issue involved is the choice of the initial separation δ N . In numerical studies, it can be set close to the computational mesh size. Our testing shows that within a wide range of δ N , λ ′ remains quite constant. Secondly, λ ′ is a function of the position (x, y) in section AN . Many particle pairs should be examined with an attempt to cover the whole flow region. For each particle pair, it corresponds to a value of λi′ which is assigned to [( xi ,1 + x i , ) / 2, ( y i ,1 + y i , ) / 2] . A collection or set of λi/ ( i = 1, …k) will reflect the complete information of the whole mixer. To facilitate the analysis, λi′ is plotted over section AN , which is termed here as λ ′ -map. The averaged value λ ′ = (1 k )∑i =1 λi k ′ over the plane reflects the overall chaotic level of the flow. It measures the ability of a mixer to disperse concentrated species into other flow domain. We applied this method to examine the current TLCCM mixer. The two models have similar structures, and the mixer unit-length of model B is one half that of model A. For comparison, the results of model B are calculated within 2-mixer-units length. Relevant results are plotted in Fig. 5.4. It is found that: (1) In both models, the — 94 — Chapter Analysis of Three-dimensional and Spatial-periodic Chaotic Mixer positive λ ′ indicates the occurrence of chaotic advection. (2) For model A and in the Re range from 0.02 to 1, the value of λ ′ remains rather constant at around 0.62. It is just as expected since the flow characteristics retain invariant at Re[...]... tracers In Jones et al (1989), a semi-linear quadratic interpolation was used to construct a 2D map for studying chaotic mixing in twisted pipe The same method was adopted by Yi & Bau (20 03) to analyze the mixing kinematics in bended micro- conduits Very recently, Mott et al (2006) also introduced a toolbox using advection maps to optimize micromixer design In this study, three different mapping schemes... the influences of the inlet and outlet conditions In comparison with the simulation of the whole mixer device, the computational efforts involved in CFD studies can be greatly reduced — 118 — Chapter 6 Perturbed Rotating Flow and Chaotic Mixing Chapter 6 Perturbed Rotating Flow and Chaotic Mixing Relevant studies have shown that rotating flow plays an important role in chaotic advection It is involved... mapping of an arbitrary point can be determined through interpolation Three methods were studied: (1) Triangular weighted interpolation (Tri-WI) The first method applied to construct the maps is triangular weighted interpolation As illustrated in Fig 5.12, a triangle that contains the interpolating point is built using three known points For every point inside the triangle, its value is determined... In the following cycles, above-mentioned behaviors occur again and again As shown in Figure 6.1 (b), the originally circles deform into a triangular shape Near the three tips, the fluids are continuously stretched and folded up The periodic perturbations on the rotating flow have led to chaotic mixing In the outer region, the originally continuous material lines are disrupted and the fluid particles... polynomial fitting (a) before and (b) after improvement (о): supporting points (●): interpolating point, and its exact position on the next plane (□): the mapping position Data are from TLCCM-A, Re=0.2 — 112 — Chapter 5 Analysis of Three-dimensional and Spatial-periodic Chaotic Mixer r0 ri rref ri interpolation point reference and supporting point supporting points non-supporting points Fig 5.16 Schematic... schemes were used to track the inert tracers on the Poincaré section Their effectiveness to predict the chaotic mixing in spatial periodic mixer, the accuracy and relevant influencing factors were examined We would like to point out that the present discussion gives primarily information from a statistical and engineering viewpoint Some assumptions adopted here may not — 1 03 — Chapter 5 Analysis of Three-dimensional... rotating flow with intermittent perturbations The general profile is illustrated in Fig 6.1(a & b) On the right side of the channel, the turning of the channel from the base-layer to the top-layer has caused a rotating flow On the left side, the fluid moves back and forth, intermittently knocks — 120 — Chapter 6 Perturbed Rotating Flow and Chaotic Mixing (a) (b) Fig 6.1 Evolution of examined fluid. .. of the fluid is driven into the left base-layer channel In the second half cycle ( z = 0.5 L0 ~ L0 ), the rotation in the right side continues, while the fluids on the other side do not change much as the channel turns 90o in the same layer Then, when the fluids move back and impinge upon each other, folding occurs as shown in section A1 From A0 to A1, the flow has passed around 1 /3 of a circle In the... obtained through inert particle tracing simulation Three methods including the triangular weighted interpolation, the Shepard’s interpolation and weighted least square polynomial fitting are investigated to approximate the mapping function Results show that one main factor affecting the accuracy is the discontinuities in the tracers’ distribution on the mapped plane This is caused by the splitting and... dispersed into a large area after 9 mixer units The inner area is relatively stable At section A9, the material line R =30 still remains continuous, but has become wrinkled with clear signs of stretching and folding The most inner circle R=20 remains intact, only showing slight deformation This is because the perturbation is introduced by the external branch stream, and it is supposed to decay in the radial . studying chaotic mixing in twisted pipe. The same method was adopted by Yi & Bau (20 03) to analyze the mixing kinematics in bended micro-conduits. Very recently, Mott et al. (2006) also introduced. weighted interpolation. As illustrated in Fig. 5.12, a triangle that contains the interpolating point is built using three known points. For every point inside the triangle, its value is determined. embedded in a chaotic area. Fig. 5.10(b) is the experimental mixing picture at Re = 0.01. The mixing is apparently incomplete. While mixing occurs on one side of the mixer, the fluids on the