^{1,2,a)}, Denis Doorly

^{1}and Dimitri Kustrin

^{1}

### Abstract

This article explores the lamination, stretching, and mixing produced by sequences cyclically permuting a cat's eyes flow structure to stir the flow. Such sequences are experimentally driven by electromagnetic forces. Their intensity is kept constant between experiments while the duration of the forcing cycles varies over a decade. Mixing observations show that the mixing processes evolve from a seesaw stirring for short cycles (due to the regular rotation of the principal direction of the cat's eyes flow structures) to a cat's eyes stirring where the seesaw stirring is complemented by the rolling occurring within eddies. The transition from seesaw stirring to cat's eyes stirring is related to the persisting of the cat's eyes flow structure during one turnover time before it is flipped. Reference cases such as steady and random forcing configurations complement this exploration for comparison with the cat's eyes flip sequences. It is shown that cat's eyes flip sequences are efficient and possess baker-like mixing properties with an exponential growth for the length of interfaces and their lamination. The exponential coefficients of the stretching and lamination rates are conserved when varying the duration of the mixing cycles and using the generic cat's eyes flow turnover time as the reference of time to build these exponents. In particular, the stretching coefficients can be assumed as nearly constant when compared to the topological entropy which varies over a decade. This is attributed to the ability of the cat's eyes flip sequences to integrate lamination during the stirring sequences. This integration of the lamination compensates the reduction of flow's unsteadiness when increasing the duration of the mixing cycles so as to conserve a good stirring and mixing performance. Therefore, the lamination, stretching, and mixing of the cat's eyes flip sequences are robust to changes of the cycles’ duration.

The authors acknowledge the Royal Society, and the Engineering and Physical Sciences Research Council (UK) Grant No. EP/D072034/1.

I. INTRODUCTION

II. FLOWS DRIVEN BY ELECTROMAGNETIC FORCES

A. Experimental setup

B. Electromagnetic forcing

III. CAT’S EYESFLOWS

A. Velocity fields

B. Geometry, shear, and strain

IV. MIXING AND LAMINATION

A. Methods and measures

1. Mixing

2. Lamination

B. Illustration of cat's eyesflow sequences

1. Mixing

2. Lamination

V. DISCUSSION

A. Lamination and stretching

1. Lamination

2. Stretching

B. Mixing

VI. CONCLUSION

## Figures

Illustration of a baker process combined with an additional stretching. The baker process cyclically alternates the production of striations and the stretching of interfaces. The reference of time is represented by the arrow labelled t *. The first column is a reference case where the line is growing like , where n a is the number of stirring cycles. The second and third columns illustrate two asymptotic changes (singular and integrative laminations) of the mixing process when increasing the period of the stirring cycles, where n b indicates the cycles’ numbers. The lines are growing like and for the second and third columns, respectively.

Illustration of a baker process combined with an additional stretching. The baker process cyclically alternates the production of striations and the stretching of interfaces. The reference of time is represented by the arrow labelled t *. The first column is a reference case where the line is growing like , where n a is the number of stirring cycles. The second and third columns illustrate two asymptotic changes (singular and integrative laminations) of the mixing process when increasing the period of the stirring cycles, where n b indicates the cycles’ numbers. The lines are growing like and for the second and third columns, respectively.

Schematic (a) and photograph (b) of the experimental rig. The 3-axis robot and the pair of permanent magnets (north and south) are illustrated in the photograph (c). The arrow labelled j indicates the direction of the electrical current crossing the brine. The arrow labelled B indicates the direction of the magnetic field (produced by the permanent magnets) contributing to the generation of horizontal body forces.

Schematic (a) and photograph (b) of the experimental rig. The 3-axis robot and the pair of permanent magnets (north and south) are illustrated in the photograph (c). The arrow labelled j indicates the direction of the electrical current crossing the brine. The arrow labelled B indicates the direction of the magnetic field (produced by the permanent magnets) contributing to the generation of horizontal body forces.

The two columns show the horizontal distribution of the body forcing, normalised by gravity, during the steady stages of the cat's eyes flip sequences, i.e., with the magnets’ pair oriented at ±15°. The chevrons indicate the directions in which the flow is pumped.

