Coalescence of viscous blisters under an elastic sheet

Abstract

We study the coalescence of identical viscous blisters beneath an elastic sheet both experimentally and numerically. Using a time-resolved synthetic schlieren technique, we measure the evolution of the thickness field of the merging blisters and more specifically the dynamics of the coalescence region. To explain this dynamics, we develop a one-dimensional model based on the lubrication approximation, from which we derive scalings to predict the growth of the coalescence neck. We also numerically solve the full non-linear equation to assess the theory and compare to experiments. Our model illustrates that, at short times, the dynamics of coalescence is mainly controlled by the bending of the elastic sheet, leading to a relationship between the coalescence speed and the radius of curvature of the interface at the coalescence neck.

Figures (4)

• Figure 1: Sketch of the experimental setup. Sunflower oil is fed under an elastic membrane (thickness $d$) through two holes (diameter $\phi \simeq 1$ mm) made in an underlying transparent plate, forming two identical liquid blisters of typical thickness $h \sim 0.1-1$ mm separated by a distance $L_{gap} \sim 3$ cm. A checkerboard pattern is placed under the plate and imaged from the top through the whole setup. Apparent deformations of the pattern enable one to reconstruct the liquid thickness field $h(x,t)$.
• Figure 2: (a) Reconstruction of the height profile $h(x,y)$ at $t = t_0 + 2 s$. The contours of the blisters are clearly visible and the spacing between their respective centers corresponds to the imposed inter-intrusion distance $L_{\text{gap}}$. Here, $V=0.5$ mL, $L_{gap}=3$ cm and $d=0.2$ mm. (b) Height profiles between the centers of the injected blisters. The contact point position $x_0$ is highlighted as a red dot. Color gradient indicates increasing times. (c) Temporal evolution of the coalescence point height $h_0 (t)$. On the linear part, we measure a coalescence speed noted $dh_0/dt (t)$ when $h_0$ reaches $h_s = 40~\mu$m. (d) Identification of the height profile $h(x,t_0)$ and determination of the radius of curvature $R$ of a small region around the minimum
• Figure 3: Left : Speed $dh_0/dt$ of the elevation of the coalescence point as a function of the radius of curvature $R$ on a log-log scale, for three different membranes thickness (($\bullet$) $d=0.2$ mm, ($\bullet$) $d=0.8$ mm, ($\bullet$) $d=1.6$ mm). Right :$dh_0/dt$ normalized by the bending modulus $B$ plotted as a function of $R$ on a log-log scale. The dashed and continuous straight lines correspond to equation \ref{['scaling_v']} with $\alpha=1$. The black dot corresponds to the numerical computation presented in section \ref{['sec:num']}, using $L = 1.5$ cm, $d = 0.2$ mm, $E = 1.6 \times 10^6$ Pa, and $\mu = 0.05$ Pa.s..
• Figure 4: Left : Successive profiles of the numerical solution of equation \ref{['eq:h_bending_adim']}, relaxing towards a uniform height $\bar{h} \simeq 2.21\times10^{-2}$. The dotted and dashed curves are theoretical composite approximate solutions of equation \ref{['eq:h_bending_adim']}. Inset : Pressure field during the self-similar evolution. The horizontal dashed line corresponds to $p=1$. Right :$dh^\star_0/dt^\star$ and $1/R^\star$ plotted in a log-lin scale as a function of time. The vertical dotted line in the right figure $(t^\star \simeq 356.5)$ separates the self-similar growth on the left (blue profiles), and the exponential relaxation toward $\bar{h}$ (red profiles).