Steady-Profile Fingering Flows in Marangoni Driven Thin Films

We present experimental and computational results indicating the existence of finite-amplitude fingering solutions in a flow of a thin film of a viscous fluid driven by thermally induced Marangoni stresses. Using carefully controlled experiments, spatially periodic perturbations to the contact line of an initially uniform thin film flow are shown to lead to the development of steady-profile two-dimensional traveling wave fingers. Using an infrared laser and scanning mirror, we impose thermal perturbations with a known wavelength to an initially uniform advancing fluid front. As the front advances in the experiment, we observe convergence to fingers with the initially prescribed wavelength. Experiments and numerical computations show that this family of solutions arises from a subcritical bifurcation.

Figure 1

(top) Time evolution (left to right) of the contact line flow with the laser perturbation active. (bottom) A schematic diagram of the experimental setup.

Figure 2

A fingering pattern produced by initial laser-heating perturbations shown in (a). Frames (b) and (c) show the persistence of the fingers in the evolution of the flow after the laser is turned off. The vertical size of each image is 5 mm, the film thickness is $0.34\mu \mathrm{m}$, and the time interval between each image is 4800 s. The inclination angle of the substrate is $85°$ from the horizontal.

Figure 3

(a) Computed steady-profile fingers for $\lambda =1.12$, $h_{\mathrm{e}\mathrm{q}}=0.03$, and $h_{\mathrm{p}\mathrm{r}\mathrm{e}}=0.0018$ and (b) the corresponding contour plot.

Figure 4

(a) Numerical simulations for fingering length vs time in dimensionless units with $h_{\mathrm{e}\mathrm{q}}=0.03$ and $h_{\mathrm{p}\mathrm{r}\mathrm{e}}=0.0018$ and 21 s as one unit of dimensionless time. (b) The linear dispersion curve for the initial growth rate of infinitesimal perturbations to the uniform contact line. $q_{\max }$ is the value of $q$ for which $\sigma$ is maximum. (c) The approximate linear dispersion curve for infinitesimal perturbations to the steady fingers.

Figure 5

Bifurcation diagram of finger length, $A_{\infty }$, versus wave number $q$ for $h_{\mathrm{e}\mathrm{q}}=0.03$ and $h_{\mathrm{p}\mathrm{r}\mathrm{e}}=0.0018$. The open circles (○) represent saturated fingering lengths measured from experiments. The solid curve shows results from numerical simulations of Eq. (2). An approximate (dashed) curve for the unstable branch is shown which connects the saddle-node point $q_s$ and the critical wave number for the linear instability of the uniform contact line $q_c$. The solid circles (●) correspond to experiments in which weak initial perturbations are applied and yield decay to irregular waves.

Figure 6

Sequences of images from two experiments starting from initial perturbations with the same wave number, $q=2\pi /(0.39\mathrm{m}\mathrm{m})$, with $q_c<q<q_s$. Top row: $t=0\mathrm{s}$, middle row: 2200 s, and bottom row: 4400 s. The left column (a)–(c) corresponds to a strong initial perturbation (a) that saturates to uniform fingers with wave number $q$ (c). The right column (d)–(f) starts with a weak perturbation (d), which uniformly decays (e) before being invaded by surrounding edge influences (f). (Each image is $7.7\mathrm{m}\mathrm{m}\times 5\mathrm{m}\mathrm{m}$ and the thickness of the film $h_{\mathrm{e}\mathrm{q}}$ is $0.6\mu \mathrm{m}$. The inclination angle of the substrate is $60°$ from the horizontal.)