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Chaotic mixing in a planar, curved channel using periodic slip

We propose a novel strategy for designing chaotic micromixers using curved channels confined between two flat planes. The location of the separatrix between the Dean vortices, induced by centrifugal force, is dependent on the location of the maxima of axial velocity. An asymmetry in the axial velocity profile can change the location of the separatrix. This is achieved physically by introducing slip alternatingly at the top and bottom walls. This leads to streamline crossing and Lagrangian chaos. An approximate analytical solution of the velocity field is obtained using perturbation theory. This is used to find the Lagrangian trajectories of fluid particles. Poincare sections taken at periodic locations in the axial direction are used to study the extent of chaos. The extent of mixing, for low slip and low Reynolds numbers, is shown to be greater when Dean vortices in adjacent half cells are counter-rotating. Wide channels are observed to have much better mixing than tall channels; an important observation not made for separatrix flows till now. Eulerian indicators are used to gauge the extent of mixing with varying slip length and it is shown that an optimum slip length exists which maximizes the mixing in a particular geometry.


I. INTRODUCTION
Increased focus on microfluidic devices has brought analysis of simple flow fields exhibiting scalar 'chaotic mixing' to the forefront in a variety of fields such as applied dynamical systems and microfluidics [1][2][3][4][5] . Consequently, several 'chaotic mixer' designs have been proposed and examined. We propose a novel design strategy relying on a simple geometry modification leading to chaotic mixing in flow in a curved channel confined between two planes. Dean vortices in curved channels are a classical example of secondary flows transverse to the axial flow, induced by centrifugal force 6,7 . The proposed mixer is composed of a repeating pattern of two alternating half cells; each half cell composed of a curved channel in which the fluid slips at the top or bottom wall, alternatingly. These half 2 cells make up a unit cell, which is repeated periodically. The slip leads to an asymmetry in the geometry which causes the Dean vortices to alternatively grow and shrink. Streamlines in alternate cells, hence exhibit transverse intersections, providing an inherent mechanism for chaotic mixing. The proposed modification to the best of our knowledge is the first which leads to a variation in the location of the separatrix between the Dean vortices.
Ottino and Wiggins 3 give an excellent overview of the different approaches used to design chaotic micromixers. This study is concerned with the design of a passive micromixer i.e. a mixer which uses the geometry of the channel to induce chaotic mixing without any moving parts. The fundamental idea behind passive micromixers is to induce a periodically altered secondary flow transverse to the axial flow, such that streamlines in different sections appear to cross. The symmetry of most passive micromixers implies that the secondary flow is composed of two vortices with a separatrix between them. Such flows are referred to as 'separatrix flows' 8 . Ottino 9 first observed that apparent "streamline crossing", in periodically repeated cells, can be considered as an indicator of Lagrangian chaos. Briefly, the approaches for "streamline crossing" in passive micromixers may be classified as either 'rotation of the vortices' 10,11 or 'movement of the separatrix' 12,13 in sections, which alternate in the axial flow direction.
Dean vortices represent a class of secondary vortices induced by a simple geometry modification i.e. curvature of the channel. Consequently, a lot of work has been devoted to designing micromixers which induce chaotic advection in curved channels 11,[14][15][16][17][18][19] . Aref et al 11 proposed the first design for the same; by taking the channel out of the plane and consequently rotating the Dean vortices 14,20 . Some of the other geometries proposed based on this idea have included: 3 dimensional L-shaped bends 15 , F-shaped units 16 , square waves 14 , spirals 18 and so on. These designs are based on the same principle: rotation of Dean vortices.
The complex geometry modifications required render easy integration into lab on chip devices unfeasible. An alternative approach has been to exploit the asymmetry caused at high Reynolds numbers due to the bifurcation of the two vortices to the four vortex pattern 17 .
This approach requires high Reynolds numbers and curvature ratio and hence, is unsuitable for micromixers.
