Transient structures in rupturing thin films: Marangoni-induced symmetry-breaking pattern formation in viscous fluids

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Science Advances  08 Jul 2020:
Vol. 6, no. 28, eabb0597
DOI: 10.1126/sciadv.abb0597
  • Fig. 1 Notation and experimental system.

    (A) Schematic of a curved convex hemispherical thin film of radius R and film thickness 2h. The air pressure difference between the interior and the exterior of the bubble is given by Δp = pepi, and g is the gravitational acceleration. (B) The film is driven toward an instability pattern with wavelength λ and height displacement ζ in the radial direction. (C and D) Convex and concave hemispherical geometries with the experimental images overlaid at the position where they are observed.

  • Fig. 2 Optimal flow profiles on bubbles.

    Optical flow profiles (A to C) of some of the images from the experiments in (D) to (F). (G) shows the evolution of the optical fluid velocity along the horizontal x axis (x0 is the rightmost x coordinate of each image), averaged over the vertical axis. Here, 1 pixel corresponds to 1.531 × 10−4m in real-life distance. The X line in (D) and (G) marks the transition front between the thin-film flow region on the right and the pattern-forming region on the left.

  • Fig. 3 Pattern formation on concave bubbles.

    Experimental images (A to D) and simulation results (E to H) of pattern formation on a concave thin film near rupture. (I) to (L) show the numerical FDs of (A) to (H), where y0 and y1 are the y values for the top and bottom edge, respectively. The contours around the curve denote the standard deviations (SD) of the FD, with the deepest color showing the bounds of (1/2) SD and the faded color giving the 1 SD bounds. The relative fractal error δ from top to bottom is as follows: 0.0417, 0.1636, 0.1716, and 0.0926. The colors in (A) to (H) represent film thickness using the same scale as in Fig. 2.

  • Fig. 4 Pattern formation on convex bubbles.

    Experimental (A to D) and simulation (E to H) images of pattern formation on a curved thin film (showing the time-dependent amplitude of the surface fluctuations) with associated (I to L) plots of the Marangoni number and the Marangoni field (M to P) on the hemisphere, where y0 and y1 are the leftmost and rightmost y values of the cross section shown in (M), respectively. In (K), regions A, B, and C refer to the pattern-forming, black-film, and thin-film flow regions, respectively.

  • Fig. 5 Pattern formation on convex bubbles II and stability diagram.

    (A to D) Experimental images of the labyrinth state on a convex hemispherical thin-film system. (E to H) FEM simulation of the amplitude equation (Eq. 4) in region A. (I) Plot of the stability diagram, where A denotes regions of pattern formation, and B and C denote the black-film and the thin-film flow regions, respectively.

  • Table 1 List of normalized parameters for Eq. 1, where L̂=La2 and M̂=M2.

    List of normalized parameters for Eq. 1, where L̂=La2 and M̂=M2..


Supplementary Materials

  • Supplementary Materials

    Transient structures in rupturing thin films: Marangoni-induced symmetrybreaking pattern formation in viscous fluids

    Li Shen, Fabian Denner, Neal Morgan, Berend van Wachem, Daniele Dini

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