Static and dynamic fluid-driven fracturing of adhered elastica

The transient spreading of a viscous fluid beneath an elastic sheet adhered to the substrate is controlled by the dynamics at the tip where the divergence of viscous stresses necessitates the formation of a vapor tip separating the fluid front and fracture front. The model for elastic-plated currents is extended for an axisymmetric geometry with analysis showing that adhesion gives rise to the possibility of static, elastic droplets and to two dynamical regimes of spreading; viscosity dominant spreading controlled by flow of viscous fluid into the vapor tip, and adhesion dominant spreading. Constant flux experiments using clear, PDMS elastic sheets enable new, direct measurements of the vapor tip and confirm the existence of spreading regimes controlled by viscosity and adhesion. The theory and experiments thereby provide an important test coupling the dynamics of flow with elastic deformation and have implications in fluid-driven fracturing of elastic media more generally.

Figure 1

(a) Schematic diagram of the theoretical model and experimental setup with the physical parameters in the system. (b) Photograph of an experimental fluid front showing the lag between the fluid front and fracture front.

Figure 2

(a) Plot of the static profiles for a constant volume $V$ transitioning from bending to gravitationally dominated regimes. Inset (i) shows numerical solution in the pure bending regime (blue dots) plotted on top of theoretical profile (4) (red curve), and (ii) the numerical solution in the gravity dominant regime with a bending tip (blue dots) plotted on top of theoretical profile (6) (green curve). (b) Dimensionless radial extent with volume. (c) Dimensionless central deflection with volume.

Figure 3

Images of an experiment with viscosity and volume flux $\mu =2.12\mathrm{Pa}\mathrm{s},Q=3.09\times {10}^{-7}{\mathrm{m}}^3{\mathrm{s}}^{-1}$, respectively, at $t=64\mathrm{s}$ taken from (a) underneath the experiment showing the radial extent and lag region (N.B.: the injection pipe is obscuring the left-hand side of the image), and (b) above at an oblique angle $\varphi$ to the horizontal showing the fluorescent line painted on top of PDMS sheet with a filtered image showing the deflected line. (c) Expanded view of the edge of the experiment shown in (a) filtered to demonstrate intensity contrast between vapor tip and the substrate. Fluid front given by the black-dashed line with lag lengths given by pairs of red dots. (d) Lag length with time for two experiments in the viscosity dominant regime with different viscosities $\mu$ and volumes fluxes $Q$.

Figure 4

Deflection profiles for an experiment in the viscosity dominant regime with viscosity $\mu =2.12\mathrm{Pa}\mathrm{s}$, volume flux $Q=3.09\times {10}^{-7}{\mathrm{m}}^3{\mathrm{s}}^{-1},\sigma =101\times {10}^3\mathrm{Pa}$ for $t=64–130\mathrm{s}$, where $\mathrm{\Delta }t=6\mathrm{s}$. (a) Measured deflection, and (b) deflection scaled with theoretical expressions (13) and (14). The black-dashed line shows the theoretical profile (4).

Figure 5

Radial extent with time in the viscosity dominant regime. (a) Measured radial extent and (b) radial extent scaled with ${\{Q^7{B}^3/\left[{\left(12\mu \right)}^2\sigma \right]\}}^{1/30}$. Black-dashed line corresponds to best fit $R_F/{\{Q^7{B}^3/\left[{\left(12\mu \right)}^2\sigma \right]\}}^{1/30}=1.40t^{3/10}$.

Figure 6

Radial extent with time in the adhesion dominant regime. (a) Measured radial extent. (b) Radial extent scaled with ${(Q/{\kappa }_{adh})}^{1/4}$. Black-dashed line corresponds to best fit $R_F/{(Q/{\kappa }_{adh})}^{1/4}=1.45t^{1/4}$. (c) (inset) Measured curvature $\kappa$ plotted against measured prefactor $c$, where $R_F=c{\left(Qt\right)}^{1/4}$. Black-dashed line given by $c=1.45{\kappa }_{adh}^{-1/4}$.