Universal wrinkling dynamics of a sheet on viscous liquid

Abstract

We investigate the wrinkling dynamics of a thin elastic sheet that is indented or compressed while floating on a viscous liquid. We show that the deformation speed controls the dynamics, leading to a wrinkle wavelength significantly smaller than that selected under quasistatic compression. Once active compression ceases, the wrinkles coarsen until their wavelength relaxes toward the equilibrium value. We develop a theoretical model coupling Stokes flow in the liquid to elastic bending of the sheet, which quantitatively predicts both the initial wavelength selection and its subsequent coarsening. We demonstrate that the same mechanism governs two dimensional and axisymmetric geometries, thereby extending classical static wavelength selection laws to dynamic situations. Although developed from controlled laboratory experiments, the model captures a generic viscous-elastic coupling and applies broadly to thin elastic films interacting with viscous environments, including the formation of surface wrinkles in pahoehoe lava flows.

Figures (3)

• Figure 1: Schematic of the experimental setup used to (a) compress or (b) indent/impact a thin PET/PDMS sheet floating on a viscous liquid. (c) Wrinkles on a rectangular PDMS sheet ($h = 50$$\mu$m) floating on silicon oil ($\mu = 5$ Pa s) compressed at $V = 5$ mm/s (top) and $V= 200$ mm/s (bottom). (d) Wrinkles on a PET circular sheet ($R=35$ mm and $h = 3$$\mu$m) floating on a fluid ($\mu = 5$ Pa s) at $t=3$ ms after being impacted by a marble at $V=2$ m/s. Scale bars in (c)-(d): 1 cm. (e) Dependence of the initial wavelength $\lambda_0$ on $\mu V$ for a circular PET sheet ($h=3$$\mu$m, $R=63$ mm) resting on silicon oils ($1 \leq \mu \leq 30$ Pa s) and impact or indented at speeds raging from 5 mm/s to 2 m/s. The rectangular PDMS sheet ($h=50$$\mu$m, $L=39$ mm) is resting on silicon oil ($\mu = 5$ Pa s) and confined at speeds from 5 mm/s to 200 mm/s. (f) Coarsening of the wrinkled pattern for a rectangular PDMS thin sheet ($h=50$$\mu$m) kept at a given compression ratio on a viscous fluid ($\mu=30$ Pa s) after being compressed at various speeds as indicated ($\lambda_S \simeq 7$ mm).
• Figure 2: Evolution of the wrinkle growth rate $\bar{\sigma}={2\mu \sigma}/{B k_S^3}$ as a function of the wavenumber $\bar{k} =k/k_S$ for various $\bar{\mathcal{F}}={\mathcal{F}}/{B k_S^2}$. With this notation, Eq. (\ref{['disp-rel-main']}) becomes $\bar{\sigma} = -\bar{k}^3 + \bar{\mathcal{F}} \bar{k} - {1}/{\bar{k}}$ (see SM). The most unstable wavenumber $\bar{k}_{0}$ and its corresponding growth rate $\bar{\sigma}_{0}$ are shown for $\bar{\mathcal{F}}=3$.
• Figure 3: (a) Evolution of the rescaled initial wavelength, $\lambda_0 /\lambda_S$, for different experimental setups as a function of the rescaled compression/indentation speed, $V/V_c$, where $V_c$ is defined in Eq. (\ref{['lambda0']}) for rectangular sheets and by Eq. (\ref{['vc_circular']}) for circular sheets. The solid curve is obtained by solving numerically Eq. (\ref{['lambda0']}) whereas the dashed-dotted curve corresponds to the asymptotic power-law (\ref{['lambda0-asymp']}). The data from Box et al. are also shown Box2017. (b) Evolution of the rescaled wavelength, $\lambda/h$, as a function of the rescaled time, $E t/\mu$, during the coarsening regime. The solid curve corresponds to Eq. (\ref{['lambda-coarsening']}) with $\nu = 0.5$.