The two columns show the horizontal distribution of the body forcing, normalised by gravity, during the steady stages of the cat's eyes flip sequences, i.e., with the magnets’ pair oriented at ±15°. The chevrons indicate the directions in which the flow is pumped.

Illustration of instantaneous velocity fields within the 15L M × 15L M mixing domain during flows driven by the steady forcing configurations at −15° and +15°, left and right hand side pictures, respectively. The white squares show the central region of size 5L M × 5L M which corresponds to the mixing domain investigated in Fig. 5 .

Illustration of instantaneous velocity fields within the 15L M × 15L M mixing domain during flows driven by the steady forcing configurations at −15° and +15°, left and right hand side pictures, respectively. The white squares show the central region of size 5L M × 5L M which corresponds to the mixing domain investigated in Fig. 5 .

Velocity fields, within the 5L M × 5L M mixing domain, for the steady forcing configurations during quasi-steady flow regimes. As indicated at the top, the column corresponds to the orientation of the magnets at ±15°. The colour map gives the flow intensity, i.e., . The black arrows (1/64 plotted) indicate the local direction of the velocity. The circles indicate the centre of eddies, i.e., elliptical stagnation points. The octagons highlight central hyperbolic stagnation points and the grey (orange) arrows indicate their principal directions. The white lines indicate streamlines where the flow direction is shown by white arrows.

Velocity fields, within the 5L M × 5L M mixing domain, for the steady forcing configurations during quasi-steady flow regimes. As indicated at the top, the column corresponds to the orientation of the magnets at ±15°. The colour map gives the flow intensity, i.e., . The black arrows (1/64 plotted) indicate the local direction of the velocity. The circles indicate the centre of eddies, i.e., elliptical stagnation points. The octagons highlight central hyperbolic stagnation points and the grey (orange) arrows indicate their principal directions. The white lines indicate streamlines where the flow direction is shown by white arrows.

Velocity fields at times t/Tcyc = 0.25 and t/Tcyc = 0.75 (first and second columns) for the cat's eyes flip sequences (rows) within the 5L M × 5L M mixing domain. The sequences are denoted CF0.25, CF0.5, CF1, CF2, and CF4 according to . The colour map gives . The black arrows (1/64 plotted) indicate the local direction of the velocity. The circles indicate the centre of eddies, i.e., elliptical stagnation points. The octagons highlight central hyperbolic stagnation points and the grey (orange) arrows indicate their principal directions. The white lines indicate streamlines where the flow direction is shown by white arrows.

Velocity fields at times t/Tcyc = 0.25 and t/Tcyc = 0.75 (first and second columns) for the cat's eyes flip sequences (rows) within the 5L M × 5L M mixing domain. The sequences are denoted CF0.25, CF0.5, CF1, CF2, and CF4 according to . The colour map gives . The black arrows (1/64 plotted) indicate the local direction of the velocity. The circles indicate the centre of eddies, i.e., elliptical stagnation points. The octagons highlight central hyperbolic stagnation points and the grey (orange) arrows indicate their principal directions. The white lines indicate streamlines where the flow direction is shown by white arrows.

Maxima between the two correlations functions and where U A and U B are two velocity fields of reference and U (t) are the velocity fields of the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the mixing domain 5L M × 5L M . (a) U A and U B correspond to the steady velocity fields during the steady forcing configuration with the pair of magnets oriented at ±15°. (b) U A and U B correspond to the velocity fields at times t/Tcyc = 0.25 and t/Tcyc = 0.75 during the cat's eyes flip sequences.

Maxima between the two correlations functions and where U A and U B are two velocity fields of reference and U (t) are the velocity fields of the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the mixing domain 5L M × 5L M . (a) U A and U B correspond to the steady velocity fields during the steady forcing configuration with the pair of magnets oriented at ±15°. (b) U A and U B correspond to the velocity fields at times t/Tcyc = 0.25 and t/Tcyc = 0.75 during the cat's eyes flip sequences.

Dimensionless distribution of Q, Q *, strain s *, pure strain λ*, and pure shear b * for the steady flow driven by the forcing configuration at +15° which is shown in Fig. 5 . The values of λ* are set to zero within the elliptic regions.

Dimensionless distribution of Q, Q *, strain s *, pure strain λ*, and pure shear b * for the steady flow driven by the forcing configuration at +15° which is shown in Fig. 5 . The values of λ* are set to zero within the elliptic regions.

