Rayleigh-Taylor-like instability in a foam film

It is well known, since the seminal work of Mysels et al. [Soap Films: Study of Their Thinning and a Bibliography (Pergamon Press, New York, 1959)], that the thinner parts of a foam film go up by gravity, whereas the thicker parts go down. Preparing a foam film in a controlled way, so that the top part of the film is much thicker than the bottom part at initial time, we show that this situation is indeed unstable under gravity. The observed instability is identified as a Rayleigh-Taylor-like instability and studied in the linear regime. The wavelength and the growth rate are measured as a function of the effective gravity, and as a function of the thick film extension. We theoretically derive the dispersion relation for the instability, taking into account finite size effects. The fastest mode is analytically determined and is in qualitative agreement with the experimental observations.

Figure 1

Scheme of the experimental setup and notations used in the text. Light blue represent the three foam films connected along the free meniscus (dark blue). They are supported by a deformable metallic frame : the red lines are the immobile edges and the black line is the mobile edge (at its initial position). The deformable film $F$ is illuminated with the white light WL and recorded in reflection with the color camera CC. The whole setup is on a table which can be tilted by an angle $\theta$.

Figure 2

Images of the film at the times [0, 67, 83, 167, 250, 284, 384, 1467] ms, in the vertical case, solution $S_b$. In the $z$ direction, the whole height is shown, from the moving meniscus at the bottom [see the white arrow on images (a–d)] to the free meniscus at the top of the image. In the $x$ direction, the image is cropped in the central part of the film, and only 10.3 mm are visible. The remaining part of the film evolves similarly, excepted close to the lateral sides. The bottom edge moves over 13 mm during the first 100 ms [images (a–d)], at a velocity of 80 mm/s.

Figure 3

Initial film thickness profile. (Top) Raw image obtained with the spectral camera, just after stretching the film. The total height in the $z$ direction is 19 mm, and the wavelength $\lambda$ varies from 375 nm at the top to 1010 nm at the bottom. The gray level is the light intensity (in arbitrary unit). The central part corresponds to the initial thin film and the two lateral parts are the thicker parts of the film, which have been extracted from the menisci. (Middle) Film thickness profile extracted from the top image. (Bottom) Sketch of the film profile and notations used in the text.

Figure 4

Determination of the wavelength. The red line is the frontier between the initial film and the top Frankel's film, as detected by the image processing. The wavelength is the average distance between two successive minima $x_j$ and $x_{j+1}$ of the curve.

Figure 5

Behavior in the linear regime for solution ($S_a$). (Top) Wavelength as a function of the effective gravity; (bottom) growth rate of the instability. The solid lines are the scaling laws given respectively by Eq. (10) with the prefactor 3 (instead of the expected 19.5) and by Eq. (11) with the prefactor 1/3.5 (instead of the expected 1/2.8). The color indicates the value of the thick film height $d_2$, in mm.

Figure 6

Behavior in the linear regime for solution ($S_b$). (Top) Wavelength as a function of the effective gravity; (bottom) growth rate of the instability. The solid lines are the scaling laws given respectively by Eq. (10) with the prefactor 4.5 (instead of the expected 19.5) and by Eq. (11) with the prefactor 1/4 (instead of the expected 1/2.8). The color indicates the value of the thick film height $d_2$, in mm.

Figure 7

Experimental wavelength (top) and growth rate (bottom) as a function of $d_2$, for solution $S_b$ with the vertical setup. The thin film thickness $h_1$, expressed in $\mu \mathrm{m}$, is given by the color chart. (Top) Dashed line: scaling law (12); solid line: scaling law (10) with the prefactor 3.12 (instead of the expected 19.5). (Bottom) Dashed line: scaling law (13); solid line: scaling law (11) with the prefactor 1/5.6 (instead of the expected 1/2.8).

Figure 8

Various forces acting on systems (1) and (2), limited by the red rectangles respectively in the left and right schemes. (Left) The force balance on system (1) involves surface tension and gravity, as viscous forces compensate on both sides. (Right) On system (2), gravity force is reduced by a larger amount of thin film, and interfacial viscous forces and inertia need to be considered.

Figure 9

Origin of the line tension. We consider the frontier width, measured in the plane normal to the frontier: The width $ds$ measured along the interface shape is slightly larger than the width $\delta$ measured in the film plane. It induces a resulting force $T$ oriented along the frontier tangent.

Figure 10

Value of $n$ as a function of the wavelength with $\rho ={10}^3\mathrm{kg}/{\mathrm{m}}^{-3}, h_1=0.5{10}^{-6}\mathrm{m}, h_2=2{10}^{-6}\mathrm{m}, {\mu }_s={10}^{-7}\mathrm{kg}/\mathrm{s}$, and $\overline{g}=9.8$. Black solid line: solution of the equation set [Eqs. (25, 26, 27), (30), and (35)–(36)]; becomes blue and red lines: asymptotic scaling laws given by Eqs. (8) and (9), respectively. The fastest mode is represented by the black dot, close to the intersection of both asymptotes. The gray dashed line is at the abscissa $d_2$ and the large wavelengths at the right of this line are forbidden. If this cutoff at $d_2$ occurs at a wavelength smaller than the fastest one, then the gray dot represents the actual fastest mode.

Figure 11

Theoretical fastest wavelength (top) and associated time scale (bottom) as a function of the upper film size $d_2$. The parameters are $h_1=0.5\mu \mathrm{m}, h_2=2\mu \mathrm{m}, {\mu }_s={10}^{-7}\mathrm{kg}{\mathrm{s}}^{-1}$, and an infinite value of $d_1$. The angle $\theta$, governing the apparent gravity, verifies $sin\theta =0.01$, 0.1, 1, respectively, for the red, magenta, and blue lines. (Top) Horizontal dashed lines: $19.5{\lambda }_{\text{th},1}$ [Eq. (10)]; black dashed line: $3.8{\lambda }_{\text{th},2}$ [Eq. (12)]. (Bottom) Dashed lines: $2.8/n_{\text{th}1}$ [Eq. (11)]; dot-dashed lines: $10.5/n_{\text{th}2}$ [Eq. (13)]. Solid lines are the numerical solutions ${\lambda }^{\text{num}}$ and $n^{\text{num}}$ of the full system [Eqs. (25, 26, 27), (30), and (35)–(36)]. The prefactors of the scaling laws Eqs. (10)–(13) are fitted on these numerical solutions.

Figure 12

Theoretical growth rate as a function of the experimental value shown in Figs. 5 ($\circ$), 6 ($•$), and 7 ($\times$). The theoretical value is determined by numerical resolution of the system, using the experimental values of $d_2, h_2$, and $h_1$ (we used $h_2=1.5\mu \mathrm{m}$ for the vertical series). The other parameters are $\rho ={10}^3\mathrm{kg}/{\mathrm{m}}^3$ and ${\mu }_s=0.8{10}^{-7}\mathrm{kg}/\mathrm{s}$. The line corresponds to $n^{\text{th}}=n^{\text{exp}}$.

Figure 13

Theoretical wavelength as a function of the experimental value shown in Figs. 5 ($\circ$), 6 ($•$), and 7 ($•$) with the same parameters as in Fig. 12. (Top) The color chart shows the effective gravity value in $\mathrm{m}/{\mathrm{s}}^2$, red corresponding to 9.8 $\mathrm{m}/{\mathrm{s}}^2$. (Bottom) The color chart shows the Frankel's film height $d_2$ in mm.