However, all these approaches preserve the symmetry of the flow about the centerline (in the direction of the centrifugal force) of the cross-section. Consequently, the separatrix between the Dean vortices does not move and is located at the centerline. The alternative 3 approach i.e. 'movement of the separatrix', thus, has not been explored for curved micromixers, despite it having been used in the design of micromixers based on other geometry modifications 4,13,21 . The general strategy here is to vary the location of separatrix between the secondary vortices from one section to the next by modifying the geometry, which leads to the requisite transverse "streamline crossing". The flow field can then be expected to lead to chaotic mixing. The justification for this hypothesis is based on Linked Twist Maps (LTMs), which have been shown to be ergodic 5,8 . The inherent symmetry of the Dean vortices at first sight might lead one to conclude that varying the location of the separatrix in a curved channel is not possible. This is probably why this mechanism has never been explored for curved channels. However, from recent work 22 , we have discovered that the location of the separatrix depends (amongst other things) on the maxima of the axial velocity. This suggests an easy way to introduce asymmetry in the Dean vortices: by making the axial velocity profile asymmetric. In the design discussed in this paper, this is achieved by introducing slip alternatingly at the top and bottom wall of the channel in a periodic manner.
The maxima of the axial velocity consequently moves near the top (or bottom) wall. Thus, the location of the separatrix varies between the two sections, leading to streamline crossing and Lagrangian chaos. Although, secondary vortices rely on inertia, chaos is observed for finite but low Reynolds numbers as the asymmetry in the proposed design is independent of inertial effects.
Our design is made feasible by recent advances in the study of slip in microchannels 23,24 .
Since, drag reduction is a major challenge in microchannels, the ability to induce slip has become increasingly important. Recent work on super-hydrophobic surfaces has shown that slip lengths of 25 μm and drag reduction of 40% can be achieved 25 . Rothstein 26 has a recent detailed review of this aspect. Despite the fact that enhanced control over "slip" is possible in microchannels, not many devices exist which exploit it for purposes other than simple drag reduction 27 . Our design marks a first step in this direction. Additionally, the slip is expected to facilitate drag reduction, both for the axial flow as well as the secondary Dean flow 28,29 .
Thus, our design combines two optimum features of microchannels to obtain chaotic mixing: asymmetric secondary Dean vortices at low Reynolds numbers and drag reduction due to slip.
However, in the present study we do not assume a particular cause of slip, so the consequent analysis is independent of the mechanism of slip.
Struman et al 8 proposed that counter-rotating Linked Twist Maps (LTMs) on the plane have better mixing properties than co-rotating LTMs for a wider range of parameters. A similar observation was made by Kang et al 15 , who found out that, for their mixer design, mixing is better when the Dean vortices are counter-rotating in successive half cells. Thus, motivated we study mixing in two different unit cell geometries which cover both the preceding scenarios: circlet and serpentine. We demonstrate that the response to introduction of asymmetry is qualitatively different in the two cases. This has practical implications for the optimization of the two designs and implies that the underlying symplectic maps are qualitatively different. Both the geometries are confined between two parallel planes. This is an immense simplification over existing designs of chaotic mixers in curved channels.
Techniques used for quantifying chaos and optimizing micromixers, range from calculating Lyapunov exponents to variance of an initial distribution of tracers. These methods require repeated computations of the Lagrangian paths of fluid particles, making it computationally intensive to efficiently and thoroughly span the parameter space 30 . To overcome this difficulty, Struman and Wiggins 31 have introduced the concept of Eulerian indicators, which may be used to predict the quality of mixing based on the velocity field alone. Consequently, for optimization of the mixer in the 2x2 {(serpentine, circlet) x (slip length, aspect ratio)} parameter space, we use Eulerian indicators in conjunction with Poincare sections. To the best of our knowledge, the variation with aspect ratio has not been studied in the literature before for separatrix flows.
The paper is organized in the following manner. In section II, the design geometry and the velocity field in the mixer is elucidated. The infinite time transport properties of the system are studied in Sec. III using Poincare sections. In Sec. IV, we study the change in the extent and nature of chaotic mixing across the parameter space with the aim of narrowing down the parameter space for optimization. In Sec. V, we briefly discuss finite time transport, in the parameter space of interest identified in the previous section. We end by summarizing the key results and discussing their significance in section VI.