Dimensionless distribution of strain s *, pure strain λ*, and pure shear b * for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) at t/Tcyc = 0.75 for which velocity fields are shown in Fig. 6 . The values of λ* are set to zero within the elliptic regions.

Dimensionless distribution of strain s *, pure strain λ*, and pure shear b * for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) at t/Tcyc = 0.75 for which velocity fields are shown in Fig. 6 . The values of λ* are set to zero within the elliptic regions.

Temporal advancement of two (black and white) non-diffusive dyes driven by the steady forcing configuration with the pair of magnets oriented at 15°. Mixing times, t*, are from left to right 0, 1, 2, and 4 (first row) and 8, 16, 32, and 64 (second row). The reference time is T flow = 11 s and t* = t/T flow .

Temporal advancement of two (black and white) non-diffusive dyes driven by the steady forcing configuration with the pair of magnets oriented at 15°. Mixing times, t*, are from left to right 0, 1, 2, and 4 (first row) and 8, 16, 32, and 64 (second row). The reference time is T flow = 11 s and t* = t/T flow .

Temporal advancement of a dyes’ interface driven by the steady forcing configuration at +15°. The chosen interface is similar to the one of Fig. 10 . The times, t*, are from left to right 0, 1, 2, and 3 (first row) and 4, 6, 9, and 15 (second row). The colour scale gives the striations’ thickness, d, within measurement circles of diameter ϕ = 1 cm, i.e., 0.25L M , illustrated at t* = 0. Only the points for which the line's length from these points to the line's extremities is larger than ϕ/2 are displayed.

Temporal advancement of a dyes’ interface driven by the steady forcing configuration at +15°. The chosen interface is similar to the one of Fig. 10 . The times, t*, are from left to right 0, 1, 2, and 3 (first row) and 4, 6, 9, and 15 (second row). The colour scale gives the striations’ thickness, d, within measurement circles of diameter ϕ = 1 cm, i.e., 0.25L M , illustrated at t* = 0. Only the points for which the line's length from these points to the line's extremities is larger than ϕ/2 are displayed.

Temporal advancement of two (black and white) non-diffusive dyes driven by cat's eye flip sequences performed in the 5L M × 5L M mixing domain. The sequences CF0.25, CF0.5, CF1, CF2, and CF4 are given from top to bottom every two rows. The times, t*, are 0, 1, 2, 4, 8, 16, 32, and 64. The associated videos illustrate the mixing process from t* = 0 to 32 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812798.1] [URL: http://dx.doi.org/10.1063/1.4812798.2] [URL: http://dx.doi.org/10.1063/1.4812798.3] [URL: http://dx.doi.org/10.1063/1.4812798.4] [URL: http://dx.doi.org/10.1063/1.4812798.5]doi: 10.1063/1.4812798.1.

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Temporal advancement of two (black and white) non-diffusive dyes driven by cat's eye flip sequences performed in the 5L M × 5L M mixing domain. The sequences CF0.25, CF0.5, CF1, CF2, and CF4 are given from top to bottom every two rows. The times, t*, are 0, 1, 2, 4, 8, 16, 32, and 64. The associated videos illustrate the mixing process from t* = 0 to 32 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812798.1] [URL: http://dx.doi.org/10.1063/1.4812798.2] [URL: http://dx.doi.org/10.1063/1.4812798.3] [URL: http://dx.doi.org/10.1063/1.4812798.4] [URL: http://dx.doi.org/10.1063/1.4812798.5]doi: 10.1063/1.4812798.1.

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Poincaré map of the cat's eyes flip mixing sequences CF0.25, CF0.5, CF1, CF2, and CF4 within the 5L M × 5L M domain. The Poincaré maps are obtained using the periodic, phase averaged, velocity fields. Sixteen numerical particles are tracked in time using forward and backward advancement over 1000 periods. Their initial positions are given in the top left picture and the colouring of the particles is chosen with respect to their initial positions.

Poincaré map of the cat's eyes flip mixing sequences CF0.25, CF0.5, CF1, CF2, and CF4 within the 5L M × 5L M domain. The Poincaré maps are obtained using the periodic, phase averaged, velocity fields. Sixteen numerical particles are tracked in time using forward and backward advancement over 1000 periods. Their initial positions are given in the top left picture and the colouring of the particles is chosen with respect to their initial positions.

