Research Article

Stretching and folding sustain microscale chemical gradients in porous media

Joris Heyman, Daniel R. Lester, Régis Turuban, Yves Méheust, and Tanguy Le Borgne
PNAS June 16, 2020 117 (24) 13359-13365; first published May 28, 2020; https://doi.org/10.1073/pnas.2002858117

  1. Edited by Andrea Rinaldo, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, and approved April 21, 2020 (received for review February 20, 2020)

Article Figures & SI

Figures

  • Fig. 1.
    Fig. 1.

    (A) Experimental dye plume with backbone filament (red lines) at selected stages of the folding mechanism (run III; Movie S4). Beads have been shrunk by 40%, and white lines depict the contact line joining bead centers; surrounding beads are not shown. (B) Cross-sections detailing the typical folding stages (run I; Movie S3). Upon passing the contact point between beads 1 and 2 (B.0), the filament is stretched (diverging arrows) along the contact plane (dashed line) and compressed in the perpendicular direction (converging arrows). After contact (B.1), the directions of stretching and compression are inverted, and a cusp forms, creating a fold (B.2) that is advected over the subsequent contact between beads 3 and 4 (B.3). Other experimental runs and numerical simulations are reported in SI Appendix, Figs. S1 and S2 and Movie S5.

  • Fig. 2.
    Fig. 2.

    Cross-stream sections of the fluorescent dye plume at increasing distances from the injection: x/d=0 (A), 3.2 (B), and 9.3 (C) (run II; Movies S1 and S2). Colors represent local dye concentrations. Beads appear in deep blue. A spline curve (dotted black lines) is used to fit the filament backbone on each cross-section image, from which the total filament length L is obtained.

  • Fig. 3.
    Fig. 3.

    Total filament elongation L/L0 with respect to the normalized pore advection distance x/d from the dye injection point. (Inset) Convergence of the mean segment length Lc=L/nc and area Sc=S/nc toward constant values, with nc, the number of cusps.

  • Fig. 4.
    Fig. 4.

    Experimental average (triangles) and SD (circles) of the local maximum concentration cmax along x. Values are normalized by the initial maximum concentration c0. Classical model predictions are shown as black dashed and dotted lines. Lamellar mixing model predictions for the average and SD of cmax are shown respectively as the green dotted line and the purple continuous line (see SI Appendix, sections B and C for derivations).

  • Fig. 5.
    Fig. 5.

    Experimental setup for three-dimensional imaging of the mixing of a fluorescent solute dye in optically matched porous bead packs. The dye is continuously injected through a thin needle upstream of the granular column, while maintaining a steady background laminar flow. After stabilization of the solute dye plume, transverse cross-sections of concentrations are obtained by displacing a laser sheet (wavelength, 488 nm; beam waist 50 μm; Oxxius) in the x direction while recording the emitted fluorescence with a camera (16 bits; Hamamatsu ORCA-Flash4.0) mounted with a band-pass filter (Midopt BN532) and oriented at an angle of 45° from the laser plane. The perspective view is then orthorectified by a projective transform.

  • Fig. 6.
    Fig. 6.

    (A) Reconstruction of filament backbone (black line) from dye distribution and localization of high-curvature regions (red circles). (B) Local curvature along the filament backbone (black line) with isolated cusps (red circles) and threshold value κ=103d−1 for cusp detection (dashed line). Note that the endpoints of filaments are considered as cusps, explaining why red circles can exist below κ. (C) Exponential growth of the number (nc) of cusps in the filament backbone as a function of the advection distance (x/d) from the injection point. The fitted exponent for nc (gray line) is similar to that for the total filament length (dashed line), suggesting stationarity of the stretching and folding processes.

Data supplements

  • Supporting Information

    • Download Appendix (PDF)
    • Download Movie_S01 (MP4) - Dye plume transverse cross-section with respect to advection distance in experimental run II . Colour scale indicates the tracer concentration, hence beads appear as low concentration regions in deep blue while the injected dye appear in yellow. The video starts at the cross-section containing the dye injection point and spans a downstream distance of 13 grain diameters.
    • Download Movie_S02 (MP4) - Dye plume transverse cross-section with respect to advection distance in experimental run VI. Colour scale indicates the tracer concentration, hence beads appear as low concentration regions in deep blue while the injected dye appear in yellow. The video starts at the cross-section containing the dye injection point and spans a downstream distance of 13 grain diameters.
    • Download Movie_S03 (MP4) - Tracking of the filament backbone elongation with advection for experimental run I. The tracked filament, coloured by the curvilinear coordinate, is superimposed over grayscale images. The total elongation L/L0 is given at various cross-sections from the injection. The estimated bead locations obtained with a Hough transform-based algorithm are indicated by yellow circles.
    • Download Movie_S04 (MP4) - 3D view of the dye plume formed in the experimental run III (see also Fig. 3 and Extended data Fig. 3a). A folding event is highlighted by a red line showing the filament backbone passing through two consecutive contacts between beads (white spheres joined by a tube).
    • Download Movie_S05 (MP4) - Video produced from laminar flow numerical simulations of crystaline packings (details of simulation parameters in (14)) showing the evolution of transverse flow directions (grey arrows) and material lines (colored lines) through the flow advection. Saddle points in the transverse flow velocity field can be seen to move from one contact to another in the cross-stream direction, resulting in the stretching and folding of material lines. They reflect the intersection of attracting and repelling manifold surface, generated at grain contact boundaries (14).
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Stretching and folding sustain microscale chemical gradients in porous media
Joris Heyman, Daniel R. Lester, Régis Turuban, Yves Méheust, Tanguy Le Borgne
Proceedings of the National Academy of Sciences Jun 2020, 117 (24) 13359-13365; DOI: 10.1073/pnas.2002858117
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Proceedings of the National Academy of Sciences: 117 (24)
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