II. MIXER DESIGN AND VELOCITY FIELD
The basic geometry of the mixer is a curved channel, with a rectangular cross section, composed of sections, half cells, with slip at the top and bottom wall, alternatingly. The section with slip at the top is referred to as half cell 1, while that with slip at the bottom wall as half cell 2. Each half cell spans an azimuthal extent of 180 0 . The two half cells together form a unit cell, which is repeated periodically to form the mixer. Two different arrangements 5 of these alternative sections are considered throughout the paper. If the direction of centrifugal force is same in the two sections, it is referred to as 'Circlet' (Fig 1 (a)); if the direction is opposite, it is referred to as 'Serpentine' (Fig 1 (b)). The top view of the geometry of each unit cell for both configurations is depicted in Fig 2. The curvilinear coordinate system (x, y, θ) used in the study is depicted in Fig 3. The origin of the x-y plane (O') is located at the center of a cross section of the channel. The azimuthal  coordinate determines the axial position of the cross section.  The asymmetry of the Dean vortices in the x-direction in channels of finite curvature or at high Reynolds numbers can also lead to chaotic mixing 17 . To ensure that chaotic mixing is not caused by these effects, we only study the small curvature ratio and Reynolds number regime. This ensures that any asymmetry is due to a change in the location of the separatrix.
The flow is assumed to be steady and fully developed along the axial direction within each half cell. Hence, the velocity field in each half cell is assumed to be independent of the preceding cell and switches discontinuously at the boundary between cells. This assumption enables us to obtain an analytical solution for the velocity field. This simplified velocity field captures the essential features of the geometry, while simplifying subsequent computations and enabling a thorough parametric study of the system. The quasi, fully-developed assumption has been used several times in the literature to model chaotic mixers [9,10,11].
Further, the development length for curved channels has been shown to be of the order of the width of the channel at low Reynolds numbers 32,33 . Hence, the quasi, fully-developed assumption is justified for the small curvature regime investigated in this work.
The non-dimensional parameters used in this work are defined as: Re is the Reynolds number (defined on the average velocity, for each half cell. We note that for the circlet geometry L must vary with each half cell i.e. it must be a spiral. However, this change will not affect the qualitative results discussed as L only determines the nondimensionalised length of a half cell. Therefore, we neglect the variation of L between different half cells for the circlet geometry, implying we ignore the effects due to the spiral nature of the geometry. Further, in terms of non-dimensional parameters, the quasi-fully developed assumption requires . We see that this holds in the regime studied.
The domain perturbation method is used to find an asymptotic solution of the governing Navier-Stokes equations for small curvature ratio and Reynolds numbers. Details of the solution procedure and expressions for the velocity field are given in the Appendix.
We first examine the velocity in the unit cell of the circlet geometry. As noted above, the maxima of the axial velocity moves down in this half cell (Fig 4 (c)).
Hence, as expected, the separatrix has moved downwards in Fig 5 (c). We note that the velocity profile in the second half cell is the reflection in the x-axis of the velocity profile in the first half cell, for the circlet geometry.  5 (b)) The velocity field in half cell 2 of the serpentine geometry may be inferred from its relation to half cell 1. For the serpentine geometry, the direction of rotation of the secondary vortices reverses from half cell 1 to half cell 2 as the direction of centrifugal force reverses. This implies the velocity in half cell 2 is a reflection in the y-axis of half cell 1.This is in addition to the change due to slip at different walls, which implies a reflection in the x-axis. Therefore, the velocity in the second half cell 9 is an 180 o rotation about the origin of the velocity in the first half cell. Finally, we note that the axial velocity in the unit cell of the serpentine geometry is the same as that in the circlet geometry (Fig 4 (b)-(c)), while the difference in the Dean vortices can be noted by comparing