Temporal advancement of two (black and white) non-diffusive numerical dyes for the cat's eye flip sequence CF1 (Video ) performed in the 15L M × 15L M mixing domain. The associated video illustrates this mixing process from t* = 0 to 32 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812798.6]doi: 10.1063/1.4812798.6.

Temporal advancement of two (black and white) non-diffusive numerical dyes for the cat's eye flip sequence CF1 (Video ) performed in the 15L M × 15L M mixing domain. The associated video illustrates this mixing process from t* = 0 to 32 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812798.6]doi: 10.1063/1.4812798.6.

Evolution of the dyes interface during the stirring processes illustrated in Fig. 12 for the mixing domain 5L M × 5L M . The colour scale indicates the local striations’ thickness. The measurement circles have a diameter of 1 cm, i.e., 0.25L M , and are illustrated with a circle at t* = 0. Only the points for which the line's length from these points to the line's extremities is larger than ϕ/2 are displayed. The associated videos illustrate these mixing sequences for durations longer than t* = 14 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812798.7] [URL: http://dx.doi.org/10.1063/1.4812798.8] [URL: http://dx.doi.org/10.1063/1.4812798.9] [URL: http://dx.doi.org/10.1063/1.4812798.10] [URL: http://dx.doi.org/10.1063/1.4812798.11]doi: 10.1063/1.4812798.7.

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Evolution of the dyes interface during the stirring processes illustrated in Fig. 12 for the mixing domain 5L M × 5L M . The colour scale indicates the local striations’ thickness. The measurement circles have a diameter of 1 cm, i.e., 0.25L M , and are illustrated with a circle at t* = 0. Only the points for which the line's length from these points to the line's extremities is larger than ϕ/2 are displayed. The associated videos illustrate these mixing sequences for durations longer than t* = 14 (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812798.7] [URL: http://dx.doi.org/10.1063/1.4812798.8] [URL: http://dx.doi.org/10.1063/1.4812798.9] [URL: http://dx.doi.org/10.1063/1.4812798.10] [URL: http://dx.doi.org/10.1063/1.4812798.11]doi: 10.1063/1.4812798.7.

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Evolution of the dyes interface during the stirring processes illustrated in Fig. 14 for the mixing domain 15L M × 15L M . The colour scale indicates the local striations’ thickness. The measurement circles have a diameter of 1 cm, i.e., 0.25L M , and are illustrated with a circle at t* = 0. The associated video illustrates this mixing sequence (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812798.12]doi: 10.1063/1.4812798.12.

Evolution of the dyes interface during the stirring processes illustrated in Fig. 14 for the mixing domain 15L M × 15L M . The colour scale indicates the local striations’ thickness. The measurement circles have a diameter of 1 cm, i.e., 0.25L M , and are illustrated with a circle at t* = 0. The associated video illustrates this mixing sequence (enhanced online). [URL: http://dx.doi.org/10.1063/1.4812798.12]doi: 10.1063/1.4812798.12.

(a) Temporal evolution of the spatial average of the lamination for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the 5L M × 5L M mixing domain. (b) Temporal evolution of the spatial average of the lamination for CF1, the steady and random sequences within the 5L M × 5L M and 15L M × 15L M mixing domains. (c) Mean values of the exponents of the exponential growths, αlam (measured when t * ⩾ 4), obtained for different orientations of the interface versus . (d) Comparison of the αlam obtained for the cat's eyes flip sequences when using Tflow or Tcyc as the reference of time.

(a) Temporal evolution of the spatial average of the lamination for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the 5L M × 5L M mixing domain. (b) Temporal evolution of the spatial average of the lamination for CF1, the steady and random sequences within the 5L M × 5L M and 15L M × 15L M mixing domains. (c) Mean values of the exponents of the exponential growths, αlam (measured when t * ⩾ 4), obtained for different orientations of the interface versus . (d) Comparison of the αlam obtained for the cat's eyes flip sequences when using Tflow or Tcyc as the reference of time.

Illustration of the height initial distributions of black and white dyes used to check the robustness of the mixing sequences.

Illustration of the height initial distributions of black and white dyes used to check the robustness of the mixing sequences.

