Stokes theory of thin-film rupture

The structure of the flow induced by the van der Waals destabilization of a nonwetting liquid film placed on a solid substrate is studied by means of theory and numerical simulations of the Stokes equations. Our analysis reveals that lubrication theory, which yields $h_{\text{min}}\propto {\tau }^{1/5}$, where $h_{\text{min}}$ is the minimum film thickness and $\tau$ is the time until breakup, cannot be used to describe the local flow close to rupture. Instead, the slender lubrication solution is shown to experience a crossover to a universal self-similar solution of the Stokes equations that yields $h_{\text{min}}\propto {\tau }^{1/3}$, with an opening angle of ${37}^{\circ }$ off the solid.

Figure 1

Schematics of the flow configuration, including a sample numerical evolution of the free surface for $h_o/a=4.34$, where $a={[A/\left(6\pi \sigma \right)]}^{1/2}$ is the molecular length scale [20]. The free-surface shapes are plotted at the four time instants indicated with blue dots in Fig. 2. The initial condition is $h_o/a(1-{10}^{-3}cos0.063x)$, with an associated rupture time $t_{\mathrm{R}}\approx 1.3207501\times {10}^4$ [Fig. 2]. The entire computational domain, which spans a streamwise length $\pi /0.063\simeq 49.87$, is not represented for convenience.

Figure 2

(a) Minimum film thickness as a function of the time remaining to rupture (solid lines) for $h_o/a=(1.37,4.34,13.73,43.42)$ and initial conditions $h_o/a(1-{10}^{-3}coskx)$, using the corresponding optimal wave numbers, $k=(0.17,0.063,0.021,0.0065)$. The results obtained with the lubrication approximation [36] are also presented for $h_o/a=43.42$ (dashed line). (b) Instantaneous exponent $n\left(\tau \right)=d{log}_{10}{h}_{\text{min}}/d{log}_{10}\tau$. The blue dots indicate the times at which the interface profiles are plotted in Fig. 1, and the red dot marks the time chosen for the self-similarity test of Fig. 4. (c) Times at which $n\left({\tau }_{1/5}\right)=1/5+0.1\times (1/3-1/5)$, and at which $n\left({\tau }_{1/3}\right)=1/5+0.9\times (1/3-1/5)$, obtained from the Stokes equations. (d) Rupture time $t_{\mathrm{R}}$ obtained from the Stokes equations (solid line), from the lubrication equation (dashed line), and estimated from Eq. (16) (thin line). The dot marks the rupture time associated with Fig. 1.

Figure 3

Comparison of the function $f\left(\xi \right)$ obtained from the self-similar solution (thick solid line) with eight rescaled profiles obtained from the numerical simulations for $h_o/a=4.34$ at times $-{log}_{10}\tau =(2.98,3.66,4.33,5.00,...,7.75$) (thin solid lines). The asymptotic shape $\xi tan{\theta }_o$, with ${\theta }_o={37}^{\circ }$ (dashed line) is plotted with a vertical offset of 0.15. The inset shows the unscaled profiles.

Figure 4

Isocontours of $U$ obtained by solving the self-similar system (19, 20, 21, 22, 23, 24) (right panel), and by using the rescaled velocity $u{\tau }^{2/3}$ for ${log}_{10}\tau =-4.62$ extracted from the simulation of Eqs. (1, 2, 3) (left panel; red dot in Fig. 1). The inset shows the values of $U$ and $V$ along the interface, together with the asymptotic far-field law ${\xi }^{-(1+\lambda )}$ for $\xi \gg 1$, where $\lambda =0.19$ according to Eq. (26).

Figure 5

Left panel: sketch of the geometry including the boundary conditions for the velocity and mesh displacement fields. Right panel: sample deformed meshes for $h_o/a=1.37$ and $t=(72.766,82.376,82.617)$ in the upper, middle, and lower panels, respectively.

Figure 6

Comparison with the self-similar solution $H\left(\eta \right)$ computed by Zhang and Lister [36] and the numerical integration of the lubrication equation performed with Comsol [76].

Figure 7

Left panel: sketch of the initial geometry used as initial guess for the Newton–Raphson algorithm. Right panel: final solution obtained after the root-finding iterations. In particular, $f\left(\xi \right)$ and ${\theta }_o$ are obtained from the calculation. The boundary conditions at ${\xi }^2+{\eta }^2\gg 1$ and $0<\eta <f\left(\xi \right)$ are also represented.