III. INFINITE TIME DYNAMICS
The flow field described above is a 2+1 dimensional, quasi fully-developed, flow field, which changes discontinuously from one half cell to the next. The equations of motion of the passive tracer particle are given by: Following the standard practice in the literature on chaotic mixers, we study the infinite-time dynamics of the system by examining the Poincare sections at the end of every unit cell. The system is, thus, reduced to a two-dimensional symplectic map with one degree of freedom.
The axial velocity is dependent on the location in the x-y plane; hence different particles take different times to reach the same cross section. However, while constructing the Poincare map, it is assumed that each particle has sufficient time to reach the cross section.  We consider the case with no slip( 0) k  as the unperturbed map, while the introduction of slip( 0) k  is treated as a perturbation. However, we note that this perturbation need not be small.
We first discuss the circlet geometry. For 0 k  , c H reduces to flow in a curved channel with no-slip at all the walls as shown by: All other points on the wall are fixed points, while all the points inside the domain lie on periodic orbits around two elliptic fixed points, one each in the top and bottom half. The above can be inferred from Fig 7 (a). With the introduction of slip, we see that both chaotic and ordered regions exist in the Poincare sections (Fig 7 (b)-(d)). The size of these regions changes with the system parameters and is studied systematically in the next section.
For the serpentine geometry, s H is the identity map, when 0 k  , since:      (Fig 7). However, for the serpentine geometry, we see that the fixed points form a number of unstable periodic orbits, which disintegrate with increasing perturbation leading to an increase in the size of the chaotic region (Fig 8). The periodic orbits are usually referred to as KAM tori. The largest of such KAM tori forms a boundary between regions with regular and chaotic motion. Further, KAM tori act as barrier across which fluid particles cannot be transported. Thus, their presence implies that mixing does not occur over the entire domain. The transport in Hamiltonian maps (even in regions which are chaotic) has many subtle features. We refer the reader to the vast literature for further details [35][36][37] . The Poincare sections in Fig 7 and Fig 8 also show selfsimilarity and symmetry, a characteristic feature of Hamiltonian maps 38,39 . All these properties have been extensively studied, and their implications for mixing, although an interesting problem is beyond the scope of present work.
From Fig 9, we observe that the serpentine mixer exhibits a larger chaotic region and hence has better mixing properties at low Reynolds numbers and slip length. We can also observe that only one KAM torus is present for the serpentine geometry (Fig 9 (b)), as opposed to the two for the circlet geometry (Fig 9 (a)). It has been shown rigorously that counter-rotating linked twist maps (LTMs) on the plane have better mixing properties than co-rotating LTMs 5,8 . This result implies that the serpentine geometry should be more chaotic. 16 They have demonstrated that the approach works well for separatrix flows, and can help one narrow down the parameter space for optimum designs for which further numerical studies can be performed. Here, "streamline crossing" is considered essential for chaotic mixing, and an Eulerian indicator,  is defined to quantify the mixing. Following Struman and Wiggins 31 , we define ( , ) xy  as the angle between the streamlines at( , ) xy . 1 12 12 .  31 . Since, the slip length() k determines the location of the separatrix,  is particularly suited to study the changes in the system with varying slip length. (Fig 10).

Variation of Slip Length
Let us first consider the effects of varying slip length for a fixed aspect ratio.
Increasing slip length, leads to a greater shift of the maxima of the axial velocity, hence one vortex grows, while the other shrinks. At first, this leads to an increase in "streamline crossing" and hence  and thus the system can be expected to become more chaotic.
However, as the vortex size keeps increasing, the other almost completely shrinks, leaving only one vortex to occupy the cross-section. Hence, overlap between streamlines reduces at large slip, thus implying an optimum slip length that maximizes chaotic mixing. This can be seen from Fig 10 (a)-(f), which shows the value of the parameter  as the slip length is varied for different aspect ratios.
We can see from   implying that lower slip leads to more efficient mixing (Fig 10 (e)). This counter-intuitive result may be explained by noting that the location of the separatrix is also governed by the relative strength of the vortices. Consider the case of slip at the top wall. Since the relative area of slip increases for wider channels, hence the strength of the top vortex also increases. Consequently, the separatrix is pushed downwards. However, the asymmetry of the axial velocity opposes this effect, causing the separatrix to move upwards. At large aspect ratios(

3)
  and low slip, the asymmetry of the axial velocity is reduced and the effect of the vortex strength dominates.
Hence, the top vortex is able to push the separatrix downwards, below the centerline, at low slip lengths (Fig 11 (a)). However, since this is an 1 () O  effect, with increasing slip the effect due to the asymmetry of the axial velocity ( (1) O ) dominates. Therefore, as the slip length is increased the separatrix moves upwards (Fig 11 (b)-(c)). Hence, the separatrix moves towards the centerline with increasing slip. The relative overlap between the streamlines thus reduces initially. This leads to the first hump in  observed in Fig 10 (e).

