1887
banner image
No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Lamination, stretching, and mixing in cat's eyes flip sequences with varying periods
Rent:
Rent this article for
USD
10.1063/1.4812798
/content/aip/journal/pof2/25/7/10.1063/1.4812798
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/7/10.1063/1.4812798

Figures

Image of FIG. 1.
FIG. 1.

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 . The first column is a reference case where the line is growing like , where 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 indicates the cycles’ numbers. The lines are growing like and for the second and third columns, respectively.

Image of FIG. 2.
FIG. 2.

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 indicates the direction of the electrical current crossing the brine. The arrow labelled indicates the direction of the magnetic field (produced by the permanent magnets) contributing to the generation of horizontal body forces.

Image of FIG. 3.
FIG. 3.

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.

Image of FIG. 4.
FIG. 4.

Illustration of instantaneous velocity fields within the 15 × 15 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 5 × 5 which corresponds to the mixing domain investigated in Fig. 5 .

Image of FIG. 5.
FIG. 5.

Velocity fields, within the 5 × 5 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.

Image of FIG. 6.
FIG. 6.

Velocity fields at times t/T = 0.25 and t/T = 0.75 (first and second columns) for the cat's eyes flip sequences (rows) within the 5 × 5 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.

Image of FIG. 7.
FIG. 7.

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

Image of FIG. 8.
FIG. 8.

Dimensionless distribution of , , strain , pure strain λ, and pure shear 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.

Image of FIG. 9.
FIG. 9.

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

Image of FIG. 10.
FIG. 10.

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 = 11 s and * = / .

Image of FIG. 11.
FIG. 11.

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., 0.25 , 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.

Image of FIG. 12.
FIG. 12.

Temporal advancement of two (black and white) non-diffusive dyes driven by cat's eye flip sequences performed in the 5 × 5 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.

doi: 10.1063/1.4812798.2.

doi: 10.1063/1.4812798.3.

doi: 10.1063/1.4812798.4.

doi: 10.1063/1.4812798.5.

Image of FIG. 13.
FIG. 13.

Poincaré map of the cat's eyes flip mixing sequences CF0.25, CF0.5, CF1, CF2, and CF4 within the 5 × 5 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.

Image of FIG. 14.
FIG. 14.

Temporal advancement of two (black and white) non-diffusive numerical dyes for the cat's eye flip sequence CF1 (Video ) performed in the 15 × 15 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.

Image of FIG. 15.
FIG. 15.

Evolution of the dyes interface during the stirring processes illustrated in Fig. 12 for the mixing domain 5 × 5 . The colour scale indicates the local striations’ thickness. The measurement circles have a diameter of 1 cm, i.e., 0.25 , 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.

doi: 10.1063/1.4812798.8.

doi: 10.1063/1.4812798.9.

doi: 10.1063/1.4812798.10.

doi: 10.1063/1.4812798.11.

Image of FIG. 16.
FIG. 16.

Evolution of the dyes interface during the stirring processes illustrated in Fig. 14 for the mixing domain 15 × 15 . The colour scale indicates the local striations’ thickness. The measurement circles have a diameter of 1 cm, i.e., 0.25 , 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.

Image of FIG. 17.
FIG. 17.

(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 5 × 5 mixing domain. (b) Temporal evolution of the spatial average of the lamination for CF1, the steady and random sequences within the 5 × 5 and 15 × 15 mixing domains. (c) Mean values of the exponents of the exponential growths, α (measured when ⩾ 4), obtained for different orientations of the interface versus . (d) Comparison of the α obtained for the cat's eyes flip sequences when using T or T as the reference of time.

Image of FIG. 18.
FIG. 18.

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

Image of FIG. 19.
FIG. 19.

(a) Temporal evolution of the stretching, i.e., / , for the cat's eyes flip sequences (CF0.25, CF0.5, CF1, CF2, and CF4) within the 5 × 5 mixing domain. (b) Temporal evolution of the stretching for CF1, the steady and random sequences within the 5 × 5 and 15 × 15 mixing domains. (c) Mean values of the exponents of the exponential growths, α, obtained for different orientations of the interface versus and using T as the reference of time. (d) Comparison of the exponents of the exponential growths obtained for the cat's eyes flip sequences when using T or T as the reference of time.

Image of FIG. 20.
FIG. 20.

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

Image of FIG. 21.
FIG. 21.

Mixing rate, ∂ /∂, for the cat's eyes flip (CF0.25, CF0.5, CF1, CF2, and CF4) and the random sequences within the 5 × 5 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.

Image of FIG. 22.
FIG. 22.

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

Tables

Generic image for table
Table I.

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 5 × 5 and 15 × 15 . The reference length scale is the size of the magnets = 40 mm, and the reference timescale is the typical flows' turnover time = 11 s.

Generic image for table
Table II.

Some characteristic values for the stretching, lamination, and mixing of different flow sequences (given in the first column) within 5 × 5 and 15 × 15 domains. α and α give the exponents of the exponential growths obtained, respectively, for lamination and stretching using T as the reference of time. r indicates the ratios between the median and mean values of lamination. Superscripts give the standard deviations over different dyes' distributions.

video/mp4,video/x-flv,video/flv,audio.mp3,audio.mpeg

Multimedia

The following multimedia file is available, if you log in: 1.4812798.mm.v1.mpg
The following multimedia file is available, if you log in: 1.4812798.mm.v2.mpg
The following multimedia file is available, if you log in: 1.4812798.mm.v3.mpg
The following multimedia file is available, if you log in: 1.4812798.mm.v4.mpg
The following multimedia file is available, if you log in: 1.4812798.mm.v5.mpg
The following multimedia file is available, if you log in: 1.4812798.mm.v6.mpg
The following multimedia file is available, if you log in: 1.4812798.mm.v7.gif
The following multimedia file is available, if you log in: 1.4812798.mm.v8.gif
The following multimedia file is available, if you log in: 1.4812798.mm.v9.gif
The following multimedia file is available, if you log in: 1.4812798.mm.v10.gif
The following multimedia file is available, if you log in: 1.4812798.mm.v11.gif
The following multimedia file is available, if you log in: 1.4812798.mm.v12.gif
Loading

Article metrics loading...

/content/aip/journal/pof2/25/7/10.1063/1.4812798
2013-07-17
2014-04-16
Loading

Full text loading...

This is a required field
Please enter a valid email address
752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Lamination, stretching, and mixing in cat's eyes flip sequences with varying periods
http://aip.metastore.ingenta.com/content/aip/journal/pof2/25/7/10.1063/1.4812798
10.1063/1.4812798
SEARCH_EXPAND_ITEM