Effect of slip on the contact-line instability of a thin liquid film flowing down a cylinder

Liquid coating films on solid surfaces exist widely in a plethora of industrial processes. In this study, we focus on the falling of a liquid film on the side surface of a vertical cylinder, where the surface is viewed as slippery, such as a liquid-infused surface. The evolution profiles and flow instability of the advancing contact line are comprehensively analyzed. The governing equation of the thin film flow is derived according to the lubrication model, and the traveling-wave solutions are numerically obtained. The results show that the wave speed increases with the increase of a larger slippery length. A linear stability analysis (LSA) is carried out to verify the traveling solutions and time responses. Although previous studies tell us that the wall slippage always promotes the surface flow instability of the thin film flow, the linear stability analysis, numerical simulations, and nonlinear traveling-wave solutions in the current study present a different conclusion. The analysis show that for a thin film flow with a dynamic contact line the wall slippage in different directions plays much more complex roles. The streamwise slippery effect always impedes the instability of the flow and suppresses the wave height of traveling wave, while the transverse slippery effect has a dual effect on the surface instability. The transverse slippery effect significantly improves the instability while the wave number of the perturbation is small, and simultaneously it reduces the cutoff wave number. The transverse slippery effect will change its role if the wave number of the perturbation exceeds a critical value, which can stabilize the contact line.

Figure 1

Thin film flowing down the side surface of a cylinder with wall slippage, and the wall has an anisotropic slippery property.

Figure 2

Asymmetric flow profiles of thin liquid films on vertical cylinders for (a) $t=20,R=[0.8,1,2]$, and (b) $t=40,R=[0.8,1,2]$. The used parameters are ${\beta }_z=0.1,{\beta }_{\theta }=0$, and $b=0.05$.

Figure 3

Asymmetric flow profiles of thin liquid films on vertical cylinders for (a) $t=20,{\beta }_z=[0,0.1,0.2]$, and (b) $t=40,{\beta }_z=[0,0.1,0.2]$. The used parameters are $b=0.05,{\beta }_{\theta }=0$, and $R=1$.

Figure 4

Asymmetric flow profiles of thin liquid films on vertical cylinders considering different precursor thickness: (a) $t=20,b=[0.01,0.05,0.1]$, and (b) $t=40,b=[0.01,0.05,0.1]$. The used parameters are ${\beta }_z=0.1,{\beta }_{\theta }=0$, and $R=1$.

Figure 5

The evolution profiles of the liquid-air interface for different substrate curvature at (a) $t=20$ and (b) $t=40$, including $R=0.8$ (solid line), $R=1$ (dash-dotted line), and $R=2$ (dashed line).

Figure 6

The evolution profiles of the liquid-air interface considering different precursor thickness at (a) $t=20$ and (b) $t=40$, including $b=0.01$ (solid line), $b=0.05$ (dash-dotted line), and $b=0.1$ (dashed line).

Figure 7

The evolution profiles of the liquid-air interface on the cylinders with different slippery lengths at (a) $t=20$ and (b) $t=40$, including ${\beta }_z=0$ (solid line), ${\beta }_z=0.1$ (dash-dotted line), and ${\beta }_z=0.2$ (dashed line).

Figure 8

Traveling-wave profiles of the liquid-air interface for ${\beta }_z=0$ (solid line), ${\beta }_z=0.1$ (dash-dotted line), and ${\beta }_z=0.2$ (dashed line): (a) $b=0.1$ corresponding to different ${\beta }_z$, (b) $b=0.05$ corresponding to different ${\beta }_z$, and (c) $b=0.01$ corresponding to different ${\beta }_z$.

Figure 9

Traveling-wave profiles of the liquid-air interface for ${\beta }_z=0$ (solid line), ${\beta }_z=0.1$ (dash-dotted line), and ${\beta }_z=0.2$ (dashed line): (a) $R=0.8$ corresponding to different ${\beta }_z$, (b) $R=1$ corresponding to different ${\beta }_z$, and (c) $R=2$ corresponding to different ${\beta }_z$.

Figure 10

Temporal growth rate $\sigma \left(q\right)$ vs wave number $q$ for ${\beta }_z=0$ (solid line), ${\beta }_z=0.1$ (dash-dotted line), and ${\beta }_z=0.2$ (dashed line): (a) $b=0.1$, (b) $b=0.05$, and (c) $b=0.01$. Red circle shows the prediction in the previous study by Lin [29]. Arrow represents the increase of the streamwise slippery effect.

Figure 11

Temporal growth rate $\sigma \left(q\right)$ versus wave number $q$ for (a) $R=1$ and (b) $R=2$. Arrow represents the increase of the slippery effect.

Figure 12

Comparison of growth rates between LSA $(q=0.85$ [solid lines] and $q=0.5$ [dashed lines]) and numerical simulations (triangle markers): (a) ${\beta }_z={\beta }_{\theta }=0$, (b) ${\beta }_z={\beta }_{\theta }=0.1$, and (c) ${\beta }_z={\beta }_{\theta }=0.2$.

Figure 13

Three-dimensional illustrations of a single-mode perturbed fluid film flowing on the surface of a vertical cylinder: (a) $q=0.5,{\beta }_{z,\theta }=[0,0.1,0.2]$, and (b) $q=0.85,{\beta }_{z,\theta }=[0,0.1,0.2]$. The radius of the cylinder is $R=2$.

Figure 14

Time evolution of a single-mode perturbed fluid film plotted in the Cartesian coordinate: (a) $t=40,{\beta }_{z,\theta }=0.2,R=2$, and (b) $t=80,{\beta }_{z,\theta }=0.2,R=2$.

Figure 15

Temporal growth rate $\sigma \left(q\right)$ vs wave number $q$ for different azimuthal slippery lengths: (a) ${\beta }_z=0$, (b) ${\beta }_z=0.1$, and (c) ${\beta }_z=0.2$. Arrow represents the increase of the azimuthal slippery effect, ${\beta }_{\theta }=[0,0.05,0.1,0.15,0.2]$.

Figure 16

Temporal growth rate $\sigma \left(q\right)$ vs wave number $q$ for different streamwise slippery lengths: (a) ${\beta }_{\theta }=0$, (b) ${\beta }_{\theta }=0.1$, and (c) ${\beta }_{\theta }=0.2$. Arrow represents the increase of the streamwise slippery effect, ${\beta }_z=[0,0.05,0.1,0.15,0.2]$.