Variation of Aspect Ratio
The dependence of the mixing efficiency of chaotic mixers based on separatrix flows on the aspect ratio of the channel has not been investigated in the literature before.
Considering variations with the aspect ratio provides an additional parameter in the problem, which as we show below, is crucial for optimization.
From Fig 10 (c) and (d), we note that the value of  is less for 2   as compared to 1   . However, the extent of chaos increases clearly for the wider channel as can be seen from the Poincare sections in Fig 13 and Fig 14. This anomaly can be explained by noting that changes in aspect ratio while affecting the location of the streamlines, however also 19 affect the strength of the secondary circulations. Since, the second affect is not captured by ; it cannot be used for a comparison between channels of different aspect ratios. (In comparisons between different aspect ratios, the cross-sectional area and the pressure drop are kept constant.) From Fig 13 and Fig 14, we see that wider channels have much better mixing for both the geometries. For wider channels the relative area of slip is more which implies that the relative area of "streamline crossing" increases, thus, the extent of chaos is expected to increase. We note that for the serpentine geometry, on increasing the aspect ratio, two KAM tori are present instead of one (Fig 14).   (Fig 16(a)). In contrast, for the unperturbed serpentine geometry, all the particles stay at their initial position ( (Fig 16(b)).
With the introduction of slip, the underlying geometrical structures governing transport become different. These are shown for the three qualitatively different cases: Circlet (Fig 17), Serpentine (Fig 18) and Serpentine-wide ( 3)   (Fig 19). The initial tracer locations for all cases can be seen in Fig 16(b). Clearly, the transport from the top half of the channel to the bottom half is more efficient for the serpentine geometry (Fig 18 (b)), after 3 unit cells, as compared to the circlet geometry (Fig 17(b)). Hence, the extent of mixing follows the qualitative discussion above. From Fig 18 and Fig 19, we can compare the transport of particles in serpentine geometry of different aspect ratios. The transport of particles from the upper to lower half is seen to be qualitatively different for the wider channel. We note that from these figures only a qualitative comparison can be made between the different cases. A systematic study of transport at finite times can be carried out using lobe dynamics i.e. the study of the geometrical structure formed by the intersection of the stable and unstable manifolds of hyperbolic fixed points 37 . However, it lies beyond the scope of present work.

VI. CONCLUDING REMARKS
In this work, we have presented and analyzed a novel micromixer design strategy for curved channels using slip surfaces. Our design exploited the 'movement of the separatrix' mechanism to cause chaotic mixing in curved channels. Until now, there was no discernible physical mechanism that could induce asymmetry about the separatrix in Dean flow. We While the results obtained in this study are independent of the mechanism of the slip, the optimization of the mixer with different slip mechanisms may yield slightly different results.
For instance, it has been observed that wider channels have a greater effective slip length when superhydrophobic surfaces are used 26 . This compliments our results on wider channels as increasing slip in wider channels will lead to even better mixing than predicted here. We further note that geometry modifications such as herringbones, used in previous passive chaotic mixers, increase the pressure drop required to sustain the flow rate. However, since slip leads to drag reduction, the pressure drop will actually be less for the proposed slip-based chaotic mixer. Further, investigations are needed to quantify the reduced drag for curved channels with slippery surfaces. This added advantage should facilitate integration of the design to lab on chip devices.

APPENDIX: SOLUTION OF THE VELOCITY FIELD
Flow field in a curved channel of circular cross section was first determined by Dean 6 , while Cuming 7 extended the solution to a channel of rectangular cross section. To analyze the chaotic mixer, we derive the velocity field in a curved channel with slip boundary conditions at either the top or bottom walls. The solution procedure closely follows Garg et al 22 and is 26 briefly outlined here. We solve for the velocity field in half cell 1 of the circlet geometry i.e.
slip at the top wall and centrifugal force directed along the positive x-direction. The velocity field for half cell 2 i.e. with slip at the bottom wall, of the circlet geometry is obtained using the transformation yy  , while of the serpentine geometry using , y y x x     .
The continuity and Navier-Stokes equations in the aforementioned curvilinear co-ordinate system (Fig 3) are: