Coalescence of Elastic Blisters Filled with a Viscous Fluid

Pockets of viscous fluid coalescing beneath an elastic sheet are encountered in a wide range of natural phenomena and engineering processes, spanning across scales. As the pockets merge, a bridge is formed with a height increasing as the sheet relaxes. We study the spatiotemporal dynamics of such an elastohydrodynamic coalescence process by combining experiments, lubrication theory, and numerical simulations. The bridge height exhibits an exponential growth with time, which corresponds to a self-similar solution of the bending-driven thin-film equation. We address this unique self-similarity and the self-similar shape of the bridge, both of which are corroborated in numerical simulations and experiments.

Figure 1

(a) Schematic ($\tilde{y}=0$) of the studied system. Two identical pockets filled with a viscous fluid of viscosity $\mu$ and density $\rho$ are created by injecting prescribed fluid volumes beneath an elastic sheet. The elastic sheet has a Young’s modulus $E$, a Poisson ratio $\nu$, and a thickness $d$. After the fluid injection is stopped, each blister spreads until reaching a radius $R_0$ when they meet, which is defined as time $\tilde{t}=0$. The fluid height profile is $\tilde{h}(\tilde{x},\tilde{y},\tilde{t})$, while ${\tilde{h}}_0(\tilde{t})=\tilde{h}(0,0,\tilde{t})$ is the bridge height, ${\tilde{h}}_{\mathrm{i}}=\tilde{h}(\pm R_0,0,0)$ is the initial blister height, and $\tilde{r}(\tilde{t})$ is the half-width of the bridge along the $\tilde{y}$ axis (into the plane). (b) Contour plot of the nondimensional height profile $\tilde{h}(\tilde{x},\tilde{y},\tilde{t})/{\tilde{h}}_i$ from the three-dimensional simulation when the two blisters make contact ($\tilde{t}=0$).

Figure 2

(a) Nondimensional cross-sectional height profiles $h(x,y=0,t)$, at different nondimensional times $t$, from an experiment with $\mu =0.5\mathrm{Pa}\mathrm{s}$ and ${\tilde{h}}_{\mathrm{i}}=2.55\mathrm{m}\mathrm{m}$, and from the numerical solutions (solid lines) of Eq. (1) with $N=59$ and $h_{\infty }=0.016$. (b) Nondimensional height $h_0(t)$, and half-width $r(t)$ of the bridge, extracted from the numerical solutions of Eq. (1). Inset: The two-dimensional bridge profile along the $y$ axis, $h(x=0,y,t)$ at the same nondimensional times as in (a). $r(t)$ is extracted numerically as the first point along $y$ where $h(x=0,y>0,t)\le h_{\infty }$, indicated with a star marker.

Figure 3

(a) Experimental bridge height ${\tilde{h}}_0(\tilde{t})$ for different viscosities $\mu \in [0.1,0.35,0.5]\mathrm{Pa}\mathrm{s}$ and blister peak heights ${\tilde{h}}_{\mathrm{i}}\in [1.9-4.2]\times {10}^{-3}\mathrm{m}$. (b) Experimental nondimensional bridge height $h_0(t)={\tilde{h}}_0(t)/{\tilde{h}}_{\mathrm{i}}$ as a function of nondimensional time $t=\tilde{t}/T$. The solid and dashed lines represent the numerical solutions of Eq. (1) for $N=59$ and $N=0$, respectively, and with $h_{\infty }=0.016$. The short-time behavior for $N=59$ is well described by $h_0(t)=h_0(0)\exp (26t)$; while for $N=0$ it follows $h_0(t)=h_0(0)\exp (17.9t)$. (c) Inset: the bridge height profiles $h(x,y=0,t)$ [denoted as $h(x,t)$] from the short-time numerical solution of Eq. (1) for $N=0$ and $h_{\infty }=0.016$. Main: self-similar representation of the height profiles ($y=0$) from Eq. (1) ($N=0$) shown in the inset. Here, we used $f(t)=0.016\exp (17.9t)$ and $\kappa =8$. The black solid line is the numerical solution of Eq. (5) with $\kappa =8$ and $\alpha =0.0175$.