(a) Temporal evolution of the stretching, i.e., l/l 0, for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the 5L M × 5L M mixing domain. (b) Temporal evolution of the stretching for CF1, the steady and random sequences within the 5L M × 5L M and 15L M × 15L M mixing domains. (c) Mean values of the exponents of the exponential growths, αstretch, obtained for different orientations of the interface versus and using Tflow as the reference of time. (d) Comparison of the exponents of the exponential growths obtained for the cat's eyes flip sequences when using Tflow or Tcyc as the reference of time.

(a) Temporal evolution of the stretching, i.e., l/l 0, for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the 5L M × 5L M mixing domain. (b) Temporal evolution of the stretching for CF1, the steady and random sequences within the 5L M × 5L M and 15L M × 15L M mixing domains. (c) Mean values of the exponents of the exponential growths, αstretch, obtained for different orientations of the interface versus and using Tflow as the reference of time. (d) Comparison of the exponents of the exponential growths obtained for the cat's eyes flip sequences when using Tflow or Tcyc as the reference of time.

(a) Temporal evolution of the mixing coefficient for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the 5L M × 5L M mixing domain. (b) Temporal evolution of the mixing coefficient for CF1, the steady and random sequences within the 5L M × 5L M and 15L M × 15L M mixing domains.

(a) Temporal evolution of the mixing coefficient for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the 5L M × 5L M mixing domain. (b) Temporal evolution of the mixing coefficient for CF1, the steady and random sequences within the 5L M × 5L M and 15L M × 15L M mixing domains.

Mixing rate, ∂C mix /∂t, for the cat's eyes flip (CF0.25, CF0.5, CF1, CF2, and CF4) and the random sequences within the 5L M × 5L M mixing domain. The right hand side gives a semi-log plot of the left hand side at early times. The dotted lines indicate exponential growth in the semi-log plots. The dashed lines indicate the times when the typical striations’ thickness (estimated using the measure of lamination) reaches the size of the counting box for concentration measurements.

Mixing rate, ∂C mix /∂t, for the cat's eyes flip (CF0.25, CF0.5, CF1, CF2, and CF4) and the random sequences within the 5L M × 5L M mixing domain. The right hand side gives a semi-log plot of the left hand side at early times. The dotted lines indicate exponential growth in the semi-log plots. The dashed lines indicate the times when the typical striations’ thickness (estimated using the measure of lamination) reaches the size of the counting box for concentration measurements.

Temporal evolution of the rescaled variance, , averaged over 16 realisations for the cat's eyes flip sequences within the 5L M × 5L M mixing domain when starting with a blob of dye of diameter LM. The initial and final distributions of dyes are illustrated for a given blob.

Temporal evolution of the rescaled variance, , averaged over 16 realisations for the cat's eyes flip sequences within the 5L M × 5L M mixing domain when starting with a blob of dye of diameter LM. The initial and final distributions of dyes are illustrated for a given blob.

## Tables

Dimensionless values of forcing times, and , and typical dimensionless root mean squares of flows’ characteristic, velocity , pure strain , pure shear , and strain for the mixing domains 5L M × 5L M and 15L M × 15L M . The reference length scale is the size of the magnets L M = 40 mm, and the reference timescale is the typical flows' turnover time T flow = 11 s.

Dimensionless values of forcing times, and , and typical dimensionless root mean squares of flows’ characteristic, velocity , pure strain , pure shear , and strain for the mixing domains 5L M × 5L M and 15L M × 15L M . The reference length scale is the size of the magnets L M = 40 mm, and the reference timescale is the typical flows' turnover time T flow = 11 s.

Some characteristic values for the stretching, lamination, and mixing of different flow sequences (given in the first column) within 5L M × 5L M and 15L M × 15L M domains. αlam and αstretch give the exponents of the exponential growths obtained, respectively, for lamination and stretching using Tflow as the reference of time. r50% indicates the ratios between the median and mean values of lamination. Superscripts give the standard deviations over different dyes' distributions.

Some characteristic values for the stretching, lamination, and mixing of different flow sequences (given in the first column) within 5L M × 5L M and 15L M × 15L M domains. αlam and αstretch give the exponents of the exponential growths obtained, respectively, for lamination and stretching using Tflow as the reference of time. r50% indicates the ratios between the median and mean values of lamination. Superscripts give the standard deviations over different dyes' distributions.

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