Journal: Scientific Reports

Article Title: Air invasion into three-dimensional foam induces viscous fingering instabilities

doi: 10.1038/s41598-024-53566-3

Figure Lengend Snippet: ( a ) Schematic of the radial viscous fingering experimental setup for air invading foam. ( b ) Gillette Foamy Regular shaving foam slowly injected using a 60 cc syringe to a sample radius, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{R}}=87.7\pm 0.5\mathrm{ mm}$$\end{document} R = 87.7 ± 0.5 mm , and ( c ) air is injected into the cell from a reservoir tank at constant pressure.

Article Snippet: More specifically, the interface is rendered as a vector of points, X , of a radius versus angle, and its discrete Fourier transform is determined, given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) = fft( X ) where the Matlab function fft is used, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) is the discrete Fourier transform, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} k is the wave number.

Techniques: Injection

Journal: Scientific Reports

Article Title: Air invasion into three-dimensional foam induces viscous fingering instabilities

doi: 10.1038/s41598-024-53566-3

Figure Lengend Snippet: Time sequences of the viscous fingering interface at different air injection pressures ( a ) 0.5 psi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {\text{t}}=2.371\mathrm{ s}$$\end{document} Δ t = 2.371 s , ( b ) 1.0 psi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {\text{t}}=0.913\mathrm{ s}$$\end{document} Δ t = 0.913 s , ( c ) 2.0 psi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {\text{t}}=0.063\mathrm{ s}$$\end{document} Δ t = 0.063 s , ( d ) 3.0 psi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {\text{t}}=0.021\mathrm{ s}$$\end{document} Δ t = 0.021 s , and ( e ) 4.0 psi, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {\text{t}}=0.013\mathrm{ s}$$\end{document} Δ t = 0.013 s . \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta {\text{t}}$$\end{document} Δ t is the time intervals between interfaces shown in the images.

Article Snippet: More specifically, the interface is rendered as a vector of points, X , of a radius versus angle, and its discrete Fourier transform is determined, given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) = fft( X ) where the Matlab function fft is used, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) is the discrete Fourier transform, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} k is the wave number.

Techniques: Injection

Journal: Scientific Reports

Article Title: Air invasion into three-dimensional foam induces viscous fingering instabilities

doi: 10.1038/s41598-024-53566-3

Figure Lengend Snippet: Fingertip radius \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{r}}}_{{\text{tip}}}$$\end{document} r tip as a function of time for repeated experiments at ( a ) low and ( b ) high injection pressures. Fitting lines for each experiment are shown as gray lines.

Article Snippet: More specifically, the interface is rendered as a vector of points, X , of a radius versus angle, and its discrete Fourier transform is determined, given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) = fft( X ) where the Matlab function fft is used, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) is the discrete Fourier transform, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} k is the wave number.

Techniques: Injection

Journal: Scientific Reports

Article Title: Air invasion into three-dimensional foam induces viscous fingering instabilities

doi: 10.1038/s41598-024-53566-3

Figure Lengend Snippet: Representative fingertip velocities \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{v}}}_{{\text{tip}}}$$\end{document} v tip and wall capillary numbers, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{Ca}}}^{*}=\upmu {{\text{v}}}_{{\text{tip}}}/\upgamma$$\end{document} Ca ∗ = μ v tip / γ , for the three flow regimes.

Article Snippet: More specifically, the interface is rendered as a vector of points, X , of a radius versus angle, and its discrete Fourier transform is determined, given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) = fft( X ) where the Matlab function fft is used, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) is the discrete Fourier transform, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} k is the wave number.

Techniques:

Journal: Scientific Reports

Article Title: Air invasion into three-dimensional foam induces viscous fingering instabilities

doi: 10.1038/s41598-024-53566-3

Figure Lengend Snippet: ( a ) Average finger width density (m/m) versus air injection pressure and ( b ) finger area density ( \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\text{m}}}^{2}/{{\text{m}}}^{2}$$\end{document} m 2 / m 2 ) versus air injection pressure.

Article Snippet: More specifically, the interface is rendered as a vector of points, X , of a radius versus angle, and its discrete Fourier transform is determined, given by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) = fft( X ) where the Matlab function fft is used, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$F(k)$$\end{document} F ( k ) is the discrete Fourier transform, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document} k is the wave number.

Techniques: Injection