Asymptotic regimes in elastohydrodynamic and stochastic leveling on a viscous film

An elastic sheet that deforms near a solid substrate in a viscous fluid is a situation relevant to various dynamical processes in biology, geophysics, and engineering. Here, we study the relaxation dynamics of an elastic plate resting on a thin viscous film that is supported by a solid substrate. By combining scaling analysis, numerical simulations, and experiments, we identify asymptotic regimes for the elastohydrodynamic leveling of a surface perturbation of the form of a bump, when the flow is driven by either the elastic bending of the plate or thermal fluctuations. In both cases, two distinct regimes are identified when the bump height is either much larger or much smaller than the thickness of the prewetted viscous film. Our analysis reveals a distinct crossover between the similarity exponents with the ratio of the perturbation height to the film height.

Figure 1

Schematic of the system we are studying, where an elastic plate is supported by a viscous film on a solid substrate, surrounded by ambient air. The elastic plate has a thickness $d$ and a width $b$. Initially, the overall profile presents a localized bump, whose profile is invariant in the $y$ direction, i.e., quasi two dimensional. Far away from the perturbation, the viscous film has a constant thickness $\epsilon$. In the bump region, the height profiles $\widehat{h}(x,t)$ and $h(x,t)=\widehat{h}(x,t)-\epsilon$ of the viscous film and the bump, respectively, vary with the horizontal position $x$ and time $t$, and remain symmetric about $x=0$. At $x=0$, we define the characteristic height $h(x=0,t)=h_0\left(t\right)$ of the bump and its typical radius $R\left(t\right)$, with initial values given as $h_0(t=0)=h_{\mathrm{i}}$ and $R(t=0)=R_{\mathrm{i}}$.

Figure 2

(a) Typical optical microscopy images showing the temporal evolution of the interference fringes due to the liquid bump capped by the elastic plate ($20\mu \mathrm{m}$ scale bar). The image at 20 minutes is uncropped, showing the invariance in the $y$ direction, while the later images are cropped at the red box. (b) Intensity profile (averaged along the $y$ direction) of the bump at a given time $t$, and corresponding reconstructed bump profile at 400 minutes. (c) Temporal evolution of the bump profile.

Figure 3

(a), (b) Bump-height profiles for an initial aspect ratio $h_{\mathrm{i}}/\epsilon =43$, from a numerical solution of Eq. (1) and an experiment at two dimensionless times $T$ as indicated, which correspond respectively to $t=37$ min and $t=416$ min. The geometrical parameters are $h_{\mathrm{i}}=2.9\mu \mathrm{m}$, $R_{\mathrm{i}}=16.5\mu \mathrm{m}$, and $\epsilon =67$ nm. To account for the experimental uncertainties in the geometrical parameters, the experimental time $t$ is divided by a free-fitting factor $\alpha =0.13$. Note that, specifically for these figures, the initial condition for the numerical solution was fixed by a curve fitting of the actual experimental profile at $t=13$ min. (c), (d) Bump-height profiles for an initial aspect ratio $h_{\mathrm{i}}/\epsilon =0.1$ from a numerical solution of Eq. (1) at two dimensionless times $T$, as indicated, chosen so that the central heights $H(X=0,T)$ match those in the top row.

Figure 4

Nondimensional bump height as a function of dimensionless time in the bending case, for various initial values of $h_{\mathrm{i}}/\epsilon \in [{10}^{-2};{10}^3]$. The colored diamond-shaped markers are rescaled data points from the numerical solutions of the dimensionless version of Eq. (1) with $N=0$, and the black circle-shaped markers are scaled experimental data points. The exponents of the two asymptotic regimes of Eqs. (2) and (3) are indicated with triangles. The inset provides a zoom in the region containing the experimental data for the three samples, with initial aspect ratios $h_{\mathrm{i}}/\epsilon =30$, 43, 56, corresponding respectively to $\epsilon =50$, 67, 26 nm; $h_{\mathrm{i}}=1.52$, 2.9, $1.48\mu \mathrm{m}$; and $R_{\mathrm{i}}=9.6$, 16.5, $9.9\mu \mathrm{m}$. The uncertainties in all experimental length scales are about 5%. To compensate for those, the characteristic time $\tau$ for each sample is multiplied by a free-fitting factor $\alpha =0.7$, 0.13, and 1.3, respectively.

Figure 5

Contour plot of the dimensionless bump-height profile $\langle H(X,T)\rangle$ as a function of both the dimensionless position $X$ and time $T$, as obtained from numerical solutions of the dimensionless version (see text) of Eq. (1), with $N=5$, and for $h_{\mathrm{i}}/\epsilon =10$ (left) or $h_{\mathrm{i}}/\epsilon =0.2$ (right). The thick solid lines indicate $\langle H(X,T)\rangle =0.03$ as an arbitrary reference.

Figure 6

Nondimensional bump height as a function of dimensionless time in the stochastic leveling dynamics, for different initial values of $h_{\mathrm{i}}/\epsilon \in [{10}^{-1},{10}^2]$. The colored diamond-shaped markers are rescaled data points from the numerical solutions of Eq. (1) with $5\le N\le 8$. Each data set is an average from a minimum of 30 numerical solutions. The solid red line corresponds to Eq. (4) with a prefactor of order unity. The insets show rescaled bump-height profiles for $h_{\mathrm{i}}/\epsilon =100$ with $T\in [1,10]\times {10}^{-3}$ (lower left) and for $h_{\mathrm{i}}/\epsilon =0.1$ with $T\in [4,4.6]\times {10}^{-3}$ (upper right).